LIBRARY OF THE University of California. Class i CONTENTS. I. Algebra. Preliminary Notions Definitions and Symbols Addition Subtraction . Brackets — Vincula Introductory Simple Equations Questions in Simple Equations Definitions continued Multiplication Division Algebraic Fractions Addition — Subtraction Multiplication — Division Grejitest Common Measure Least Common Multiple . Simple Equations in general Simultaneous Simple Equations Theory of Exponents Square Root of Poljmomial Cube Root of Polynomial Surds — Imaginary Quantities Binomial Surds Operations with Imaginary Quanti ties .... Quadratic Equations, Rule I. ,, ,, Rule II. Theory of Quadratic Equations Simultaneous Quadratics Theory of Proportion Variation ... Arithmetical Progression Geometrical Progression . Harmonical Progression . Piling of Balls and Shells Square Pile — Rectangular Pile General Rule for all Piles The Binomial Theorem . Limitations of the Theorem The Exponential Theorem Theory of Logarithms , Construction of Logarithms Exponental Equations Compound Interest Annuities Increase of Population Permutations Combinations Probabilities . Life Annuities Life Assurances PAGE 1 2 4 6 7 8 10 12 13 16 19 20 21 22 24 25 30 36 37 39 40 42 45, 46 49 50 52 58 60 61 62 64 67 68 69 70 77 78 79 80 84 85 86 87 88 91 93 99 102 Theory of Equations Transformation of Equations Limits of the Roots Equal Roots . Rule of Descartes . Criteria of Imaginary Roots Newton's Rule Theorem for the Biquadratic , Solution of Equations . Newton's Approximation Approximation by Position Cardan's Method for Cubics Decomposition of a Biquadrati Recurring Equations Binomial Equations Vanishing Fractions Maxima and Minima Indeterminate Equations Indeterminate Coefficients Summation of Finite Series The Diflferential Method Construction of Tables . Interpolation Summation of Infinite Series Recurring Series . Reversion of Series Convergency of Series II. Plane TRiaoNOMETRV. Definitions, &c. Measurement of Angles . Sine, Cosine, Tangent, &c. Fundamental Relations . Solution of Right-angled Triangles Oblique-angled Triangles Miscellaneous Problems . Quadrature of the Circle Unit of Circular Measure French and English Degrees . Extension of Definitions Sine and Cosine of (A+B) Expressions involving Two Angles Single Angles and their Halves Multiple Angles — Ambiguities Applications of Formulai Inverse Trigonometric Functions Solution of a Quadratic by Tables Solution of a Cubic by Tables Construction of Trigonometric Tables Developments of sin ^, cos ^ . CONTENTS. Euler's Expressions for sin 6^ cos 6 . De Moivre's Theorem Developments of cos"^, sin"^ . Developments of sinw^, cos?i^ . Development of ^ in powers of tan ^ Euler's Series : Machin's Series Developments of y, of sin fi, and of cos ^ Wallis's Expression for ^w Imaginary Logarithms The Numbers of Bernoulli Summation of Trigonometric Series Imaginary Roots of Unity Construction of Log Sines and Co- page 213 213 215 216 216 217 218 218 219 220 222 225 226 III, Spherical Trigonometry. Preliminary Theorems . . . 227 Fundamental Formulae . . . 231 Formulae for Sides . . . .233 Napier's Analogies . . . 235 Right-angled Triangles : Napier's Rules 236 Quadrantal Triangles . . . 238 Solution of Oblique-angled Triangles 240 Examples of the Six Cases . 240-250 Area of Triangle : Spherical Excess . 252 IV. Mensuration. Area of Parallelogram and Triangle ,, ,, Triangle : three sides given Areas of Quadrilaterals . Equidistant Ordinates . Regular Polygons .... The Circle and its Sectors Segment of a Circle Circular Ring .... Inscribed and Circumscr. Triangle . Prism and Cylinder Pyramid and Cone .... Frustum of Pyramid or Cone . Surface of a Frustum The Sphere and its Surface Theorem of Archimedes . Volume of Segment and Zone . Equidistant Sections . . . Weight, &c., of Shot and Shells . Weight of Powder in Shells . ^. Applications of Maxima an^l Minima V. Analytical Geometry. Equations of a Point Equation of a Straight Line . Straight Line subject to Conditions Problems on the Straight Line The Circle : Rectangular Axes The Circle : Oblique Axes Tangent at a Point Tangent from a Point 255 257 258 260 261 262 264 265 266 267 268 269 270 271 279- 273 274 275 277 279 285 287 291 296 300 301 302 302 Locus of an Equa. of Second Degree Construction of Loci Problems on the Circle . Polar Co-ordinates The Conic Sections Equation of the Ellipse . The Principal Diameters Change of Co-ordinates . Conjugate Diameters Change from Oblique to Rectangular Co-ordinates .... Tangent to the Ellipse . Subtangent, Normal, Subnormal Polar Equation of the Ellipse . Radius of Curvature Chord of Curvature Area of the Ellipse The Hyperbola .... Equation of the Hyperbola Principal Diameters Conjugate Diameters Tangent to the Hyperbola The Asymptotes .... Equation with Asymptotes for Axes Conjugate Hyperbolas The Parabola .... Parabola referred to Conjugate Axes Tangent, Normal, &c. . Properties connected with Tangents . Polar Equation of the Parabola Area of the Parabola . " . Radius of Curvature Locus of Equation of the Second De- gree ...... The Different Curves represented Determination of particular Loci Problems on Loci .... VI. Mechanics : Statics. Conspiring and Opposing Forces The Parallelogram of Forces The Triangle of Forces . Composition of Forces . The Polygon of Forces . Problems in Statics The Principle of Moments Parallel Forces acting in a Plane Centre of Gravity of a Rigid Body General Equations of Parallel Forces Equilibrium in general . Problems on Equilibrium Mechanical Powers Lever — Balance — Steelyard Combination of Levers . Wheel and Axle : Toothed Wheels Pulley : Systems of Pulleys . Inclined Plane : Screw . The Wedge . Mechanical Powers in Motion . Principle of Virtual Velocities Friction .... 304 305 306 309 310 311 313 315 316 318 319 322 326 327 328 329 330 331 333 334 336 336 339 340 341 342 344 345 348 349 350 351 354 355 357 361 362 364 365 367 367 372 375 377 383 387 390 396 396 401 402 404 407 410 412 418 410 CONTENTS. XI VII. Dynamics. Uniform Motion .... 424 Variable Motion .... 425 Accelerated Motion 426 Falling Bodies : Gravity- 429 Motion on Inclined Planes 430 Parallelogram of Velocities 431 Motion of Projectiles 432 Circular Motion : Centrifugal Force . 439 Moving Forces .... 442 Principle of D'Alembert 449 Moment of Inertia 450 Impact of Bodies . . . . 451 VIII. Hydrostatics. Transmission of Pressure 456 Hydrostatic Paradox 457 Levelling Instrument 460 Explanation of Symbols, &c. . 460 Centre of Pressure 465 Kesultant of Fluid Pressures . 466 Equilibrium of a Floating Body 467 Equilibrium of a Rotating Fluid 469 Specific Gravities of Bodies . 470 Hydrostatic Balance . . 472 Common Hydrometer 472 Nicholson's Hydrometer . 473 Elastic Fluids : the Atmosphere 476 Law of Mariotte and Boyle . 477 Altitudes by the Barometer . 479 The Wheel Barometer . 482 The Thermometer . . . . 482 The Syphon 482 The Common and Forcing Pumps . 483 The Diving Bell . . . . 485 Air Pumps . . . . . 486 The Condenser . . . . 487 IX. Differential and Integral Calculus. Differentiation .... 489 Investigation of Rules . . . 494 Algebraic Functions . . . 495 Applications to Geometry . . 497 Log. and Exp, Functions . . 498 Trigonometrical Functions . . 600 Inverse Functions . . . .600 Integration of Particular Forms . 603 Areas of Curves : Definite Integrals 610 Volumes of Revolution . . .613 Lengths of Curve Lines . Surfaces of Revolution , Successive Differentiation Theorem of Leibnitz Maclaurin's Theorem Taylor's Theorem .... Limits of Taylor's Theorem ,, ,, Maclaurin's Theorem Compound Functions Implicit Functions Vanishing Fractions Maxima and Minima Max. and Min. of Implicit Functions Functions of Two Variables . Change of Independent Variable Failure of Taylor's Theorem . ,, ,, Maclaurin's Theorem Asymptotes to Curves Spiral Curv^es Circular Asymptotes Sectorial Areas Contact : Osculation Rad. of Curv. in Rect. Co-ordinates Rad. of Curvature in Polar Curves . Chord of Curvature Consecutive Lines and Curves Enveloping Curves Singular Points .... Integ. of Rational Fractions . ,, ,, Irrational Functions Integration by Reduction ,, ,, Parts ,, ,, Series Successive Integration . £14 516 619 620 621 625 627 529 630 633 634 641 645 548 652 653 655 655 658 660 661 663 665 568 669 673 576 576 581/ 685 687 690 694 696 X. Applications to Mechanics. The Centre of Gravity . . .597 Theorems of Guldinus . . .699 Moment of Inertia : Gyration . 600 Centre of Oscillation . . .603 Centre of Percussion . . . 604 Attraction of Bodies . . . 605 Velocity : Acceleration . . .608 Equations of Motion . . .609 Cycloidal Pendulum . . . 613 Simple Pendulum .... 614 Problems on the Pendulum . . 615 The Ballistic Pendulum . . .618 The Cycloid 619 The Catenary . . . .620 Note on Interpolation . . . 622 Answers to Examples . . . 624 ERKATA. Page 163, Ex. 8, for = read +. P. 183, line 2, for 88 read 80 ; and line 4, for 33 read 25. P. 206, Ex. 17, for tan read sin ; and Ex. 19, for tan b read tan B. P. 381 in the diagram, the point e should be on OE. P. 446, line 18, for exactly read very nearly. P. 449, line 22, after denominator put -^g. P. 571, last line but one, omit comma after C. Types fallen out or defective. Page 27, Ex. 10, -. P. 64, line 6, the exponent n-fjp. 7) 25 P. 72, Une 8, -. P. 77, line 19, — . P. 169, line 29, the minus sign. q 2 P. 224, line 1, the exponent w. P. 225, line 16, the exponent . n P. 297, Prob. II., |. P. 499, Ex. 5, exponents of ar, z-1. P. 515, line 8, ^. P. 533, Ex. 1, .^±^. \' A CODRSE OP ELEMENTAHY MATHEMATICS, ETC., ETC. The preparation necessary for the profitable study of the following course of Mathematics is — a knowledge of common Arithmetic, and some acquaintance with the principles of Geometry, as taught in Euclid's Elements. A student ignorant of these initiatory, but most important departments of elementary science, would scarcely seek his first lessons therein from a book such as this. The Elements of Euclid is a work by itself ; universally known and esteemed, and everywhere to be easily procured : — to transfer its pages to the present performance, could be of no possible advantage to the learner. And the same may be said of common Arithmetic : — both this and Euclid are more conveniently studied from the ordinary manuals in popular use. We shall therefore commence the volume now in the hands of the reader, with a treatise on Algebra — the indispensable foundation of the entire fabric of modern analytical science. I. ALGEBRA. 1. Preliminary Notions. — Algebra may be regarded simply as an extension of the principles of Arithmetic. In the latter science the symbols of quantity, to which its rules and operations are applied, are limited to the nine digits or figures 1, 2, 3, 4, 5, 6, 7, 8, 9, together with the cypher or zero, 0. And not only is the notation of Arithmetic limited to these ten symbols, but each symbol is employed by every computer in the same sense : — the character or symbol 4, for instance, stands for/owr, always ; 6 for six; 8 for eight, and so on : the symbols of Arithmetic are thus fixed in meaning, as well as limited in number. It is otherwise in Algebra: in this science the symbols of quantity comprehend not only the figures of arithmetic, but also the letters of the alphabet : — the figures being, as in arithmetic, of invariable signification, but the letters admitting of arbitrary interpretation. It is this latter circumstance — namely, the possession of a set of symbols which we may employ to represent anything we please — that gives to Algebra its pecu- liarity and its power. In Arithmetic, known quantities only can bo denoted by symbols : in Algebra a quantity altogether unknown, in value, at the outset of an inquiry, may be represented — an alphabetical letter serving this pui-pose, — and then the rules of the science, to be hereafter developed, will enable us ultimately to interpret its meaning, consistently -84592 2 DEFINITIONS— SYMBOLS OF QUANTITY — SIGNS OF OPERATION. with the conditions which connect it, in that inquiry, with the known quantities concerned, 2. Definitions— Symbols of Quantity—Signs of Opera- tion. — As noticed above, the symbols by which the quantities operated upon in algebra are represented, are the figures of ordinary arithmetic, and the letters of the alphabet : the marks or signs by which these opera- tions are indicated, are called signs of operation : the principal of these are the following : — + , phis, the sign of addition, implying that the quantity to which it is prefixed is to be added. — , minus, the sign of subtraction, denoting that the quantity to which it is prefixed is to be subtracted. Thus 5 + 2, which is read 5 plus S, signifies that 2 is to be added to 5 ; and 5—2, which is read 5 minus 2, indicates that 2 is to be subtracted from 5. In like manner a-\-b, or a plus b, implies that b is to be added to a, that is, that the quantity represented by b is to be added to that represented by a. And a—b, or a minus b, implies that b is to be sub- tracted from a. Of course we cannot actually perform the addition and subtraction operations thus indicated, till we know what numbers or quantities a and b stand for. It may be remarked here, that although the letters a, b, &c. are but the representatives of quantities or numerical values, yet, for brevity of expression, we refer to them as the quantities themselves. The crooked mark 'v. placed between two quantities denotes the dif- ference between those quantities : thus a~6 means the difference between a and b, whether that difference be the result of subtracting b from a, or a from b. X , the sign of multiplication, when placed between two quantities, implies that those quantities are to be multiplied together : thus 4x6, or 6 X 4 means that 4 and 6 are to be multiplied together, and axb,or bxa, implies in like manner the product of a and b. Instead of the sign x , a dot placed between the factors is often used for the sign of multiplication: thus 4.6, or 6.4, and a.b, or b.a, each implies the product of the quantities between which the dot is placed. It must be observed, however, that the dot should range with the lower part of the figures or letters, and not with the upper part, to avoid con- founding it with the decimal point, as, in the case of figures, might otherwise happen : thus 6.4 means 24, but 6'4 means 6 and 4 tenths. In the case of letters however, the dot is usually dispensed with alto- gether, and the factors simply written side by side, without any inter- vening sign at all : thus, ab, ex, bxy, abxz, &c. mean the same as a x 6. cxx, bxxxy, axbxxxz; or as a.b, c.x, b.x.y, a.h.x.z, &c. This suppression of the intervening sign of multiplication between the fac- tors is not allowable when those factors are numbers, as is obvious : if 6 X 4, or 6.4 were written 64, sixty-four would be implied, and not 24, as intended. But when a single numerical factor enters with the letters, then the multiplying sign may be omitted, since no ambiguity can arise : thus, 6xax6 or Q.a.b, may be more conveniently written 6a6, which means 6 times the product of a and b, or as it is' more briefly read, 6 times a, b. It is proper, as here, always to place the numerical factor first, and the literal factors afterwards ; and also to arrange these latter in the order in which they succeed each other in the alphabet. The numerical factor, thus placed first, is called the coefficient of the quantity DEFINITIONS SYMBOLS OF QUANTITY — SIGNS OF OPERATION. 8 multiplied by it: thus 6 is the coefficient of ah in 6afe, and 15 is the coefficient of xyz in \bxyz. -r, the sign of division, when placed before a quantity, supplies the place of the words *< divided by," so that 8-~2 means 8 divided by 2, 12-r3 means 12 divided by 3, a-^-h means a divided by h, and so on; but, as in common arithmetic, division is more frequently indicated by writing the dividend above and the divisor below a horizontal bar of separation, thus: — •7- is the same as a-^-h, and ~ is the same as 3ajy-r2a6. zoo 8. The four signs now explained —indicating the four fundamental operations both of arithmetic and algebra — are, of course, those of most frequent occurrence in calculation. Algebraists, however, economize their signs of operation as much as possible, and never introduce them need- lessly. This has been already exemplified in the case of multiplication : the absence of sign between letters, placed side by side, as much implies the multiplication together of the numbers those letters represent, as if each were separated from the others by an oblique cross, or a dot. In like manner, when a row of additive and subtractive quantities are con- nected together by the proper signs, if the Jirst of these quantities be additive, or plus, the sign + is suppressed as superfluous; thus : a—b-\- c+d — 4 is the same as -f «— fc + c+(?— 4, and implies that a, c, and d are additive ; or, as they are more frequently called, positive quantities, and that 6, and 4, are subtractive, or negative quantities. If the letters a. 6, c, d stood respectively for 2, 4, 3, 8, the interpretation of the expression just written would be 5. 4. The term coefficient has been already defined : it is the numerical multiplier of the algebraic quantity to which it is prefixed : when this numerical multiplier is simply 1, it is not inserted : it is superfluous to introduce unit-factors; a-|-5, is as well understood to be once a plus once fc, as la + 16; but if the question were asked — What is the coefficient of a or of 6 in the expression «+ 6? the answer would be, not nothing y but 1. 5. =, equal to, is the sign of equality: it implies that what is written on one side of it is equal to what is written on the other, thus : — 7+4=11, 7-4=3, 7-4+1=4, Sx-\-2x-x=ix, &c. 6. Any quantity — of how many letters soever it may consist — is called a simple quantity, or a quantity of but one term, provided it be not sepa- rated into distinct parts by the interposition of a plus or minus sign ; thus, each of the following is a simple quantity, or a quantity of but one term : — _ _ ^ _ 2ax 14 oaoa;, 7aocy, -^j-, - — , &c. 7. Each of the following, however, is a compound quantity : the first consists of two terms, the second of three terms, and the third of four terms : — 4 2a+8&, 6a— 26+c, 5a5+2cc?— 3m+-. 8. We shall now add a few exercises by which the learner may satisfy himself as to whether he correctly understands what has already been explained or not. 4 ADDITION OF ALGEBRA. Exercises on the Definitions. — In the following exercises we shall suppose a=4:, b^S, c=6, and d=7. Find the values in numbers of the following expressions : — Expression. Interpretation. Vakie. Baj-2b-c 12+6-5 13. 5b-Za-\-2d 15-12+14 17. 4d^2c-6a+l 28+10-24+1 16. Sc-d-Zb+ia 40-7-9+16 40. 2a6+5ccZ-16 24+175-16 183. Consequently 3a^ + 26— c= 13, 56 — 3h, —Aaxy—2b, 8axy-\-Qb. (6) 5aa;+l, 3aa;-2, 6aa:+4, -aa;+3, —7ax-5, 4ax+d. (7) 8cz+2x-4, — Zcz—7x, 5cz—ix-{-l, _9c2+3a:-8, 6a;+2, —7. (8) Saxy-^2bz-4.c, —6axy—Bbz-{- 7c, liaxy+5bz~Zc, 2axy—bz—Qc, —iliaxy—bz-12c^ —ldaxy+^bz-\-5c, axy-\-ibz+ 13c. 11. Case II. — When the quantities are not all like quantities. EuLE. — Collect the like quantities from the several expressions, and add them together : to the sura connect, with their proper signs, those of the quantities which have no like. Note. — Athough it is of no consequence which set of like quantities be added first, yet the custom is to commence with the quantity at the top of the Jirst column on the left, and to put down, under that column, the sum of all the quantities like it; then to collect the quantities like that at the top of the second column, and so on, as in the following examples: — (1) 2x—7y-\- iz Sz + 2x— y — 2y4-52+ a Ax-Zz-\- 7y %x-Zy+14z-\-a (2) 7ax—2by-{-z 36y+92 —ax — Qz -\-2ax+by 6by-Bax-\-6 (3) 4:xyz—dxy+ 2yz 6xy -j-Syz — 7ccyz —9ax +xyz'^ll —iyz -\-7xy— 8 5ax->r7hy-\-4z-\-Q —2xyz+9xy-\- Qyz—dax+d 12. — ExEECisEs TO be WORKED. — In the following examples the several expressions may be taken as they are, and placed one under another as above ; or the arrangement of the terms may be changed, so that when placed in vertical columns, the like terms may stand one under 6 SUBTRACTION OF ALGEBRA. another, as in Case I. ; or lastly, without transcribing, and writing them one under another, we may pick out the like quantities from the ex- pressions as they stand, and write down the sum of each set at once, afterwards connecting, with their proper signs, the unlike quantities. (4) 4.ax-2y-{-7y dy-8+ax, i2-Bax+p. (6) 6ar-8y+5a, 4y+2x, 20-6a?+7y, 3a-l-6. (6) abx-{-2, 6ay-7-^2abx, 14:— ay, 6ay—9. (7) 8mx—Sny, 5az-^27nx, Iny-azy 2Z-imx+Qny, ias-B. (8) 2--7-+3, 9^-8, 7?+5, 3aa;-?+2, 8^+ y X X o ox 9-+1. (9) 7dbx—Qaby-^as, 4az—Zay+dbx, ^ax-^7dbx—9aby, 2dbz^6aX'-aby, — 12ay—4az-\-x, 2dbz—5ax—dbz. 13. Subtraction of Algebra. —The operations of Arithmetic are all performed with positive numbers or quantities. In Algebra, quantities both positive and negative equally enter into our computations : we must know therefore how the operation called subtraction is to be per- formed, whether the quantities operated upon be positive or negative. To subtract a quantity is to take it away from some other quantity : this we can actually do, provided we can split this other quantity into two parts — one of which shall be equal to the quantity to be taken away : the other part is, of course, the remainder. This simple truth suggests the rule for algebraic subtraction, thus : 1. Suppose we have to subtract 5 from 12 : instead of 13 we may write its equivalent 7 + 5, so that actually taking the 5 away, the remainder is 7. 2. Suppose we have to subtract —5 from 12 : then, instead of 12, writing its equivalent 17—5, and then taking the —5 away, the remainder is 17. 3. Suppose we have to subtract 5 from —12: then, instead of —12, writing its equivalent —17 + 5, and then taking the 5 away, the remainder is -17. 4. Lastly, suppose we have to subtract —5 from —12: then, instead of —12, writing its equivalent —7—5, and then taking the —5 away, the remainder is — 7. Having thus taken all the possible varieties, as to the signs, of the two numbers — which numbers have, of course, been taken at random — we safely infer the truth of the following results of subtraction : — From 12 12 —12 -12 Subtract 6 — 5 5 —6 There remains 7 17 —17 — 7 and we moreover see that the very same results would have been obtained if we had changed the sign of each number to be subtracted, and had then added.* * If we use general symbols instead of figures, the following results, namely, From a a —a —a Subtract 6 —6 b — & Remainder a— 6 a+6 —a— 6 — a+6 are shown to be true, by substituting for a the following equivalents, namely : — a=a— 6+6, a^a-jr^—bf — a=— a— 6+6, — a= — a+6— 6. BRACKETS — VINCULA. 7 We conclude therefore that algebraic subtraction may always be con- verted into algebraic addition, by simply changing the sign of the quantity to be subtracted. Hence the following rule. 14. EuLE. — Conceive the sign of every quantity that is to be sub- tracted, to be changed : and then, with this supposed change of signs, proceed as if the operation were that of addition instead of subtraction : the result will be the remainder. (1) (2) (3) From 7aa;-i- %y 5xy— 8az+2 —Saxy-]- hz—2x Subtract . Zax— 5by _ Sicy-f 2az— 1 5axy-\-7bz-Sx — 6 Remainder iax-\-liby 8xy-~10az+Z ^8axy~6hz-\- x+Q Note. — It must be carefully borne in mind that when terms occur that have no like, those in the upper row must be annexed to the remainder with their proper signs, and those in the lower row with their signs changed. (1) From iax—2by-\- 4 subtract 2ax—6hy—Z. (2) From 2abx-\-Zay—2z subtract 5abx—Say+2z. (3) From —7xyz-{-5xy-\-Q suhtract —2xyz — Sxy. (4) From —Zmxy — 6nyz+2asyihtT2iCt27nxy-\-nyz+S. (5)' From Saz—Aby-{-Sx subtract —6x-{-2by—7az-[-l. (6) From 2^ax-Sby-{-17 subtract 2d-Uiby-lx-\-2z, (7) From 6^py—9az—m subtract 17az—lpy-{-7n. (8) From —5ca;s— 7ey-f^2-f a subtract 2<;^—3A;2-fca;«—6-f-4. 15. Brackets — Vincula.— The signs of operation hitherto em- ployed have been prefixed to simple quantities only : when we intend them to apply to compound quantities — or to quantities consisting of two or more simple terms — it is necessary to inclose the terms within hraclcets, such as ( ), or { }, or [ ] ; or else to cover all with a mnculum, or bar , to imply that the several terms thus tied together are to be treated as one whole. For instance, by writing or tying together the terms in this way, we may express the subtractive operations above thus : — 7ax+ 9hy— {Sax — 5hy) = iax-\- 146y, 5xy—8az+ 2 — {—Sxy -{-2az-'l} =8xy— 10a2-f 3, —daxy-\-bz-2x—[5axy+7lz—dx~-Q]'=—8axy^6bz-\-x-\-6. 16. The minus sign before each of these bracketed quantities implies that the sign of every inclosed term is to be changed on the removal of the brackets : when a bracketed quantity is connected to other quantities by a plus sign, the removal of the brackets leaves the signs of the quantities undisturbed. The following instances of the management of bracketed quantities will be easily understood : — (a+&)-(a-6)=a-f6-a+5 = 26, a-{b-c)-{b-{-c)=a-b-{-c-b-c=a-2b. a-{-{a—b)—{c—(a—c)}=a-\-a—b—c-{-{a—c) = Sa—b—2c. 2ax—{ax—{2y—dax)}=2ax—ax+2y—Sax=2y—2ax. 17. A multiplier, or coefiicient, placed before a bracketed quantity, implies that the compound whole — or, which is the same — that each individual term is to be multiplied by that coefficient, thus, 4= {a-{-b)= 4a-t-46, 6{a-b)=6a-6b, 3(4a-6 + S)-6^a-Ht-~8)=12a-36-f 6--6a- 66i-48=6a— 9i + 54. 8 SIMPLE EQUATIONS. 18. The equivalence of the following expressions the learner is left to prove for himself — (1) 4{3a-(6-a)}=4(4a-6). (2) 2{2x-{x-^y)-l}^2(x-y-l). (3) 5{a-^x- 2{x-a)]=5{2a-x). (i) Z{{x-4:)-2{dx-{-2)-5{l-x)\ =-S9. (5) 6{{a-2x)- . d{Ax-2a)}-Z{{ix+a)-{9x-a)}=d6a-69x. 19. Simple Squations. — An equation is merely a declaration, expressed in the characters of algehra, combined or not, with those of arithmetic, that two quantities are equal. Thus : if a; be such a number that a;— 3 is equal to 8, then the algebraic statement of this equality, namely, a;— 3=8 is an equation. It is plain that to satisfy Ihis equation, a must represent the number 11. In like manner, if x be such a number that 4a;— 3 is equal to 2^7 + 7, we shall have the equation 4a?— 3 = 2a; + 7. The finding the value of the unknown quantity — in this case x — that is, the discovering what the unknown quantity really stands for, is called the solution of the equation. 20. By help of the few principles established in the preceding pages, the more easy kinds of simple equations may be readily solved. It will be only necessary to observe the following particulars. 1. Transposition. — Any term may be taken from one side of an equation and carried over to the other side, provided that, when thus transposed, its sign be changed. For by thus transposing a term, the balance, or equality of the two sides of the equation, remains undisturbed: the result is still an equation. Suppose, for instance, the term 'Sah occurs on one side of an equation, and that we wish to remove it from that side without disturbing the equality of the two sides. All we have to do is to add — 3a& to both sides, which addition, it is plain, removes the 3ah from the one side, and transfers it, with changed sign, to the other. In like manner, if the term —5a;, standing on either side of the sign of equality, is to be removed, we have only to add 5a? to both sides, which addition merely transposes the —5a; from one side to the other, on which other it re-appears with changed sign. And it is plain that if the two sides were equal before the transposition, they must be equal afterwards. For example, take the equation above; namely, 4a?— 3=2a; + 7. By transposing the 3, we have 4a;=2;i? + 7 + 3 : and by transposing the 2ar, 4a; — 2a;=7-|-3, that is, 2a?=10 : consequently a;= 5, which is the solution of the proposed equation 4a?— 3=2a? + 7, as it is easy to see, for 4.x, that is, 4 times 5, is 20 ; and 2a?, that is, twice 5, is 10 : each side of the equation is therefore 17. 2. Clearing Fractions. — Whenever a fractional term occurs in an equation, we may free the equation from the fraction by simply multi- plying both sides by the denominator of that fraction. And in this way may fractions be removed one after another : the final result still being 2a; an equation. For example : if the equation be — +4a;=7, by multiplying o each side by 3, we convert it into the equation 2a; + 12a;=21, that is, 21 3 14a;=21, so that x———-, or 1^. 14 2 21. In fact, the principles brought into operation in the solution of a simple equation are all justified by the following axiom ; namely, that whether we equally increase or equally diminish, equally multiply or TO SOLVE A SIMPLE EQUATION, ETC. 9 equally divide the two sides of an equation, the result is still an equation; or, generally — whatever operation we perform on one side of an equation, if we perform the same on the other side, the result must be an equa- tion. 22. To solve a Simple Equation containing only one unknown Quantity. — The symbolical representation of an unknown quantity is usually the letter x, or y, or z: the earlier letters of the alphabet are employed, almost exclusively, to represent known quantities. A question may be proposed in which it is declared that certain quantities are known, although their actual values may not be specified : in such circumstances, we should represent the known quantities by a, b, or c, etc. The rule for solving a simple equation, in which all the quantities but one are known, is as follows : — Edle I. If a fraction occur in the equation, clear it away by multi- plying both sides of the equation by the denominator. 2. By transposition, bring all the terms containing the unknown quantity to one side of the equation, placing the known terms on the other, so as to get an equation in which the quantities on one side are all unknown, and those on the other, all known. 3. Collect the quantities on each side into a single term : there will then be a single unknown quantify on one side of the equation, and a single known quantity on the other. 4. If the X (or the y, or tke z, as the case may be) in this unknown quantity, have a co-efficient, other than unity, divide each side by that coefficient : then the x will stand alone on one side of the equation, and a known quantity, which is its interpretation, on the other. 23. Examples. — (1) Find the value of a; in the equation 9^—5 = 3ar + 19. As there are here no fractions to be cleared, we commence with the second precept of the rule, and transpose: we thus get 9a;— 3a;=19 + 5. Collecting the terms, 6x=24. Dividing by 6, a:=4 : hence the quantity X, at first unknown, is found to be 4: we see that 9 times 4 minus 5 is equal to 3 times 4 plus 19, as the equation affirms. (2) Find the value of x in 4— — -+2a;=ll. In order to clear the fraction, we multiply by 6, and thus get 24— 5a?+12.r=66 This, by transposition, becomes — 5a;-|-12ic=66— 24 which, collecting the terms, is 7ii?=42 60 that, dividing by the co-efficient 7, we have finally x=Q. (3) |-|+5=x-5 Trans., ^-|=a:-10 2x Mult, by 2, x—-^ =2a;— 20 o Mult, by 3, Zx-2x=:Qx—Q0 Trans., Zx—2x—Qx= — QQ Collecting, — 5a;=— 60 or Trans., 60=5a; -r 5, 12=.x. <'> —+6-4=^ Mult, by 2, a;-l+|-|=0 Zx ^ by 3, Zx-Z-\-x- -=0 by 2, Qx—&-{-2x-~Zx=0 Trans., 5x—Q, therefore a;=l^. (5) ?^^+6=3(.:-3) Mult, by 2, 3(a;--2)+12=6(x-3) that is, 3:r-6+12=6x-18 Trans., 3a;-6.r=-18+6-12 that is, — 3.^;=-24 or Trans., 2i=^Bx -i-3, 8=;^. 10 SOLUTION OF QUESTIONS BY SIMPLE EQUATIONS, ETC. 24. Examples for Exercise. (1) 8-3a;-|-12=30-5^+4. (2) 4a;+3=3(a;+4). (3) 2(ar-3)=3(6-a;)+2. (6) ^+|-f=to-17. (7) ^_?+^+l=o. ^'^ 3 4^5^2 (8) aa;— 6=cx4-»>i. (9) (a+&)ar=6;»+4. ao) 5 or {aa^—hx^-^-c}^, or aa?—hx^-\-c^ Or ^{ax^—lx^-\-c), .y\ac(?—'bx^-\-c], »/ {ao^—hx^-{-c}, .yax^—hx^+c 30. The operations of addition and subtraction, hitherto applied to simple quantities only, may be easily extended to more complicated ex- pressions, such as these. So long as the expressions are like, they may be just as readily incorporated, whether they are simple or compound: it has been sufficiently seen that it is with the coefficients only of these like quantities that we deal. Any multiplier prefixed to an expression is the coefficient of that expression ; and this term coefficient is extended to mean the prefixed multiplier, whether it be numeral or literal. Thus in 2>{x'-^y+z), a{x'—':iy + z), {a + h){x''-'ily-\-z), (2a-5)(a7-— 3 '^y-\-z), the multipliers 3, a, a-\-b, 2a— 5, are the respective a coefficients of (o;^ — 'iy-\-z). If the above four quantities are to «+5 be added together, then proceeding with these coefficients, as in ^°^~^ the margin, we should find the sum to be (4a-i-6— 2)(a;'^— 4aj_j_2 2.V+^). MULTIPLICATTON. 13 Examples. — Add together the following expressions :— 1. 7(^+2/), {a+h){x^y), {2a-h){x-^y), {2,h-^){x^-y). 2. 4— 2v'(a;— a), 5^/a;— a+1, Z{x—a)^—Q, 2—[x—af^. 3. ^{x-y)-^{x-^ry), a^{x^y)+6^{x-y), {Za-{-<2)V{x-y)-'l{x-\-y)'i. 4. Subtract Zz+4:x'^-2y'^-\-h from 2x^-6y^+^z-Ji-a. 5. Subtract ^{x—y) — {a—h)xy-^Qz from 3a(a;— j^)— 4a:y+22. 6. Subtract —^x-\-y)-\-^x^—y'^)-{h+c)zivomQ{x-\-y)—a{x'^—y'^) — (p-\-c)z. 7. Subtract — 6av'y+35v'a;+a+d from 4aa;^— 5%^-^a— 5. 31. MultiplicatioxiM — In writing down the product of two algebraic factors three things must be attended to : first the sign, then the coefficient^ and, lastly, the letters. If the signs of the factors be like, that is, both plus or both minus, the sign of the product is plus. If the signs of the factors are unlike, the sign of the product is minus. This may be shown as follows : — Take any two numbers at random for factors — say 8 and 3 — then all possible varieties, as respects the signs of these factors, will be compre- hended in the four following, — 8 - 8 8 - 8 3 3 __ 3 - 3 24 -24 -24 24 The first, being the case of common arithmetic, requires no remark. In the second case the product must be— 24, because multiplication by 3 means repeating the thing multiplied three times. To discover the cor- rect product in the third case, we may proceed thus : add 4 to the multi- plier; then it is plain, that by operating with this altered multiplier, the product we shall get will be 8 x 4, or 3Q too great. The altered multiplier will be — 3-t-4, that is 1 ; and 8x 1=8. Sub- tracting, then, the 32, we have 8— 32 =— 24 for the correct product. In like manner in the fourth case, the increased multiplier being — 3 + 4 = 1, and the product by it — 8x 1 = — 8, we have by subtracting —8 x4 or — 32, 24 for the correct product* I. When the factors are simple quantities. — Rule. 1. If the signs of * The proof is the same when letters are used instead of numbers ; thus : — 6 - b h - h a a —a — a db —ah —ah • ah The first and second results are obviously true. Taking the third, add a-\-l to the multiplier, then to correct the product, we must subtract from it a+1 times h, that is, a times 6 and onceb, which is ab-\-h. Now the product from the new multiplier is once h ovb ; and subtracting ab-\-h, the result is —ah. Increasing the last multiplier in the eame way, the product is —h, and subtracting a+1 times —b, that is, —ab—h from this, the result is ah. To explain the meaning of an isolated negative quantity, the learner may be reminded that -\- and — stand in opposite relation to each other. If -f^4 represent a gain, then — £4 represents a loss. If +4" express the elevation of an object above the horizon, then — 4® expresses its depression below the horizon ; if -\-a denote time after any epoch, then —a denotes the same time before that epoch; and so on. 14 MULTIPLICATION. the two factors are like, put 4- for the sign of the product ; if they are unlike, put — . 2. After the sign write the product of the coefficients. 3. To this product annex the product of the letters, that is, write the letters in both factors one after another, without any intervening sign. Note. — The letters of the product are usually arranged in the order in ■which they follow in the alphabet ; but when the same letter recurs, the notation for powers is used ; thus, instead of aaa, we write a■^ instead of i»V\ or which is the same, asx x a;xx, we write a^, the sum of the ex- ponents in the factors being the exponent in the product : thus, generally, a?'»xaj'*=a?'"+", whatever integers m and n may be. (1) (2) (3) (4) Multiply 5ax — 8a;'y 7ahx^ — Gab'^xY by Zab 2ai/ — iax"^ — Aa^xi^^ Product Ua'ix -IQaxY -2%a%a? 24a^iV3/6 (5)9cajyx-2aca;j/-=-18ac%y. {^) -\x^yy,ia3^=i—~'^x>/z II. TFAew <^5 dividend is a compound quantity, and the divisor a simple quantity. — Rule. Find the quotient of the divisor and each term of the dividend, as in last case, and connect these several quotients together by their proper signs. ._, Sa^x^v*— 4.i;3y4 , „ „ ^ ,^^ iax-^—Sa^xz^—4xz ^ „ 1 ^^^ |^27-^=4aV-2:ry. (2) —^ axz-2a^--. Examples foe Exeecise. {1) a*ar^-Zahx^+5ax*^ax^. (2) ?^f-=|^^±l?. (Z)-12ahx^2+%^]/^z-eb!^-^-Zlz, (4) 3{2:«;-(164-a;)+l}-r5. (5) -2ia''x''y-Baxy-\-6xy-^-Zxi/. (6) labc^X'r-Wb''cxk III. When dividend and divisor are both compound quantities. — Rule. 1. Arrange the terms both of dividend and divisor, so that the powers of some letter common to both may follow each other in ascending or descending order. 2. Divide the first term of the dividend by that of the divisor ; the result will be the first term of the quotient. Multiply the whole divisor by this first term, subtract the product from the like terms of the dividend, and to the remainder annex a new term of the dividend, or two terms if necessary; regarding the result as a new dividend. 3. Proceed with the divisor and this dividend as at first, and continue the operation till all the terms in the original dividend have been brought down. If there be a final remainder, it must be written, with the divisor uudemeath, and annexed to the quotient, as in arithmetic. Note. — When divisor and dividend are seen to have a factor in common, this common factor may be cancelled from both before pro- ceeding with the operation. (1) Divide ISa;^ — 13^*— 34^^40.1? • by ix^'—lx. Here it is seen that the factor x is common to both quantities : expunging this common factor, therefore, we proceed by the rule as follows, the terms being already ar- ranged according to the descending powers of x: — Ax-1)\2x^-lZa?-Zix^-^i0x{Zx'-\-2x^-5x-\-j£jj 12a;*-21a;3 %x^-Ux^ -2Qx^-\-iO« -20x^+Z5x Remainder 5x 18 DIVISION. (2) Divide 4a''a;+5laV+10;c*— 48a:»^-15a* by 4ax-6x'' + Sa\ Arranging the terms in order, according to the powers of x, the opera- tion will be as follows : — -6a;2+4ax+3a2)10a:*-48aa;3+51aV+4a3a;-15a*(-2u;H8aa;-5a2 10a:*- Saar'- 6a V -40a;c3+57aV+ 4a^x -40aa:3^32aV+24a'a: 25a V- 20^3^- 15a« (3) Divide 1 by 1+ar. 25a^x^-20a^x-15a* 1+^)1 il-x+z^~^ l+ar —X —x—x^ The division in this third example may be carried on to any extent ; so that (omitting the remainder) — -=l— 4;+a;'— aj'+a;*— aj^+a?"— a?''4- &c. But at whatever term of the quotient we may choose to stop the opera- tion, the subsequent remainder, with the divisor underneath, must be connected, with its proper sign, to the quotient. \ 3? a:* Thus, — — -=1— aj+a:*— — — , or z=:\—x-\-3?—s?-\--——^ or =1— ar+a;*— a;^+a:*— 1+a; \-\-x \-\-x — — -, and so on. Examples for Exercise. (1) Divide 6a:2+13a:+6 by 3a;+2. (2) „ 6a;*- 96 by 3a; -6. (3) „ «^-/ by x-y. (4) ,, a^—ar^ by a— a;. (5) Divide 25a;''-a;*-2a:3-8a;2ty5«3_ 4a;2. (6) „ 6a^+9a;2_20a;by3a:2_3^^ (7) „ Ibyl-ar. (8) „ :^-\-lx'^-\-cx^d by x-a. and show that the remainder is the same as the dividend when the x in the latter is replaced by a, 33. Note. — In this last example the truth of the property mentioned is supposed to be arrived at by actual division, the remainder from that division being a"^+6a^+ca+ci; the property, however, is perfectly gene- ral, and may be established without going through any division process. It may be enunciated as follows : — If any algebraic expression with terms containing x, when divided by either x—a^ or a; -fa, leave a remainder free from x, that remainder will always be what the dividend becomes when the x in it is replaced by a, or —a, according as the divisor is «— o, or x-\-a. For, from the nature of division. Quotient x Divisor -j- Remainder=Dividend. ALGEBRAIC FRA.CTIONS. lO And in the case before us, the Eemainder, being free from a?, must continue unaltered whatever value we put for a: in this equation. Let a be put for o), if the divisor be a— a, ov —a ii it he a-\-a; then the above equation will be Quotient X + Remainder = Dividend. But any finite quantity multiplied by nothing produces nothing ; hence the equation last written is simply Remainder = Dividend; that is, when the proposed substitution for x is made in the dividend, the changed ex- pression is the same as the remainder. The following are illustrations of the application of this theorem: — (1) Is ^*— 3;k"^— 5^— 14 divisble by ;»+2? Putting —2 for a?, the dividend becomes 16 — 12 + 10 — 14, which is equal to 0. .-. Remainder =0, .-. the expression is divisible by ^+3. (2) Is x^^^ar^ + ^x—4: divisible by ic— 2 ? Putting 2 for ^ in the dividend, we have 8—8 + 6—4=2, the rem. Hence the proposed expression is not divisible by ^—2 ; the division leaves a remainder 2. The following also are useful inferences : — Whenever n is odd \ ^"-*" ^^ divisible by x-a, but not by x-]-a ( a;"+a'* ,, x-\-a, but not by x—a TTTi. • { x^ — a" ,, both a; — aaindx-\-a Whenever n is even ] " . , ^"' { x^-\-a^ ,, neither a;— a nor a;+a By the principle above, the remainder, in this last case, is 2^". 34. Algebraic Fractions. — Fractions in Algebra are treated ex- actly in the same way as those in Arithmetic: whether the symbols employed be figures or letters the rules of operation are just the same. This the learner will be prepared to expect, because algebra becomes con- verted into arithmetic so soon as the letters are replaced by numbers. The Rules therefore need here be but very briefly recapitulated. Reduction of a mixed quantity to an improper {or single) fraction. — Rule. Multiply the integral part by the denom. of the fractional part: connect the product, by the proper sign, to the num., and write the denom. underneath: thus — (l)a+^=^. (2)a+._?5=^^±^?^^. (3) ^^+^=^±^^=.^. ^ ' 6 b ^ ^ ^ y y ^ ' ^ ' ^X s/X ^X (A\ <^°+^' a:'-2ah-\-h'' _ {a-lf ah ah ah Examples for Exeecise. — Reduce each of the following mixed quan- tities to an improper fraction : — W-+;- (^)l-T (3)1-^' (4)^%2. (5).»-3.+2+^l Reduction of an improper fraction to a whole or mixed quantity. — Rule. Divide num. by den. If there be a remainder, annex it, with the denom. underneath, to the quotient, as in all cases of division. ^ ' 6 by y ^ ' 2ax ^2ax ^ 2ax 2 5J0 ADDITION AND SUBTRACTION. Examples for Exercise. — Keduce each of the following to a whole or mixed quantity : — ^^^"T"- ^^^ x-y ' ^^K-\-b- ^^^ d^x • ^^K-^-3x+2' ^^^ 4ax • '^' 4x^-7x ' Iteduction of fractions to a common denominator. — Rule. 1. Multiply each numerator by the product of all the denominators except its own, the several new numerators will thus be obtained. 2. Multiply all the denominators together, the product will be the common denominator. That the values of the fractions remain unaltered by this change in their forms is obvious ; for, take whichever of the original fractions we may, we see that the change is effected by multiplying its num. and den. by the same thing ; namely, the product of the denominators of all the other fractions. The common denom. found as above is, of course, a com- mon multiple of all the original denominators : if it be not the least common multiple, the changed fractions will not be in their most simple forms. By glancing at the original denominators, the least common multiple of them may often be readily found : — if common factors enter two or more of the denominators, they should be retained in but one : — the repetitions should be struck out; the product of the denominators, thus deprived of common factors, will be the L.C.M. (least common mul- tiple). As to the numerators, we have only to multiply each by the quotient arising from dividing the L.C.M. by the corresponding denomi- nator, so that, as before, the values of the changed fractions remain undis- turbed. The object of this reduction of fractions to a com. den. is merely to fit them for addition and subtraction, the rules for these operations being as follows. Addition. — Rule. Reduce the fractions to a common den., and write this com. den. under the sum of the changed numerators. Subtraction. — Rule. Write the com. den. under the difference of the changed numerators. (1) Eequired the sum of - — , — , j. Appljring the rule, the new numerators are 2x8a?, 5xl2aa?, cY^^au^^ and the com. den. is 3aafX2d?X4:: hence the changed frac- 16j? Vlahx Qacx^ . , . , .i ^ « • « i? j. • ^i. tions are z, s> 5> m which we see that 2x is a superfluous factor in the 24a^ 24«j?2 2iaar num. and den. of each. We might have foreseen this by running the eye along the row of denominators : — there is a repetition of the factor x and of 2 : suppressing, there- fore, one a and one 2, we have 3a,»x2x2=12aa? for the L.C.M. of the denomina- tors. Hence multiplying the numerators in order by 4, 6a, Sax, and adding the results, 8+6a64-3ac/r , ^, we have for the sum. 12ax 2x4-3 X 1 (2) — — — I — I — . Here the denominators have no com. factor; hence by the rule, 4 6 3 30a;+454-123;+20 42a;+65 . ^^ 60 = -60- ^ *^' '^- (3) -^ — — I — ^^^. Here 28 is the L.C.M. of the denominators. ^ ' 4 7 28 14a;-21-12a;-16+5-2a; 32 8 ,^ .*. = = — , the sum. 28 28 7' MULTIPLICATION AND DIVISION. 21 x+y 3^—y^ {x+y){'^—y) v?—y^ x^—y^ x^—tf Examples for Exercise. a±^ a-x (2) %x^ 14' (3) ^+^-^^2=. (4) 34-2.g 2__ 1-0^ \-x~' (i>) -S 2"^ 1 =• (6) Prove that -^+-^=-'^ ^. x-\-y x—y x—y x-\-y (7) Prove that — ?^=_^ !_. Multiplication and Division. — For Multiplication. — Rule. 1. Multiply the numerators together: the result will be the num. of the product. 2. Multiply the denominators together : the result will be the den. of the product. For Division. — Rule. Invert the terms of the divisor ; that is, turn the fraction upside down, and then proceed as for multiplication. The truth of these rules may be proved thus : — Let -, - be two fractions, which represent by x, y respectively : then Cti c 4?=-, and y=^- . '. hx=a, and dy=zc . '. hdxy=:ac o d •" ac . , a c a .•..ry=-,thatis,-X-=-. Again : resuming the equations bx=a, and dy=:c, we have hx a X ad ,, . a c a d ——- .-. -=— ; that IS, -^-=-X-. dy c y oc a c These results suggest the foregoing rules. Note.— Before performing either operation, simplify the fractions by cancelling whatever factors may be seen to be common to both num. and den. of either. If the num. of one fraction and the den. of another have a factor in common, then, before multiplying those fractions, the common factor may be suppressed. 6^ 2^ l_10a^ 3^ y 2a_ 3a;XyXl _3a;y ^' 4^2^"6"^7~"aX5x7""35a' (3) a+h h a-\rh 2a_2a{a+h) 2a-\-c ' 2a 2a+c b b(2a-\-c) /.v /o 3J2\ /a2 \ 3(a2-62) a{a-b) ^a^-b^) ^ h U{a+b) (4) (^3a--;^(^--a^=— ^^H— ^=— ^X^^— ^— -^^. „, 2a;— 3 Zx ,„. 6ax2 2 ex ,„ (3) ?i^x4-=. ^ ' a >/x Examples for Exercise -a;2 a-\-x b^—y^ ' b-^y (5) x+2a ' 4a3+8a a;2_y2 a; 1 _ %^ GREATEST COMMON MEASURE. (7) 12.{M-2.]=. (9) (10) (^-IH^4)=- (11) 35. Greatest Common Measure.— A common measure of two or more quantities is any expression that will divide them all ; and it is the greatest common measure when it is made up of all the factors which are common to the quantities measured : the symbols for the great- est common measure are G.C.M* Let A^, B^, be two quantities in which ^ is the greatest factor common to both, so that A and B have no factor in common ; and AS being taken for the greater of the two quantities, let the successive divisions be carried on, as below, till the work stops for want of a remainder. Let the last of the quotients be d, then since Quotient x Divisor + Rem. = Dividend, we shall have the following series of equations on the right of the operation. 2)5 .-. dEz=D cD-\-Ez=C lC-\-D=D aB^C=^A From inspecting these equations in succession, it is seen from the first that S is a factor of D, and therefore, from the second, that it is a factor of C, and therefore, from the third, that it is a factor of B, and therefore, finally, that it is a factor of A. But, by the original condition, A and B have no factor in common, except unity; therefore £"=1: hence the iinal divisor ES is simply ^, the greatest common measure of the two proposed quantities. If this final divisor should happen to be 5=1, we should then conclude that the quantities have no common measure, unity not being regarded as a measure. The above, then, is the process for finding the G.C.M. of two quan- tities, whether they be numbers or algebraic expressions. And it is plain, from the general type of the operation, that what is the G.C.M. of the first dividend and divisor (AS, BS) is equally the G.C.M. of every other dividend and divisor to the end. If, on arriving at any dividend and divisor, we find a factor in one which is not in the other, it may be expunged ; or a factor may be intro- * It is very desirable that the old term measure, in reference to the present subject, should be abolished, and the term divisor always employed instead. The above symbols would then be replaced by G.G.D., which are the more to be preferred, because L.C.M. already stands for least common multiple. GREATEST COMMON MEASURE. 25 duced into either provided it be excluded from the other; for these changes cannot affect the common measure of the two. (1) Required the G.CM. of 1x^—12x-\-5, and a^— 6a;+5 a;2-6a;+5)7a;a-12a;+5(7 7a;2-42a;+35 3003—30 or expunging the factor 30, x— l)x^^Qx-\- 5(x — 6 -5a;+5 Hence x—1 is the G.CM. (2) Required the G.CM. of ia?-2oc*-Zx+l and Zx^-2x-l. Multiply the first dividend by 3, then the second by 2, in order to avoid fractions. 3x'-2a-l)3x 4a^-6x2_9a;+3(4a> -8x^-ix 2a;*-5a;+3)2x3a;2_ 4a:-2(3 lla?-ll, or x-l)2x^-5x-\-Z{2x-Z 2^-2-5^+3 Hence a;— 1 is the G.CM. By finding in this way the G.CM. of the num. and den. of a fraction, and then dividing the two by that G.C.M., the fraction will be reduced to its lowest terms, so that no further simplification will be possible : thus, taking the two examples above, we have x^-6x+5 x-5 , Za^-2x-l Sx+1 —.-z=;: ^> ^^^ ~ 1x^-l2x+5 7a;-5' 4x'-2ip2_3^+l 4x24-2^-1* the num. and den. of each of the fractions being divided by the G.CM., 36. It may be as well to notice here that algebraic expressions are distin- guished from one another by certain designations which refer exclusively to the number of terms they contain, and to the highest power which the principal symbol in them reaches : thus ax is a monomial of the first degree, because it consists of but one term, and x occurs only in the first power ; ax"' is a monomial of the second degree, ax^ one of the third, and 80 on. Again, ax + h\s a. binomial of the first degree, ax'^-i-b a binomial of the second degree, ax'*-\-b one of the third degree, and so on. Simi- larly, Sx-—^x—] is a trinomial of the second degree, Ax^—^x' — Sx+l is a quadrinomial of the third degree, and so on. But instead of multi- plying particular names for the number of terms, the word polynomial is usually employed to denote an expression of several terms, without explicitly implying how many. It is worth remembering that when an expression of the second degree, or a quadratic expression, as it is sometimes called, is known to be divisible by a binomial of the first degree, the quotient may be at once obtained by simply dividing the first term of the dr Jrlend by the first term of the divisor, and the last by the last : thus referring to the quotients above, we see that x—6 may be got by dividing x"- by the x, and 5 by the —-1. In like manner, Sx-j-i may be got by dividing 3^ by the x, and —1 by the 24 LEAST COMMON MULTIPLE. —1. Similarly, knowing, as we do, that a;^— 5^— 24 is divisible by a!-\- 3 (see 33) we may get the quotient by dividing or by the aj and — 24 by the 3 ; this quotient being x—S. If we wish to find the G.C.M. of three expressions, we find the G.C.M. of two, as above, and the G.C.M. of this and the third, and so on, if there be more quantities than three. 37. Least Common Multiple.— The least common multiple of two or more quantities is the least quantity, or the quantity of lowest degree, that is divisible by each of them. The symbols for the least common multiple are L.C.M. If two quantities have no common measure their product is their L.C.M. ; for, since the factors of the two quantities are all different, these factors must all be contained in every expression which is divisible by them all, the least expression thus divisible is therefore simply the pro- duct of all the factors. Let, then, the two quantities have I for their G.C.M. : we may denote them by A^ and B^, where A, B, have no factor in common. It is plain that the L.C.M. will be AB^, that is, there cannot be any superfluous factor in AB^ ; for if this be divided by -45, tlie quotient is B, and if it be divided by B^, the quotient is A, and these quotients have no factor in common ; so that no factor can be spared from ABL Hence, to find the L.C.M. of two quantities, we divide their pro- duct by their G.C.M. And the L.C.M. of three quantities may be found by taking the L.C.M. of two of them, and then the L.C.M. of this and the third, and so on. (1) What is L.CM. of 12 and 18 ? Here the G.C.M. is evidently 6. .-.-4— =12X3=36, ihQ L.C.M. o (2) Find the L. C. M. of x^—y^ and x''—y'^. Each of these is divisible l>y ^— 2/ (^^t- 33) • the quotient from the second isx+y (page 15), by which quantity the first is not divisible (33). Hence the G. C. M. is x — y .: {ay'—f){x'—y')-^{x-y)={x'—y\x-\-y)=x'^^x'y—xf-^y\ the L.C.M. In a similar way, we find that the L.C.M. of x'+l, and {x+iy is a^-\-3?+x-\-l. „ 2x'-\-x'-2x-\, and 2x^-x''-2x-\-\ is 4a;*-5r^+ 1. ,, ic'— 1, a;— 2, and a;'— 4 is x^—5x'^-\-4t. 38. Before leaving the subject of fractions, we would invite the special attention of the student to the following property of two equal fractions : he will find it of frequent application in the solution of equations. Let there be two equal fractions, as t=;^. Adding and subtracting unity, we have — 3+1=^+1, that .»,—=—...[l]. a , c _ a—h c—d ^-l=-_l, „ __=_... [2]. Dividing [I] by [2], we have the property alluded to; namely, that T- a c a+6 c-\-d , a—h c—d If 7=^, then also — r= — -, and .'. r= . b d a—b c—d a+b c+d SIMPLE EQUATIONS IN GENERAL. 25 And it is plain that if, instead of the minus sign before the 1 in [2], we had used the sign ~, implying difference, the inference would have been, that — „ a c ., - a+6 c-\-d , a~6 Cf-^d If T=-5> then also, — -= -, and - = -. a aryJ) C'-^d a-\-o c-\-d 39. Simple Equations in General-— In a former page of this work, some illustrations were given of the application of the first four rules of Algebra to the solution of simple equations. These illustrations were proposed at that early stage of the subject, in order that the learner might gain an insight, as soon as possible, into the use and efficiency of the symbols of algebra in certain inquiries respecting numbers. We are now prepared to take a more enlarged view of the doctrine of simple equations, and to employ methods of solution preferable to some of those adopted in the former article (art. 22). To SOLVE A Simple Equation with one unknown Quantity. — Eule. 1 . Clear the equation of fractions, if any enter. This is done thus : re- gard every integral term as a fraction with I for denominator, and then multiply each numerator by all the denominators except its own, exactly as in finding the new numerators in the operation for reducing a set of fractions to a common denominator (34). But here the com. den. is suppressed : — the results furnishing an equation without fractions. Or, the equation may be cleared by multiplying every term by a common multiple (the smaller the better) of all the denominators. 2. Clear the equation of radical signs, if any enter. This is done thus : by transposition (20) cause the radical that is to be removed to stand alone on one side of the equation, the other terms all occupying the other side : then perform on each side the operation which is the reverse of that indi- cated by the radical ; thus, if the radical be V, square both sides; if it be '\/, cube both sides, and so on. [It is plain that the square of i^k is k, that the cube of i/k, is k, and so on, whatever k may represent : — the reverse operation merely removing or clearing the radical.] Note. — The order in which the general precepts for the solution of an equation may be most conveniently applied will be suggested by the example itself; but as fractions are more troublesome to deal with than integral quantities, it is usual to clear them away early, as in the following specimens. But the learner is not to expect that the neatest and best method of solving every equation that may be proposed can be explained by written instructions. He can become practically familiar with the various artifices resorted to, in particular cases, only by observing the purposes effected by them in the examples worked out for his guidance and imitation. Thus, in ex. 7, page 26, the second step in the process of solution directs that 1 be added to each side of the equation : this step is suggested from observing that, if 1 be added, the denom. of the fraction is such that, upon reducing the mixed quantity to an improper fraction, the whole of the original numerator becomes cancelled, and that thus a considerable simplification is effected. Clearing fractions (regarding —x, and (1) |-|+6=a;-5. (See p. 9.) Trans. ~—^-x=: — 10, it o —10, as — - and —r-) 3a;— 2^— 6ic=— 60, or — 5a:=-60. Dividing by —5, a;=12. SIMPLE EQUATIONS IN GENERAL* (2) ■V^.- ^=0. (See p. 9.) 2 ■ 6 4 Mult, by 12, the L. C. M. of 2, 6, 4, 6;c— 6+2a:— 3a;=0. Trans., 5a;=6 . •. x=l^. 7a;+8 9:p-12 3x+1_29-8z 5 (3) 4 8 5 10 Multiply by 40, the L. C. M. of the den. 70ar+80-45a;+60=24a;+8-116+32^. Trans., 70a;-45a;-24a;-32a;=8-116 -80-60 .-. -31a:=-248.-. ;c=8. Trans., and mult, by 2, 3v/a:=6 .'. v''^=2 ; Squaring, a;=4. (5) v'(3+4a:)+8=2a:+9. Trans., ^(34-'4a:)=2a;+l. Squaring, Z+ix=ix'^+ix+l. Trans., 2=4:^2 . •. 4=a;' .-. a:=^i. (6) v^(4+x)+v^a;=4. Trans, the ^/a:, in order that v'(4+a;) may stand alone on one side, -/(4+^)=4-^ar. Squaring, 4+a:=16— 8v^a:+a:. Trans., 8v^a;=12.-. 2v/a:=:3. Squaring, 4a;=:9 . '. x=:2\. a— a; Adding 1 to each side, 2ax-x' "« 2 +l=t' +1, or {a-xy -^=v'(i'+l) {a-xr a—x 1 V{h''-\-l) H-3_7 a;-3""5' By the principle at (38), 2x 12 a ^/(^^=^+l) (8) ^;= (9) ^_,_e.-..=i8. 1+a;- ^ {2x-\-x') ^^^ "~1- By the principle at (38), v'(2a:+x') 3 . 2x-\-x' 9 -^^P^=-.-. Squaring, ^-^^-^=-. Subtract each side from 1, then 1 ^16 , 1 _4 {\-\-xY 25' 'l+ar—S' Reversing the terms of each fraction, (10) ,\l-\rx=-- .' . X-. 4 5a;=-9 a:^/5-3 _1 "4* =1. Xy/5-^Z 2 Here the den. of the first fraction is the sum of two quantities, and the num. the difierence of their squares ; hence (p. 15) the equa. is the same as x^ 5-3-l(a:^/5-3)=l, that is, \{x^^-Z)z=l . • . arV'^- 3=2 . • . ar v'5=5 .•.a;=— =^5. Multiplying by 4, 4a;+3 43— 4ar =1 4a; that is, -^+1 3 3 129-12a: 11 =1. (by mult, the terms of the second frac- tion by 3), 4x 129-12ar ^ ••• ¥ !!-='■ Clearing fractions, 44a:— 387+36ar=0. Trans., 80a;=387.-. cc=4—. 80 (12) a-\-x=^ {a^+x^{b^-\-x')}. Squaring, a2+2aa;+a:2=a2+a:v/ (h^'+a^) Subtracting a", and dividing by ar, 2a-\-x=s/{b*+x'^. Squaring, 4a^-\-4:ax-\-x^=P+x^. Subtracting ar^, and transposing, Aax'=h^—ia^ . •. x=. =- a. 4a 4a ^a—lax-{-a^ is known to be the square of x-^a, and it is easy to see that this root may be evolved from its square thus ; the square-root of the leading term is x, the first term of the sought root ; and x^ being subtracted x'^-\-2ax-\-a'\x-\-a from the proposed trinomial, leaves 'Hax-^-d^ for • x^ remainder, the leading term of which, if divided 2a;-f a] 2ax+cfi by twice the first term x of the root, already ob- -' 2ax-\-a? tained, will give the second term a. We may regard ^x as a trial or partial divisor of the remainder ^ax-^a^y and place it against this remainder, as in the margin ; and having from the trial divisor got a, we have only to connect this a with the trial divisor to get the true or complete divisor due to the quotient a. 2. The polynomial x^ -\-':iax^ +{a^ ^^h)x^ J{.^abx4-h'^ is the square of x'^-{-ax-^h. And proceeding in imitation of the process above, doubling what may already be in the root-place, for the trial divisor expected to give the next term of the root, we find the complete root as before, thus : — 2a;2+a^| 2ax^+ (aH 26)a^ 2ax^+a'^x^ 2lx^-\-2abx-\-W- 47. There was scarcely any necessity to exhibit this last operation ; the efficiency of it might have been inferred from the preceding one; for since(;»Ha^-f&)2={(rcH«^) + 6F=(^' + «^f + 2%' + «^) + fc'(p.l5),the proposed square is no other than the trinomial just written. The root of the first term, that is the portion x'^-\-ax, of the whole root is got as pre- viously explained, and thence the remaining portion h. And by the same uniform process is the root of the polynomial furnished by {x^-\-ax'-\-hx -\-cf or (x^-\-ax^-^'bxY-\-^c{x^-\-ax--\-hx)-\-c'^ obtained; and so on: the rule for the operation is, therefore, this : — JRuLE. 1. Arrange the polynomial as if for division. 2. The square root of the first term, will be the leading term of the sought root ; place the square of it under the first term, to which it is of course equal, and, having drawn a line under it, bring down the next two terms of the polynomial ;i- regard these as a dividend, and for a trial divisor of it, write twice the root-term just found. 3. Find the quotient of the leading term of the dividend by this trial divisor, and connect it, with its proper sign, both to the root-term, and to the trial divisor ; the divisor will then be completed. * It is scarcely necessary to premise here that x may be anything — even 1. + When terms are absent from the arranged polynomial, their places should be supplied by zeros: thus x'^-\-Qx"-\-2x-^l should be written si^-\-(ix^-\-Qa?-\-2x-]rl, CUBE ROOT OF A POLYNOMIAL. 39 4. Multiply the complete divisor by this second root-term ; subtract the product from the dividend, and connect with the remainder two more terms of the polynomial for a second dividend. 5. Proceed now exactly as before, taking twice the part of the root already found as a trial divisor, for finding the third term of that root, which third term, when thus obtained, completes the divisor; and so on till all the terms of the polynomial have been brought down. If after this, there be still a remainder, we may be sure that the poly- nomial is not a complete square. (1) ix^-ix^-\-ldx^-6x+9\ 2x^- x+B (2) 4x*-12x^+25x^-2ix-\-lS\'2f-Zx-{-4: ix* 4a;* 4x^-x\ -4^3+13^^ 4a;2-3£| -12a;3-f 25a;2 ~ _4a;3-fa;« -12^3+9^ 4x^-2x-i-d\ 12x^-6x+9 4a;^-6a;-f4| 16x^-2ix+18 12x^-6x-\-9 16x^-2ix+16 We infer from there being a remainder in this second example, that the proposed polynomial is not a complete square. Examples for Exercise. (1) Ax*-As(^-dx^+2x-\-l. (2) 9x*-eax^-\-a^. 3) 9x^-12a^+10x*-2S3(^-\-17x^-8x-\-lQ. (4) Ax^+12x^-\-5x*-2x^-{-1x^-2x-\'l. (5) ix'y*-12a^/-{-17x*f^l2x^y-\-ix^ (6) ^+2L^+2(-+^)+3. 48. Cube Root of a Polynomial.— Examining the constitu- tion of a cube, we find that {x-]-af=x^-^{2x^-\-Sax-{-a^}a [See page 15]. &c. &c. Taking the first of these cuhes namely, a?-\-Za^'^Za^x-\-a^, the process in the margin, for evolving the root, is suggested. The first term x of the root being found, a?-\-dax^^da^x-\-a^\x+a and the dividend, consisting of three terms, „ '- brought down, we write 3 times the square _ of the root-term as a trial divisor for find- Zx'^-{-Zax a^ Zax^-{-Sa^x+a^ ing a ; we then complete the divisor by an- Sax^-\-Sa^x-\-a^ nexing three times the product of the last — — found term and the preceding, and also the square of the last found term. Hence the ollowing rule. Rule. 1 . The terms being arranged as in the square root, find the root of the first term, place the cube of it under that term and brmg down the three next terms for a dividend, the trial divisor of which will be tbree times the square of the preceding part of the root. 2. By aid of the trial divisor find the second term of the root, and complete the divisor by annexing thrice the product of the last root term and what precedes it, as also the square of this last root term. 3. Multiply the divisor by the new root term, subtract the product trom the dividend, and to the remainder annex three new terms from the poly- nomial. And proceed, as before, to find a third root term; and so on. 40 IMAGINARY OR IMPOSSIBLE QUANTITIES. Ex. 27x'^-5ia^-{-6Bx*-Ux^-\-21x^-6x-\-l\Sx^-2x-{-l 27a;4-18a;3+4^| -5ix^-\-eSx*-Ua^ ~~ ^ 5^x^-\-Z6x*- 8x^ 27x*-B6x^-{-21x^-ex+l\ 27;t*-36a;3+21a;2 -6x+l =d{dx^-2xy-Z{Zx^-2x)-j-l 27x*-B6a^-\-21x^-6x+l Examples for Exercise. (1) x^-ex^^-^-lBx* -20x^+15x^-6x^1. (2) x^+Qx^-^i0s^-^96x-Qi. (3) 8aP-{-d6x^-^5ix-j-27. (4) x'^-3ax^+6a^x^-7a^x^-\-Qa*x^-Za^x-^a'^. From the preceding general rules are derived the methods for extracting the square and cube roots of numbers, as given in most books on Arith- metic. But as respects the extraction of the cube root of a number, a much more simple and convenient process will be explained hereafter. 49. Surds. — ^When a quantity having the radical sign, or a fractional exponent attached to it, is such that the indicated root cannot be accu- rately extracted, the expression is called a surd, and sometimes an ir- rational quantity. Quantities which are not surds are called rational quantities. The following are all surds, namely, a/2, -v/5, 1^/7, -v/S, &c., &c., because there is no number whose square is accurately 2, or 5, or 7, or 8, &c. ; so also \/G, \/8, &c., are surds, because there does not exist any three equal numbers which, multiplied together, will accurately make 6, nor four equal numbers which, multiplied together, will make exactly 8, &c. The above are arithmetical surds: the following are algebraic surds; ^/a, V^^ V^^' (^+^)^' <^c-» because no letter, or letters, free from radical signs, can, by the reverse operation of squaring, cubing, &o., produce a, a^, &c. On the contrary, ^/4, ^/9, ^8, V"" ^^' V«^ ^/«^ •^{a+h)\ &c., are all rational quantities : the indicated operations can be accurately executed, and the results correctly exhibited: they are 2, 3, 2,-3, a^, a*, [a-^bf, &c. 50. It must be noticed, however, that the result of a square root ope- ration upon a positive quantity is, in strictness, of a twofold character : it is either +, or — : thus v 4 is as much —2 as +2, because, which- ever of these we multiply by itself, the result is still 4 : hence, employing the double sign, we are justified in saying that \/4=±:2, that is, that the square root of 4 is either plus or minus 2. In like manner, -s/9=±3, v/36 = ±6, and so on. And this twofold form equally belongs to every even root of a positive quantity, because an even number of factors, whether they be all positive, or all negative, equally gives a plus product ; thus, ^16 = ±2, because (2)*, and (—2)'' equally give 16. Any odd root of a quantity is, however, unambiguous, because any odd number of factors, with the same sign, produces that sign: thus V8=2, y — S— — '2, &c., because 2-^=8, (—2)*=— 8, &c. : and it is plain that no other real values of V^, V-8, &c., exist. 51. Imaginary or Impossible Quantities.— But when an even root of a negative quantity is indicated, there is implied an impossi- bility ; such root is called an imaginary root : no even power of a number, whether the number be positive or negative, can ever produce a negative result: thus, such operations as V— 4, ^—16, is/ —a, &c., are all of REDUCTION OF SURDS. 41 impossible performance : there is not only no number which, multiplied by itself, will produce —4, but there is no number which, squared, can even approximate to —4. It is true that no number, squared, can pro- duce 2, or 5 ; but we can assign a number which, when squared, shall approach as near to either of these as we please, as arithmetic shows : thus, v^2=l*4142...., and \/5=2'23606... But imaginaries cannot even be approximated to. All that can at present be said, as to the interpre- tation of imaginary quantities, is, that whenever they necessarily enter into the result of any problem, that problem may be safely inferred to be of impossible solution, or to involve impossible conditions. 5-2. It is desirable to notice that every single imaginary may be always represented by two factors, one of which is real, though not always rational, and the other the imaginary, -s/ — 1 oyX/ — 1, &c. :* thus ^ —A = V(4X-1)=2>/ — 1, V-16=2V— 1, ^-5 = ^5^^ — 1, V'-«= V'a^/ — 1, &c. Especied care, too, must be taken not to regard such expressions as /v/( — 5)-, V(— 2)*, ^(^af, &c., as imaginary: in these, the operations indicated by the radical and the exponent, merely neutralize one another, and the quantity to which they are attached remains unaffected by their joint influence: thus, ^(-5f=-5, V(-^)'=-2, v/(~a)^=— a, &c. 2. 4 2. In fact, these expressions are the same as (—5)2, (—2)^, (— «)^, &c. (43). 53. Reduction of Surds. — A surd may be reduced to another of simpler form, whenever the quantity under the radical has a factor upon which the extraction of the root indicated can be actually performed : thus, since ^3 8= ^(9 x2).•.^/18=3^/2. In like manner, since ^(a^— a^^)= ^{a\a^a;)] .-. j[(i^—a'^x)=a^\a—x). If the surd be fractional, it can be reduced to an integral form by multiplying numerator and denominator by such a quantity as will make the new denominator a complete power corresponding to the root : thus, — T 1-1 / ^^ / 4.2a;^a; 1x /2x 2x /Qxy_ 2x In hke maimer, ^^^^=^__^=— y-=-^y ____.^6.:2,. Examples for Exercise. r2a^ a) Vh (2) ^60=. (3) ^-54=. (4) ^/12aV/=. (7) {Sa%^-^=. (8) ■i/{ax'+!>x')=. (9) ^{Za'x+eabx+Zt'x)— (10) («+-)v/|^=- 54. Besides reducing a surd to its simplest form, it is necessary, when two surds are to be multiplied or divided,' the one by the other, that both be reduced to a common surd-index. This is done by using fractional exponents instead of the radical sign, and then bringing the fractions to a common denominator, which denominator is, of course, the surd-index common to both expressions : thus, the surd-indices in ^a and ^^b, are ♦ It may be proved, though not conveniently in this place, that any root of —1 is reducible to a form involving no imaginaiy, but y/ —1. 4<2 BINOMIAL SURDS. different, namely, 2 and 3 : to change the surds into equivalent ones with the same index, we change the fractions i, J, in a^, b\ into their equiva- lents f, I : the expressions are then a«, b^, or \/a^, \/fe^ so that ^a, X/b, are thus reduced to equivalent surds with a common surd-index, namely, the index 6. 55. Addition and Subtraction of Surds.— EuLE. l. Reduce the surds to their simplest forms. 2. If the surds be like or similar, add or subtract the coefficients, and annex the common surd part to the result. But if the surds are not similar surds, the addition or subtraction can only be indicated — (1) ^/72+v^l28=,/(36x2)+^/(64x2)=6^/2^-8^/2=14^2. (2) 9V4-Vl08=&V4-V(27x4)=9V4-3V4=6V4. (3) 4v/f-3^2V=^^/15-f,/f=|v'15^i^/15=^yl5. (4) Add together 3/24, 3^32, and ^192. Here '^2i=%/{SxB)=2l/B, and 8^32=33/(8 X 4)=6V4, also Vl92=V(64 X 3)=4V3 : hence the sum is 2^y/Z-\-6l/i-\- 4V3=6(V3-fV4). Examples for Exercise. (1) v'32+v/T2=. (2) 3V§-2V5V=. (3) ^/27a*a;+V3a%= (4) 2^18-Z^8-\-5^50z (5) ^o?x-s/^=. (6) .Ja%^^ah^-l^ab=. (7) In/I-WI +n/15=. (8) ^(a2_£2)(a_J)+6^(a+&)=. (9) 3^-16-5^-4=. (10) 2^/72+V24+aV6a;2=. 56. Multiplication and Division of Surds.— BuLE. Reduce the surds to equivalent ones expressing the same root (54) : then multiply or divide, as required — (1) V2 X V3=2^ X 3*=2^X 3^=6/(23X 32)=V72. (2) ^/8xV16=8^Xl6*=2.2^x2.2^=4x2^x2^=4.2*=46/32. (3) (3V2-6V2)--V3=3v|-5.2^--3^=^/6-5V^^/6-|vi08. Examples for Exercise. (1) 2^/3x33/4=. (2) •V4x7V6xv|=. (3) (is/3)3=. (4) (23V2)*=. (5) V12-T-V24=. (6) 2V^xv|xV16-T-V32= (7) (3V2-5V2)-r-V3=. (9) Va-^Va=- (10) (^V18+^V8)xV2=. 57. Binomial Surds. — In the foregoing examples, every divisor has been a monomial surd: when it is a binomial surd, and of one or other of the forms ^/fl^db^/&, %/cL±:%/b, a multiplier of it may always be found that will render the product rational, so that when a fraction occurs with a denominator of such a form, that den. may be rationalized by multiplying both terms of the fraction by the suitable multiplier. When the form of the binomial surd is ^a±:^b, the rational- izing multiplier at once suggests itself, from the property, that the sum BINOMIAL SUEDS 43 multiplied by the difference of two quantities, gives the difference of their squares: — it is always either ^a—^b, or ^a-\-sJh, thus: — (n/«H-x/Z>)(n/«— v/Z>)=a— &, and {s/ct—,jh){ja-\-s/h)=a—h.,....{^A'\ But when the binomial consists of two cube roots, the rationalizing multiplier is not so obvious : in this case it is trinomial, and consists of the squares of the two surd terms, and of their product with changed sign. This will be sufficiently seen by inspecting examples (4) and (7) at page 15. As in those examples, x and y may each be anything, we may replace them by %/a and \/b, when we shall obviously have i^s/a—X/b) i^^a^+yab + yb'')=a-b,wadi{^s/a-\-yb){X/a-—yah-\-yb~) = a + b.,,\B\ These expressions [^J and [B] furnish all needful directions for ration- alizing binomial surds of the form ^a±Ls/b, or \/a±L\/h. ,,v 3_ 3(2^5+3^7) _ 3(2s/5+3V7) _ 3 ^ ' 2V5-3V7"'{2V5-3V7)(2V5+3V7)~ 20-63 43^ ^ "^ ^ '' io\ 2 2(V6-V5)_ 2 ,„. 1 _ V4-V 2_ (V4-V2)(2+^2) ^ ^ V4+V2~^/4-v'2" 2 <1) 3^(2v'7-3v^5)=. (2) (2+^/3)^(3+^3)=. (3) ^/6-^(^/8+^/3)=. (4) x-^{a—^Jx)=., (6) 2^/5-3^/2)+(2^/2+3^/5) (6) 3^(V5+V3)=. Examples for Exercise. (7) 2+(V3-V2): a (8) Vx-{-yy 58. To extract the square root of a binomial, one of whose terms is rational, and the other a quadratic surd. Put «+ s/b=(x-{-yY, and a—^b={x—yf, then a=x^+y^..,[l] Also, (a+N/6) [a-^b], a^-b^ix'^-yj .-. ^(a;'-b)=x;'-y\..[2]. From [1] and [2], by addition and subtraction, // , /j\ / a+v/(a^-5) , / g-V(a2-5) . , x-y=s/{a-s/b)=^ 2 V 2 • • • L*J Consequently, whenever a^—b happens to be a square, the complex surd ^/{a^sjb) may be expressed by the sum or difference of two simple surds, but not under other circumstances. 59. When in any particular example the above-mentioned condition has place, these general formulae will enable us at once to write down the desired simplified expression for the square root of the binomial. But it is nearly as easy to go through the process by which the general results have been obtained, thus : — Eequired the square root of 16- V 87. Here the necessary condition is fulfilled, for 16^-87=169=132. Put then 16- V87=(^-2/r...[lJ, and 16+ V87=(a:+2/)-...[2]. 44 BINOMIAL SUEDS. Multiplying these together, 266-87=(a;^— y^ .-. n=a!^-if and adding and subtracting 16=a?^+2/'^ 29 1 3 1 .-. 29=2^;^ and 3=2/- .'. a!=^-=-s/6S, and tj=^-=-^/Q. .•.a;-2/=^/(16-^/87)=|(^/58-^/6). Examples fob Exercise. (1) ^/(2+^/3)=. (2) s/(8+v'39)=. (3) >,/(10-V96)=. (4) v^(7-2V10)=. (5) V(8+2v^7)=. (6) V(42+3V174?)=. (7) V(6+n/20)=. 60. The following are interesting properties of quadratic surds. 1 . The product of two dissimilar quadratic surds cannot be rational. If possible let »ya+ \/b= a rational quantity =p, then ab=p^. .-. 6=^=4a, and .-. ^hzJ-^a. a a" a 60 that nja and t^h are really similar surds, having the same surd-factor y/a. 2. A quadratic surd can never be equal to the sum or difference of a quadratic surd and a rational quantity. If possible, let ,^a=b±.^c.'.a=h^-±^h^c-\-c,'. ±.^c= — -7 — , so that the surd ^/c is equal to a rational quantity, which is absurd. 3. In every equation of the form a+s/b=a;+^y, if >^b and Vy are surds, then necessarily, a=a;, and .•.b=y. For since a-\->^b=x+ ^y .'.s/b=x—a-^ ,^y .• .x—a=0, or x=a, otherwise ^b would be = a rational quantity and a surd; .'. also b=y. If a-^s/b=0, then, necessarily, a=0, and 6=0. 4. If >^(a-^^b)=x-{-^, [x, y being one or both quadratic surds) then also must ^(as/b)=x^y, provided that ^6 be a surd. For since a-\-^b=zx''^-\-^xy-{-y', we must have, equating the rational and surd parts, a=x'^-\-y\ ^b = '^xy .'. a—^b=x'—'ilxy-\-y^={x—yy .'. ^(a--y/b) =x-^y. Note. — Any two quantities may always be expressed by the sum and difference of two others ; for, assuming this sum and difference to be x-\-y and x—y, we can always arrive at determinate values for x and y. The above, therefore, is only a particular case of a perfectly general theorem. It is from the proof of the property in this single case that the expressions [3], [4] at art. (58) are deduced in all the books ; so that they are limited to the condition of ^/b being a surd. But they are entirely unrestricted, being absolute identities ; and the investigation of them, given at (58), is quite independent of the condition that »yb is a surd. Introducing this condition, however, the process for getting s/{ft±.y/b} may be conducted a little differently; thus, taking the example worked at (59) we may proceed thus : — Pat V(16+ ^/87)=^/a;4-v'y, one of these, at least, being a surd, then 16+^87= a;+y+2*y^2/ .-. a;+y=16, and 1sjxy=.^^l .'. 4a^=87 29 3 .-. {x+y)^-ixy=16^-87={x-yf .-. x-y=lB : hence, x=—, y=-. A A .'. >/(16-V87)=yf -y/2=i(>/58~V6). IMAGINARY QUANTITIES. 45 The objection to investigating the formulas [3], [4] upon this plan may he removed by referring, not to the principle 4, above, to prove that ±:^'s/xi/=±jh, but to the fact of the double sign of the radical, whether ^fe be a surd or not. 61. Imaginary Quantities. — All the preceding rules of opera- tion equally apply, whether the surds be real or imaginary ; but in dealing with the latter, it will be convenient to replace the imaginary ^ — 1 by the symbol i ; that is, to put — l=i^ in order to avoid all confusion as to signs. Thus, — (1) (4+^/-3)(3+^/-5)=(4+^/3^/-l)(3+^/5^/-l)=(4+V3)(3^-^V5)= 12+3V3+4V5+*V15=12-V154-(3v'3+4V5)n/-1. (2) (2+^/-2)(3-V-4)=(2+V2)(3-2t)=6+3V2-4i-2iV2= 6+2^/2+(3^/2-4)^/-l. i->+GV2+^V3)V-l. 4+V-2 _ 4+V2 (4+V2)(2-V3) _ ^' 2+^/-3~2+iV3"" 4-3i2 (8+V6+ 2(^2-2^/3)^-1)^7 (5) (a+ 60 (c+ di)—{ac--lcl) + {ad+hc)i. a-\-U {a-\-hi){e—di) _ ac-\- hd be— ad. Note. — If expressions of the form a-\-b^ — l be united together by the additive or subtractive signs, or by both, it is plain that the result of the combination will always be of the same form, namely, A+B^ — }. And from the last two examples, it appears that when such expressions are combined by multiplication or division, or by both, the results are still of the same form. Examples for Exercise. (1) (2_^_3)V-7=. (2) (2_V-5)(3-f V-2)=. <^) (7+^>/-i)h-(1+2V-1)=. (4) (2+ 3V-2)(3-2V-l)=. (5) ' N/-S+V-6 (6) ^+^-1 3^/-2-2V-3 (7) (3-2^-4)3=. (8) V(a±V-l)=-* 6*2. We shall conclude this subject by giving an interesting example of the use of imaginary quantities in investigating certain properties of real quantities. [As above, i is put for ^ — 1, and .-. i^= — 1.] To prove that the sum of two squares multiplied by the sum of two squares, always gives the sum of two squares for the product. Let the factors be a^+P, and a'^ + i'"^. Each of these may be produced from a pair of imaginary factors; for they arise from {a + bi)(a—bi), and (a'-f6'f )(«'—&'«), so that the product of the original factors is the pro- duct of the four factors a-\-hi, a—bi, a'-\-bH, a'—b'i. Now, by actually multiplying the first and third of these, and then the second and fourth, • The result will be found to be of the same form : hence the square root of that result will be of the same form, and so on. (See foot-note, page 41.) 46 TO SOLVE A QUADEATTC WITH ONE UNKNOWN. we have the following pair, namely, {aa' —bb^)-{-{ab^ + bay, {aa'—bb^^ {ab' -\-ba')i, of which the product is [aa'—bby-\-(ab^-{-ba'f; .'. (a^ -{-h'){a'''^ -\-b'~)={aa' — bb'f-\-{aV-\-ba'f, which proves the theorem. For example: let a=5, 6 = 2, a'=3, 6'=4 : then (5- + ?--) (3^4-4-) = (5.3_2. 4)-+(5.4 + 2. 3)- = 7- + 26-=725. Let now a and b be inter- changed, that is, put a=2, & = 5; the first side remains unaltered, but the second side is =(2. 3-5. 4f +(2. 4 + 5. 3)'^=14- + 23-=725. If a' and b' had been interchanged, we should have got the same results. There are thus two ways in which the product of the sum of two squares may be expressed by the sum of two squares : in the instance here ad- duced,(5'- + 22)(3H4-)=7H26'=142 + 23'.* 63. Quadratic Equations. — A quadratic is an equation into which the square of the unknown quantity enters, and enters in such a way as to be removable only by the application of a special rule for the pur- pose. In simple equations, as has already been seen, the square of the unknown sometimes occurs; but then its elimination could always be effected by transposition, division, or some other of the ordinary opera- tions performed upon simple equations. The general form of a quadratic equation with one unknown quantity is this, namely, ax^- ■\-bx^=c ; if the second term be absent, the form is ax~=^c, which is called a pure quad- ratic, the more general one, ax~ -^bx=^c, being an adjected quadratic. A pure quadratic requires no special rule for its solution; for from ax'=^c, we at once get x -Vr 64. The rule for solving an adfected quadratic is deduced from the fol- lowing considerations. We already know that x''-{-2ax-\-a^={x + ay, and that x~—^aa)-\-a^={a)—af whatever be represented by a; and we see that the third term, in the first member of each of these identities, is nothing but the square of half the coefficient of the second term. Con- sequently every expression consisting of two terms of the form x^'+mx, or x"^ — mx, can always be made a complete square by merely adding (I) for a third term. Thus the expressions x+Qx, x'^—8x, x^' + Sxy ar—^x, &c., become in this way converted into the complete squares x'''-\- 6a; + 9=(:» + 3)'; a?--8a; + 16=(a;— 4f ; a;H3^ + (ff=(^ + |)-; x^'—hx^ (^ff={x—^^~, &c. The root of each square consisting of two terms, namely, the root of the first term [or), and half the coeff. of the second term taken with its proper sign. Hence the following rule. 65. To solve a Quadratic with one unknown.— Eule. I. Bring the unknown terms to one side of the equation, and the known terms to the other. 2. If the unknown square have a coefficient, other than unity, divide by that coefficient. 3. Add the square of half the coefficient of the simple unknown to each side of the equation ; the unknown side will then be a complete square. 4. Take now the root of each side, affixing the double sign ±: to that The property in the text may be more briefly proved as follows : — QUADRATIC EQUATIONS WITH ONE UNKNOWN. 47 of the known side (50), and the quadratic will thus be reduced to a simple equation. for a;": in the above example w=4. By this change [A] becomes y^-\-ay=.h , . . [B] (5) (a;-12)(;c4-2)=0. By actually multiplying, the equation is the quadratic a;2_10a;-24=0, which may be solved as above. But the factors of the first member being given, the equation is satisfied by equating either factor to 0; thus, from ic— 12=0, we get x==.12, and from a;+2=0, we get a;=— 2, which are the two values of x. Whenever we know the factors of an expression, if that ex- pression be =0, the equation may always be solved by equating each factor to 0, since the whole can become only by one or other of the factors becoming 0. (6) a;-fV(5:K+10)=8. Trans., V(5;r-fl0)=8— ar. Squaring, bx-^lQ=Qi—lQx-\-x^. Trans,, x^—21x=.—^L Completing the square, o «. /21V «r. . 441 225 Extracting the root, 21 15 21+15 ,„ X — = ±-r- .*. X=. =18, (1) Given 3«2=12a;+15 to find the values of x. Trans, and dividing by 3, a;^— 4:c=:5. Completing the square, by adding 2^ to each side, a^— 4a;+4=9. Extracting the root (that is, writing for the first side, a;— 2), we have ic— 2=±3 .-. a;=2+3, ora;=2-3, that is, a:=5, or —1. If we substitute 5 for x in the given equation, there results 75=60+15 ; and if we substitute —1 for x, there results 3=— 12+15; the equation being true for either value of x. (2) 2a2+4a;-3=ll. Trans., 2x^-\-ix=li. -i.2, x^-\-2x=7. Completing the square, x^-{-2x-\-l=8. Extracting the root, a;+l=V8=2V2. .•.ar= -1+2^^2, or -1-2^/2. (3) 3a;2-14a;+15=0. Trans., 3a:2— 14a;=— 15. 14 ^3, ^'-y^=- ■5. Completing the square. x^ 14 /7N.2 ■5= Extracting the root, 7+2 „ 2 ^-3=±3- (4) X s/x_B 3 2~""2' Multiply by 6, the L.C.M. of den., 2x—S^x=9. Put y for >^x, .'. 2f—Zy=z9. . 2_? _? • • y 2^~2* Completing the square, 9 9 9 "2^"^ 16 2^16 16 Extracting the root, 3 9 3 9 „ 3 y--=±-.'.y=.-±-=Z,or--. .-. a;=y'=9, or -. Note. — Every equation of the form x^"-\-ax"=.b [A] may be solved as a quadratic by putting y 2 -2 X2 <^) l(. 2 -2 or 3. -^y-^- Put y for -, then the equation is y(2/'-i)=2(3'-i). Dividing by y— 1, it becomes y(y+i)=o» or " Hy=2- Completing the square, 1113 Extracting the root. ]4^3 4 2' .'. x=2y=-l±^S. (8) ar-4-9ar-2=-20. Tut xr-^=y .:y'^-9y=-20. Completing the square, /9\2 81 1 48 QUADRATIC EQUATIONS WITH ONE UNKNOWN. Extracting the root, that is, —5=5, or 4 .*. a^=r, or -, a? 6 4 .•.«:=±v/5,or±- There are thus four values for x, either of which will satisfy the proposed equa- tion, which is in fact of the fourth degree. /gv CT+a; _ a— x s/a-\-^{a—x) >/a — v^(a— a;) ^ a^\■x _ ^ya-\-s/{(l—x) a—x s/a — sj{a—x)' Applying the principle at (38) a_ v/a . "^ _ ^ X i^{a—x)"x^ a—x' Clearing fractions, a^—a'^x=:ax^. Dividing by a and trans., x^-\-ax:=a^. Completing the square, x'^+ax+~=a^-\-j=-^. Extracting the root, iC-f , a —a±ai>J5 0TX=-{-l±^5). (10) x^-2x-\-6y/{aP-2x+5)=ll. Add 5 to each side, then (a5_2x+5)+6v'(^2_2;c-|.5)=16, or, putting 2p for x^—2x-]-5, ^^+6^=1 6. Completing the square, y'^-^67/-\-9=z25. Extracting the root, y+3=±5.-.y=2, or -8 .'.y'^=x^-2x-{-5=i . . . (1). OT f=x^-2x-^5=6i . . . (2). From (1), x^-2x=-l. Completing the square, a:^— 2j;+1=0. Extracting the root, a;— 1=0 .*. ic=l. From (2), a;2-2a;=59. Completing the square, a;2-2a; 4-1=60. Extracting the root, a;-l=±s/60=±2v'15 /. a;=l+2V15. (11) ax^+bx=c. -r a, x^-\—x=-. a a Completing the square. Extracting the root, b_ / 4ac-f62_^^(4ar+&2) 2a .'. x=- 4a2 2a •5±N/(4ac4-&") 2a This is a general formula for the solu- tion of every quadratic equation, a, b, c, being any values whatever. (12) a;-l =<^0- Subtract 1 from each side, then a:-2=l+-4- Add now - to each side, X ... a;_2+l=H-?-+i. X sjx X It is easy to see that each side of this equation is a complete square, the equa. being the same as .*, x=.sJx-\-2f or, putting y for pJx, and trans., f-y=2. Completing the square, Extracting, y-^±^--'y=-% or-l .'. x^y'^=:zi^ or 1. 66. The learner may, perhaps feel disposed to inquire : how is it that the double sign is prefixed to the square root of the known side of the equation, and not to that of the unknown ? The answer is, that it is because one side is unknown that no sign is prefixed : we look to the known side for the complete interpretation of the unknown : this inter- pretation furnishing sign as well as value : in the absence of it, we know nothing about the unknown side, and therefore suppress all prefix to it. SECOND METHOD OF SOLVING A QUADEATIC. 49 Examples for Exercise. (1) a;24-21a:=100. (2) a^-26a;+105=0. (3) 3a;2+ 52^=256. (4) 5a;2_i2a;+2=ll. (5) 16a;2-72a;+17=0. (6) a:+v'(10a;+6)=9. (7) x-\-^{5x-10)=8. (8) ^x+2=^{7+2x). x-{-l x—1 x—l~xT^' (10) x*-8x^=9. (11) x^-7x-\-^(x^-7x-\-lS)=i2i. (12) 3a;2-9a:-4v'(a:=^-3:c+5) + ll=0. (13) i^x+^y-5fx-\-?^=:H. (9) n:i_tz±=i. 3a; fX+1 (15) -v/^-2Va;-a;=0. (17) V(^+21)+N/(:r+21)=12. (19) ^/a;'^+v'a;3=6V'a?. (20) a;*-2a^+a;=6. (21) ^+i+;,+^=4. (22) ^2+i+4(.:+i)=-3. 67. Second Method of Solving a Quadratic-— When an equation, by the necessary preliminary operations, is reduced to a quad- ratic of the common form, the preceding method of solving it, by com- pleting the square, frequently introduces fractions, though none occur in the equation itself. The method now to be explained is free from this objection, and is, therefore, well deserving of the student's attention. If each side of the general quadratic ax'^-{-bx=c, be multiplied by 4^^, the result may evidently be written thus: — (2axf-\-2b{^aa;)=iac, the first member of which becomes a complete square, without fractions, by adding to it fc^ for the third term: we thus have {^axf + ^b{2aa;)+b'^ 4ac + 6^ or, (2aa?+fe)2=4ac^-&^ .'.^ax-\-h=^(4rac + b^). And in this way may any quadratic be reduced to a simple equation at a single step, and at the same time the introduction of fractions be avoided. The formula expressed in words is as follws: — Rule II. 1. Double the coefficient of x^ in the proposed quadratic: this will give the coefficient of a? in the reduced simple equation. 2. To the first term of the simple equation thus found, connect, with its own sign, the coef. of x in the quadratic : the first side of the simple equation will then be complete. 3. For the second side, multiply the second side of the quadratic (namely c) by 4 times the coef. of a;^ add the square of the next coef. to the result, and prefix to the two terms thus found the radical sign. Note.— If the first and second coefficients, or the second and third {b and c), be both divisible by 2 or by 4, the division had better be per- formed ; for though a fraction may thus be introduced into the other term, it will disappear from the resulting simple equation. This rule merely describes how from and <, the former of which placed between two quantities implies that the first is greater than the second, the latter, that the first is less than the second. If b'^=4:ac, the roots are real and equal, for then x= \-0, or , and . ' ^ ' 2a-~ ' 2a' ^ 2a 6^>4ac, ,, ,, real and unequal, for then b^—iac is positive. [This condition is always fulfilled when c is negative.] J-<4ac, ,, ,, imaginary, for then b^—4ac is negative, consequently, the character of the roots of a quadratic may always be ascertained without solving the equation. Thus, taking the three 1 7 equationsjust solved, namely, 6x^ + 5j?— 1=0, -;??-— 3;r—-=0, bx^—lx-h 3=0, we see that the first two necessarily fulfil the second condition, c being negative, but in the third 7- < 4.5.3 .-. the roots are imaginary. b c The sum of the two values of a; in (A) is — , and their product is - : ^ a ^ a we infer, therefore, that when any quadratic is divided by the leading co- b c eflBcient, and is thus in the form aP-\ — ar-j — =:0, that the coef. of the a a second term with its sign changed, is always equal to the sum of the roots, and that the third term, without change of sign, is always equal to the pro- duct of the roots.* We may, therefore, readily frame a quadratic that shall have an assigned pair of roots: thus, let the assigned roots be 3 and 5, then the quadratic to which these roots belong will be a?'-— 8d; + 15=0. Let the roots be —3 and 5, then the quadratic is ic-— Qar— 15 = 0. Let them be — 3 and —5, then the equation is a;--f 8;c-}-15=:0. Generally, let the roots be r, /, then the quadratic to which they belong is o)' — (r+/) a;-\- rr'=0, and as the first member of this is the product {x—r){a!—r'), it b c follows that every quadratic expression of the form x'^-\--a;-\ — , cra?- + ma}-\-n, is compounded of two simple factors, found thus. Equate the expression to : find the roots r, / of the equation, then the simple factors are {x — r), (a;—/) .* . ax"' -\-bx-\-c=a{x—r){x—r^). Note.— From the formation of a square, it is obvious that (x—aY may always be converted into (aj+a)^ by adding 4ax ; and that {x+ay will * In speaking of the roots of an equation, it is always understood that the leading coefficient of the equation, when all the terms are arranged on one side, is positive. GENERAL THEORY OF QUADRATIC EQUATIONS. 51 become (a?— rt)- by subtracting Aax; and from this fact we may easily solve an interesting problem, namely: — To find two integral squares whose sum shall be an integral square. The solution is ^<^)H^y where x is any odd integer whatever. When x is odd, x^ must be odd, and, therefore, a;'-±l must be divisible by 2. Suppose x=l, then we have 72 + 242=252. Suppose x=9, then 9^+402=412. The root of the third square always exceeds that of the second by unity, and the sum of these roots is always the first square. 69. Questions producing Quadratic Equations with one unknown Quantity. (1) A company of travellers engaged a conveyance for SI. 15s.; but at the end of the journey two found themselves without money ; in conse- quence of which each of the others had 10s. more to pay : how many 175 persons were there in company? Let x be the number, then 175 shillings is the fair share of each, and is what those who contributed x—^ actually paid : hence, by the question, 175 175 r= +10 .-. 175x=175a;-350+10a?J-20a: /. 10rc2-20a;=350 .*. x'^-2x=Z5. Completing the square, a?'— Sa; + 1=36 .-.a;— 1 = ±6 .-. a:=7, or —5. Consequently there were seven persons. The solution a;=— 5 is, of course, inadmissible : either value of x satisfies the equation, but as the question is such as to restrict the answer to a positive number, the nega- tive solution must, of course, be rejected. And similar remarks apply to the values of x in the next question. (2) Find a number, such, that if 3 times its square root be taken from 4 5 times its fourth root, the remainder may be -. o 4 Let X be the number, then 5\/x—B»yx=-, or putting y for /^x, o l57/-9f=i .'. 9f-15y=-L .-. Rule II. Page 49, 18y-15=V(-144+225)=V81=±9 15±9 4 1 , 256 1 ,^, ,. o,-, 1 ••• ^=-18-=3' '' 3 ••• 2^ -^=-8r' °' 81' *^"' ''' ^^^' ''' 81- (3) A and B leave the same place, at the same time, to travel a dis- tance of 150 miles. A, by travelling 3 miles an hour faster than J5, completes the distance 8h. 20m. before B ; at what rate per hour did A and B travel ? Suppose B goes x miles an hour, then A goes a;+3 : the time occupied by A is, therefore, — r: hours, and by B, — hours ; and by the question, 150 . „, 150 , . 150 . 25 150 . .. 6 1_6 __+8-_ that xs, ^3+y=-> or +25, ^+3=", clearing fractions, 18:c+a;2+3^=18a:+64 .*. x^-]-3x=5i. .: Eule II., 2a;+3=>/(216+9)=v^225=±15 .-. x= ~' ^ =6, or-9. Hence B goes 6 miles an hour, and A goes x-^3=9 miles an hour. £ 2 52 HOMOGENEOUS QUADRATICS. (1) Two travellers, A and B, start from the same place, at the same time, on a journey of 90 miles. A rides one mile an hour more than B, and arrives at his destination an hour before him : at what rate per hour did each travel? (2) A wine-merchant sold 7 dozen of sherry and 12 dozen of claret for 50Z. : he sells 3 dozen more of sherry for 10^. than of claret for 61. Required the price per dozen of each. (3) Divide 20 into two parts, such that the product of the whole and one of the parts may be equal to the square of the other part. (4) Is it possible to divide 1 4 into two parts, such that their product may be 50? (5) In a certain right-angled triangle the difference between the base and hypotenuse is 8 inches, and the difference between the perpendicular and hypotenuse is 4 inches : required the three sides. (6) Divide 1 1 into two parts, so that the sum of the cubes of those parts may be 407. (7) Twenty work-people, men and women, receive 2?. 8s. for a day's work: the men 24s. and the women, 24s. ; but the men receive Is. each more than the women : how many men were there ? (8) A and B are employed to dig a trench : A alone digs half, and leaves the other half to B ; the work is finished in 25 hours. They are afterwards employed to dig a similar trench, when, both working together, it is dug in 1 2 hours : in how many hours could each alone dig such a trench ? (9) What two numbers are those whose sum is 39, and the sum of their cubes 17199? [Additional questions, solvable by quadratics with one unknown, will be found at page 62.] 70. Simultaneous Equations. When one is a simple Equa- tion and the other a Quadratic. Rule. 1. Solve the simple equation for one of the unknown quantities. 2. Substitute the expression for this unknown in the quadratic, from which that unknown is thus eliminated, and then solve the equation for the other unknown. ^ ' I From the first, x= — - — . Substituting in the second, .-. 23^2+73^=39 .-. (Rule II. p. 49.) 43/-|-7=^/(312+49)=±19. ...2/==^=3, or -61 ....=1±?^=5, or -9 J. Hence the pairs of values are either x=5, y=S ; or else a:=:— 9j, y= — 6^. If both the equations are quadratics, then the elimination of one of the unknowns will frequently lead to an equation of higher degree than the second, the solution of which would require more advanced principles. But there are two classes of simultaneous quadratics, which may always be successfully treated by the rules already established. They are respec- tively homogeneous quadratics and symmetrical quadratics. 71. Homogeneous Quadratics. — A pair of equations is said to be homogeneous when every unknown term is of the same dimensions, that is, when each term has the same number of simple unknown factors. For instance, the terms ix\ 3^2/', 6ary, lif, are each of three dimensions, S1MMETRTCAL QUADRATICS. ' 53 because each contains three simple unknown factor?, namely, xxx, xyy, ^^y^ yyV' ^^ homogeneous quadratics each term is of two dimensions. Rule. ]. For either of the unknowns put an unknown multiple of the other ; that is for x put zy, or for y put zx, then the square of this other will necessarily enter every unknown term. 2. Deduce an expression for this square from each equation ; equate the two expressions, from which all but z will be eliminated, and solve the resulting quadratic in z, and thence deduce the values of x and y. .-. 832+192=14 .-. (Rule II. p. 49) 62+19=v/(12.14+192)=+23. —194-23 2 4 1 1 - =-, or -7 .-. 0?=-—-—=% or — .-. a;=±3, or +-^^2, 6 3' 2-32+2^ ' 18 _7 and y=zzx=±.2, or 4-^\/2- The values of x and y, properly paired, are therefore as follow :- x=Z\ (x=—Z'\ 2^=2 r^ b=-2|- l._ 7 ,y=-^>/2 -=-^V2 .4s/2 72. And it may here be noticed that, as there are always two unknown factors in each term of the proposed equations, whatever pair of values is discovered for them, there must always be a second pair arising from simply changing the signs of the former pair. We have spoken above of the values of x and y being properly paired. The student must be care- ful to observe consistency in this respect. Whatever value of z gives x, that same value of z must be employed to get the corresponding y, as above. 73. The above method may always be employed with success in the solution of a pair of homogeneous quadratics, but it is not always the shortest method. Rules, perfectly general in themselves, may often be advantageously departed from in particular instances. The exercise of a little independent thought and ingenuity will often suggest facihtating expedients which cannot be embodied in formal rules. These should in general be resorted to only when the penetration and skill of the alge- braist fail to discover to him easier processes. 74. Symmetrical Quadratics.— An equation is said to be sym- metrical when we may change the places of the unknowns without altering the equation or interfering with the condition expressed by it. Thus, the following are symmetrical equations, namely, x+y=a, ax^—hxy-\-ay'^=c, axy—x—y=h, &c. ; for, although we change every x into y, and every y into X, the equation itself remains substantially unchanged. The first and second of these equations are homogeneous as well as symmetrical. The rule for solving a pair of symmetrical quadratics— which rule, indeed, will often succeed for symmetrical equations of higher degrees — is as follows : — Rule. Substitute for x and y the sum and difference of two other unknowns ; that is, put it + v for x, and u—v for y, in each equation : the result will be a pair of equations with the new unknowns u and v, which pair will always present themselves in a solvable form. 51 MISCELLANEOUS QUADEATTCS. (3) a?-\-y^-x-y=U xy-\-x-\-y=\^. Putting u-\-v=.x, and u—v:=y, and remembering that {u + v) ^-\- {u — vY=^ 2{u^+ir), and that {u-{-v){u—v)=u^—iPj we have u^+i>^-u = 9 .... [1] w2_^_l_2w=19 .... [2] 2u^-\-u=28 EulelL, 4«+l=V225=±15 .*. M=r, or —4. [1], v2=9+w-w2=7, or -11 .•.r=:+l or+V-ll 7±1 ,'. x=u-{-v=—~-, or — 4±>/— 11 .:x=i, 3, or _4±^/-ll y=u—v=-—-, or — 4+)^^— 11 .-. y=3, 4, or -4+ V-11- The values of y need not have been worked out ; because of the symmetry of the equations, if x=:a, y=b, be one pair of values, then x=b, y=a^ must be another pair ; hence from x=a, a;^6, &c. we deduce y=b, y=^a, &c. (4) o^+y^+x-\-y=18 xy=6 Putting u-\-v=:x, and u—v=^yy w2+i-2+M= 9 .... [1] w2_-y2 _- 6 _ |-2] ;•. Rule II., 4it+l=N/121=±ll 5 .'. w=-, or -3. .-. [2], r2=ti2_6 1 or 3. .•.■»=±2' or±V3 .*. X=:u-\-V=. 5±1 or -3±^/3, 2 that is, a;=2, 3, or — 3±,y/3, and therefore^ y=B, 2, or — 3+/y/3. In the following miscellaneous examples we shall show how both these last may be worked without Eules. 75. Miscellaneous Examples in Simultaneous Quadratics, dc (1) aP+y^-x-y=lS [1] xy+x-]-y=l9 .... [2] 1. Add [1] to twice [2], then a>^-\-2xy-\-y^-^x-\-y=i56 that is, (ar-j-y)2-j-(a;-j-y)=56 .-.Rule II., 2(a:+y)+l=^/225=±15 -1+15 '. x+y= — - — =7, [A] 2. Subtract twice [2] from [1], then a^—2xy+y'^=:Z{x-{-y)—20=l, or —44. .\x-y=±l, or ±2^-11 • • • [B] Adding and subtracting [A], [B], 2;c=7±l, or -8±2^/-ll 2j/=7=Fl, or _8=F2v/-ll .-. xz=i, 3, or -4±^/-ll y=d, 4, or _4=FV-11 (2) x^-]ry^-\-x-^y=18 .... [1] xy= 6 . . . . [2] Adding twice [2] to [1], {x-\-yy+{x+y)=ZO ... [A] Subtracting twice [2] from [1] (x-2/)2+(^+y)=6 . . . [B] From [A], x+y= ~ ~ =5, or -6 .•. From [B], {x-y)^=l, or 12 .-. x-y=i±\, or ±2^/3 /. 2a;=5±l, or -6+2^3 2y=5+l, or -6+2^^3 .'.x=d, 2, or — 3±>/3 2^=2, 3, or -3=FV3 (3) ^_2?+i=o. ^ ' y X a;— 2/-l=0. Trans, and clearing fractions, x^—f=—xy .... [1] a;-y=l .... [2] Dividing [1] by [2], (See ex. 4, p. 15). x'^-^xy-\ry'^z=z—xy .'. x^-\-2xy-\-y'^=0 .-. x-{-y=0 ... [3] Adding and subtracting [2], [3], 1 1 ^=2' 2'=-2- Or thus, x=y-\-l, from 2nd equa. .-. from 1st, (y-\.lf-y^+y2^y=0 that is, 32/2+3y+l-f-y2+y=o that is, 42/2+ 42/+ 1=0 or, (22/+ 1)2=0 .•.22/+l=0.-.2/=-i.-.a:=- 2 (4) x-y=lj a;3-2/3=:19. MISCELLANEOUS QUADRATICS. 55 Cubing the first (p. 15), a?—'i^—Zxy{x—y)=.l that is, a?—y^—Zxy=l. Subtracting this from the second, 3^:3^=18 .-. 4xy=24, Also from 1st, oi?—2xy-\-y^=z 1 .'. a;+y=±5, and since a?— y=l, .-. a;=3, or —2 ; y=2, or —3. (5) ar»-|-3/3_i89, a:'y+a;y2=l80. Adding 3 times second to first, x^+f-\-Zxy{x-iry)=n^. that is (p. 15), (a:+2/)'=729 /. ;r+y=9 .-. (a:+y)2=81 .... [1] But since {x-{-y)xy=:lS() .'. a:y=20 .-. ixy=80 .... [2] Subt. [2] from [1], {x-y^-l . . [3] .-. [1], [3], x-^y=9, x-y=±l .*. x=5j or 4 ; y=4, or 5. (6) 2x^-7 f=-l, xy=Q. Put a;=«y .-. SvY- 7y^= - 1, ry2=6 "^ 7-3i^= v' .-. 36i;+l=V3025=±55 .-. ?>=■ " '.^=^,0T 36 .-...,= ^-=±2, or ±v/-y. .-. x=vy=±2, or if — ^_y- (7) a;+y=a, a;6+y>=&. Put a:=w+v,y=w— V, then w=-. 2 a*=w'>+5M*i;+10ttV+10wV+5«?;4-f^* .'. 10%v*+20wV=J-2%5 lUw .-.«*+ 2mV+%4: 5-}-8u5 lOw ••^=-"'=tv/-10^ But M=-, a , a , / f a^ . /46+a5) (8) Find a pair of values for x and y that will satisfy the equations iC3/4+y=4 From the first, xY='^^-y^. (4— y)2 From the second, x^y^=. r— .-. 3/6-103/H3/2-82/+16=0, or, (/_2/4-3/')-8(/+3^-2)=0 which may be put in the form (3/'-#-8{(/-l) + (y-l)}=0 or, f{f-\Y-8{{jy^-l) + {y-\)}=Q. This is evidently divisible by y—l; that is, y— 1 is one of the factors of the ex- pression on the left; hence that expression becomes for y— 1=0; in other words, y=l satisfies the equation, Solutions may often be obtained by thus decomposing the expression equated to into parts, each of which are seen to be divisible by a known factor. Adding 2 to each side of the first, \y^x/ Ay^x) 50' ^ 3 203 ^-r=-5o- • Equations of this form are called reciprocal equations, because the roots, -, are the reciprocals of each other ; that is, if r be one root, then - must be another. If, as in ex. 10, the coeflficients of the terms, when all arranged on one side, are the same when read in order from right to left, as when read from left to right, the equation is always / 25 \ a reciprocal equation. Thus, the equation referred to is a;*— 2r'+^2— ^^^a;^— 2a;+ (25 \ ^~144y' ^^' ^^' 56 MISCELLANEOUS QUADRATICS. o ^_ //40^,9\ /1849_ 43 • •^'"2- V V-25"+iy=V Too— -10 29 7 ic 2/ 29 7 Hence, the first of the given equations is either x^ .f 87 . 103 641 «2"^a:2 20~^'50 100 y or -5+ [A], [B]. 103_21_ 6_ ic2~60 10~ 100 Subtracting 2 from each side of [A], (X y\2 441 ^ ic y 100 '2/ X *10' But^+2'=??. 3^ « 10 a5_6 .*. (Second equa.), 4a; 2 or-.- *=2^' or -y. 3^=2, or 4x— 5?/=-y 5^=102/— 5y=l0 8 5y=10 60 5 ^ 2 20 ...3,=-_...x=-2/=5, orx=-y=--. And proceeding in a similar way, an- other pair of values may be got from the equation [B]. (10) 12(ic-l)3^=5« y^-x^=.\. From the second, 2'='s/(*^+l)- Substituting in the first, 12(x-l)V(x2+l)=5x. Note. — When dealing with a Squaring, 144(a;-l)V+l)=25a;2, that is, multiplying out, 144(x4-2a;3+2x2-2x+l)=25x2. Dividing by 144x2^ we have Adding 2 to each side, / IV «/ 1\ 25 . Putting 2 for xH — , and completing the square, ---^-^^ 13 ■1=±12 1 25 25 1 '=12' '' -12 '•*+~=T7>» O'^ *+"=— T^ 12 12 25 Completing squares, , 25 , /25V 49 ^-12^+V24>>=2r2 ^12 ^242 242' 1 3 From the first of these, a;=l-, or -. ^ 12'a;- -=1|, or -\\. pair The other values are imaginary. of simultaneous equations, and having obtained a value of one of the unknowns, we substitute it in one of the given equations, and thence deduce values for the other unknown, it will not always happen that each of these values, taken in conjunction with that substituted, will satisfy the other given equation. As an easy exemplification of this, suppose we havea;— 2/ = l, and ar—y^=S ; then, by division, we get ;»+y=3, from which, by subtraction, we get y=l. Sup- pose, now, we substitute this value of y in the second equation, we then get a;=±2; but the values x=—2, y=l, fail to satisfy the first equation. 76. The student must bear in mind that the sign of a radical is ambiguous only when there is no overruling condition to control it. In ex. 10, for instance, worked out above, if we had substituted the values of a, namely, a:=l^, and a?=|, in the second of the given equations, we should have got y=^{x^ + l)z=+l^, or ±1|, but the first equation ren- ders it necessary that the minus sign be rejected in the first value of y, and the plus sign in the second value. It is proper, therefore, at least as regards real values, to verify our results by actual substitution in both equations, when such ambiguities occur. The last term, 1, is with propriety called a coefficient, since the term may be written W, or afi. * See foot note, p. 55. MISCELLANEOUS QUADRATICS. 67 (1) (x'^-\-xy=12 V (2) j 2x^-3x7/-\-y"=4: \aP-2xj/-^df=9. ^ ^ \ X7j=Q. (4) l^'-^=ro^^ ^^) \x-\-y=l (6) . 12. (8) i^-3'=2 (9) |F"^?=^^ ia;+2/=12. (10) f ^-2\/^y+y=N/^-N/y ^ ^1 y+-=5. n2X f(^H/)(^-2')=51 (11) (13) (a:+2/)'-27(a:-3/)=0 \{x ^—xy-\-y'^){x—y)=x+y. if-6i{x-y)= -\-f){x+y)=7Q. (U)i{x-\-yr-6i{x-y)=0 ^ ^l(x^ 77. Miscellaneous Questions requiring Simultaneous Quadratics. (1) Fiud three numbers, such that their sum may be 33, the sura of their squares 467, and that the difference of the first and second may exceed the difference of the second and third by 6. (2) Find two numbers, such that their sum, product, and the difference of their squares may all be equal. (3) What number is that which, if divided by the product of its two digits, the quotient is 2, and, if 27 be added to the number, the digits will be reversed ? (4) When 962 men were drawn up in two square columns, it was found that one column consisted of 18 ranks more than the other. What was the strength of each column ? (5) When 732 men were drawn up in column, the number in front and the number of ranks together made up 73. How many ranks were there ? (6) The fore- wheel of a carriage made 6 revolutions more than the hind- wheel in going 120 yards ; but if the circumference of each wheel were to be increased 1 yard it would make only 4 revolutions more than the hind-wheel in going that distance. Kequired, the circumference of each wheel. ^ (7) Find two numbers, such that the difference of their squares is equal to their product, and the sum of their squares equal to the diffe- rence of their cubes. (8) A person rows a distance of 20 miles and back in 10 hours, the stream flowing uniformly in the same direction, and he finds that he rows 3 miles with the stream in the same time that he rows 2 miles against it. Required, the time of going and returning, as also the velocity of the stream. (9) By adding m times the sum of two numbers to the sum of their squares we obtain the square of m, and by adding the product of the numbers to the sum of their squares we obtain the same square m^ : what are the numbers ? (10) The hypotenuse of a right-angled triangle is to be 13 feet, and 68 THEOBEMS IN PEOPORTION. the measure of its surface is to be 30 square feet, what must its sides be? [Note. — The surface of a triangle is found by taking half the product of its base and height] (11) Find the sum of the squares of the roots of the equation ar'-hpx-\-q=0, without solving it. (12) Find the sum of the cubes of the same roots. (13) How many sides must a polygon have in order that it may have n diagonals ? (14) The men in each of two columns of troops A, B, were formed into squares : in the two squares together there were 84 ranks. Upon changing ground A was drawn up with the front that B had, and B with the front that A had, when the number of ranks in the two columns was 91 ; what was the number of men in each column ? (15) Can two real numbers be foufid, such that their sum, product, and sum of their squares shall all be equal ? (16) The product of four consecutive positive whole numbers is 3024: find them. (17) A certain number of two figures is such that the left-hand figure is equal to 3 times the right-hand one, and if 12 be subtracted from the number, the remainder will be equal to the square of the left-hand figure : required the number. (18) Find two real numbers whose sum is a and product b, and show that such numbers cannot exist if 6 >(~K ) • (19) Find two numbers, such that their sum, product, and difference of their squares may all be equal. (20) There are three whole numbers, such that the squares of the first and second together with their product make 13, the squares of the second and third together with their product make 49, and the squares of the first and third together with their product make 31 : what are the numbers? 78. Proportion. — When one quantity is divided by another of the same kind, the quotient is also called the ratio of the two quantities : the dividend, or numerator of the fraction expressing the ratio, is called the antecedent of that ratio, and the divisor, or denominator of the fraction, the consequent. In expressing the ratio of a to b, which is no other than the quotient a-^b, the small line between the dots is suppressed, and the ratio is written a : b. Proportion is an equality of ratios ; thus if r = ;^. or which is the same thing, if a:b=c:d, the four quantities a, b, c, d are said to be in proportion ; but instead of the sign of equality, it is usual to write four dots, thus, a : b : : c : d, which is read as a is to b, so is c to d. From these explanations it is plain that any two equal fractions may be converted into a proportion, and any proportion may be replaced by two equal fractions. The following are the principal theorems respecting proportion. 79. Theorems in Proportion. — l. The product of the extremes is equal to that of the means, and conversely if two products are equal, the pair of factors from which one is produced will be the extremes, and the pair of factors of the other product, the means of a proportion. THEOREMS IN PROPORTION. 69 For a\l: '.c \d implies that -=- .■. ad=.lc. d » • •* » T .1 ad Ic a c ^. ^ . . , Again, if ad=hc, then -—=—-.•. -7=-, that is, a:o::c'.d. Id od d If the two means are equal, that is, if a : b : : b : d, then ad—b'^, that is, if three quantities, a, b, d are in continued proportion, the product of the extremes is equal to the square of the mean. And since also -=- .-. ( multiplying by - J, -=-=—, so that the first is to the third as the square of the first to the square of the second, or as the square of the second to that of the third. 2. If four quantities are in proportion, they are also in proportion when taken inversely, that is, the second is to the first as the fourth to the third. a c h d Let a lb: :e : d .'. ,=^ •'• -=- .'.b : a: : d :€, d a c 3. If the quantities be all of the same kind, they are also in proportion when taken alternately, that is, the first is to the third, as the second to the fourth. Let a : J: :c : d .'. -=- .*. ( mult, by - ), -=- .'. a : c: :6 : d. b d \ c/ c d Note. — As there cannot be any ratio between two quantities different in kind, alternation can be applied only to quantities all of the same kind. 4. If a : 6: : c :cZ then also a+mb : a-{-nh: :c-\-md : c-\-nd ^ a c a , c , , a , c , *** I" each of these For since i-=-j •'• T'T^^^j-r^) ^-nd -+71^-4-71. expressions, as well as in d b d b a ^jjg equation below, the .„ ,. .,. a-^mb C+md . , , ^ , , , . sign ,^ may be put for +. By dividing, r= ;.'• a-\-mb : a-\-nb: :e-\-md : c+nd. a-\-rd) c4-nd Note. — The above is only a particular case of a theorem much more a c general, for if two fractions are equal as - =-, then we may write any combinations of a, b, for a new num. and any other combinations for a new den., provided we only take care that each term shall be of the same dimensions as respects a and b, and if in the new fraction thus got we change the a and b for c and d, we shall get another fraction equal to it : 5a^-^ab-\-U^ _ 5c^-^7cd+Z^ * ^^' 6(a+i)^ "" 6{c-]-df For let VI be a quantity such that c=ma, then necessarily d=mb. Now, if in the second fraction these substitutions be made, and then num. and den. divided by m^ the first fraction will be the result. 5. If any number of quantities of the same kind are proportionals, then as one antecedent is to its consequent, so is the sum of all the ante- cedents to the sum of all the consequents. heta:b::c:d::e:f, &c. Put ?=3=i, &c. =»i .'• a=mb, c=^md, e=mf, &c. b d f &+^+/+ &c. '^ .\a:b: -.a+c+ei- &c, : b+d+f+ &c. .-. a+c+e+ &c. =m(5+c?+/+ &c.) .'. ^^^^^^ ^o ='^=b=d ^'' 60 VARIATION. 6. If several sets of four be proportionals, then the products of the corresponding antecedents and consequents will be proportional. mi. ..a c e g h m aeh &c. cam &c. Thus, if 7=-, 7=T, 7=-, &c., then — "^ h w&, then necessarily mc>wc? ma=:nb, ,, ,, 'mc=:7id ma\, the terms continually increase in magnitude. If the number of them be indefinitely extended, the sum, therefore, must at length exceed in magnitude all finite limits; but if r_Wff-ijTT-#_21_l ■ ^ -3 96 32 1.1 . „ . 1 (3) Required the sum of the infinite series 1+q+q+ • • • • ^^^^ °'^^' ''^3' ^~^ 1 3 (4) Find a geometrical mean between - and — . Here ^/{l^i2/6' *^® ^^^^' 12 1 _K 7—?—^ 4 (5) Insert three geom. means between - and -. Here »=2» '*— ^» q~~^ ' . yi^z . »2_? .. ^—j -=-J6 : hence the means are 9 3 ^3 3^ 1/211/2 1 /« 1 1 /fi 2V 3' 3' 3V 3' '' 6^^' 3' 9^^* * (6) To find the value of the decimal NFPP . . ., in which PPP . . . represents an infinite series of recurring figures, and N any number of figures preceding the recurring 64 HAEMONTC PROGRESSION. decimals. Let N consist of n figures, and P oi p figures, then the value of the pro- posed decimal will be N / P P P ^ \ Where it is plain that the terms within the brackets form an infinite geometric series, of which the common ratio is . The sum of this series is therefore 2=- * ^ ■* ^ 10 n+p ' \ ioV~ iV(lO^-l)+P . 10 (lO^-l) .*. 'NPPP . . . = — ^i , where 10 —1 is p nines. lO'^lO^-l) Hence the fraction, equivalent to the proposed decimal, has for num. iV times the number formed by^ nines, plus P; and for den., that same number of nines followed by n ciphers. In multiplying by nines, we of course merely annex so many zeros to the number multiplied, and then subtract that number. Ex. Eequired the fraction equal to •54123123.... Here iV=54, w=2, P=123, ^=3. Also 54 x 999=54000-54=53946. . .54123123 _ 53946+123 _54069^18023 99900 99900 33300* [This process may be conveniently described in words, as in Weales Rudimentary Arithmetic, p. 138.] Examples for Exercise. (1) Find the tenth term of 7, 10, 13, &c. (2) Find the sum of six terms of 5, 15, 45, &c. (3) The first term is 7, and the common ratio ^, find 35. (4) Find the sum of 2| — ^+^— .... to infinity. (5) Find the sum of the infinite series IH — | — ^+ .... (6) Insert three geom. means between 2 and 10|-. (7) Insert five geom. means between 2 and 1458. (8) Sum the infinite series 2— ^+f — .... 86. Harmomc Progression. — Quantities are said to be in harmonic progression when the reciprocals of them are in arithmetic pro- gression. The designation arises from the fact that musical strings of equal thickness and tension, in order to produce harmony, when sounded together, must have their lengths as the reciprocals of the arithmetical series 1, 2, 3, &c., that is, as ], |^, ^, &c. If three quantities are in harm. prog, the first is to the third as the difference of the first and second to the difference of the second and third. For if a, h, c, be in harm, prog., then -, -, -, are in arith. prog. a c X o .*. 7 = — 7 .'. mult, by dbc, ac—bc=ab—ac ,-. a : c: :a^h : h—c. bach J ^ And if a similar property hold with respect to four quantities a, b, c, d, that is, if a : d :: a—b : c—d, the four are said to be in harmonic pro- portion. HARMONIC PROGRESSION. 65 From the first property, b= is the har. mean between a and c, and (I ~f"C ah ^a-h is a third har. proportional to a and h. There is no general formula for the sum of any number of terras of an harmonic progression. But as, by taking the reciprocals of the terms, we change the harmonic progression into an arith. prog, many particulars respecting such series may be easily deduced : thus, — (1) Add two more terms to the harmonic series 1, |, f . The reciprocals are 1, |, |, an arith. prog., in which the com, diff. is — ^ .*. two additional terms are f, §, that is, ^, ^ : hence the required harmonicals are 2 and 3. (2) Insert three har. means between 2 and 3. Here we have to insert three arith, means between \ and ^. Since a=^, w=5, and 1=^ .*. ^=|+4cZ .*. c?=— ^^ .*. the -.1, 11 10 9 , arith. means are — -, — , — , and Atc Ai Jti. '. the har. means are 24 24 24 11' 10' 9 ' or 2^ 2f, 2| Examples for Exercise. (1) Insert two har. means between 3 and 12. (2) Insert three har. means between 4 and If. (3) The arith. mean between two numbers is 2, the har. mean l^ : find the numbers. (4) The sum of three numbers in har. prog, is 13, and the sum of their squares 61 : what are the numbers ? 87. Miscellaneous Questions in (1) British wine at 5s. per gal. is mixed with spirits at lis. per gal. in such proportion that the mixture at 9s. per gal. may pro- duce a profit of 35 per cent., what is the proportion ? Let the prop, of the wine to the spirits be X : y, then 5x-\-lly=: cost of x+y gal. and 9x-\- 9y= selling price, /. Ax— 2y= profit. Hence by the question, 5x-\-lly : ix-2y: :100 : 35 or 20 : 7. Multijjlying extremes and means, and equating the results (79), ••. S0x—4:0y=Z5x+*l7y .•. 45a;=117y .'. 5x=:ldy. Consequently (79) a; : y: :13 : 6, so that the mixture must be at the rate of 13 gal. of wine to 5 gah of spirits. (2) In a mixture of brandy and water, it was found that if there had been 6 gal. more of each there would have been 7 gal. of brandy to every 6 gal. of water; but if there had been 6 gal. less of each, there would have been 6 gal. of Proportion and Progressions. brandy to every 5 gal. of water, what was the original proportion? Let 7x — 6 represent the number of gal. of brandy, and 6x—6 the number of gal. of water, then the first condition will be at once fulfilled; and for the second we have 7a;-12: 6a;-12: :6 :5 .-.(p. 59), X : 6x-12: :1 : 5 .-. 6a:— 12=5a; .*. xz=12 .'. 7a;— 6=78 gal. of brandy Gx—6=66 gal. of water. (3) The product of three posi- tive whole numbers in geom. prog, is 64, and the sum of their cubes is 584, required the numbers. Let them be x, xy, a;/, then c(^y^=zQit and x^-\-xY+x'f=6U. , 64 ^ 4096 From the 1st, f=—^ .'. f=-^ ,'. by substitution in 2nd, ^3+64+i^'=584 ... a;8-620a;3=:-4096 .-. a.-3=8 .-. x=2 .: y=2 .'. the numbers are 2, 4, and 8. 66 QUESTIONS IN PROPORTION AND PROGRESSION. (4) Given the sum S, of n terms of an arith. prog., a-\-b-{-c-\-...-\-l, in which the common difference is S, to find the sum S.^, of n terms of the series of squares a'^-\-b'-rc'^ + ...-\-l^. Since h=a-\-^, c=b-\-^, d=c-)r^, . . . ., Z=i-j-S, we have c3=63+3&-^S+3iB2+§^ Adding all these n equations together, omitting the quantities common to both sides, we have Suppose the arith, prog, to be 1+2+3+...+^, where a=l, and 5=1, also l=n, then Si=:^n{n-\-l), and by substitution in S.^, we have 8^=- -^ ^ — , that is, o 2 o _ n^-\-SnP+S7i n(n-}-l) n_ n^-\-Zn^-\-2n n(n-\-l) ^"" 3 ~ 2 ~3~ 3 2 • But the numerator, n{n^-\-S7i-\-2)=n{n-^l){7i+2). [See art. 68.] So that _ 2n-l 2~ 6 ' S2=7i{n+1)^~ -J, and since ~ -= .-. S„ or lH22+32+...+^2=^Vtlp±l). o This formula will be found of useful application hereafter (p. 68). If, instead of the cubes, we take the fourth powers of b,c,d, &c., we may, in like manner, find S^ in terms of /Sg, S ^; and similarly, we may find S^, Sgy &c., each in terms of the sums of the inferior powers. {\) A starts from a certain place, and travels so that his second day's journey shall be twice the first, the third three times, and so on. B starts 4 days after to overtake him, and travels uniformly per day 9 times the distance A went the first day : when will B come up with A ? (2) A hundred stones are placed in a straight line, at intervals of 1 yard : how far must a person walk who shall bring the stones, one by one, to a basket, at the place of starting, 10 yards from the first stone? (3) If the first term of an arith. prog, be n^—n-{-l, and the common difference 2, prove that the sum of n terms will be n'\ (4) Show that the series of cubes V, 2■^ 3'^, 4"\ &c., are severally ob- tained by summing the following portions of the arithmetical series of odd numbers, namely 1 | 3, 5 | 7, 9, 11 | 13, 15, 17, 19 | &c. (5) A brigade of sappers ' carried on 15 yards of sap the first night, 13 the second, and so on, decreasing 2 yards each night, till 3 yards only were left for the last night. How many nights were they employed, and what was the whole length of the sap ? (6) A servant agrees to serve his master for 12 years on condition that he is paid a farthing for the first year, a penny for the second, fourpence for the third, and so on : what would be due to him at the end of the 12 years? PILING OF SHOT— TRIANGULAR PILE. 67 (7) What fraction is equivalent to the recurring decimal -27543543 . . . . ? (8) Two vessels A and B each contain a mixture of wine and water. In A the wine : water : : 1 : 3, but in B as 3 : 5 : how much must be taken from each vessel to make 14 gal. of mixture containing 5 gal. of wine and 9 gal. of water ? (9) If a-}-b'(X a—h, prove that a^A-b^ a ab. 10 If the sum of three numbers in arith. prog, be 9, and the sum of their cubes 153 ; what are the numbers? (11) Find three numbers in geom. prog, such that their sum may be 7, and the sum of their squares 21. (12) Find four numbers in geom. prog, such that the sum of the means may be 36, and the sum of the extremes 84. (13) What two numbers are those whose difference, sum, and product are as the numbers 2, 3, and 5 respectively ? (14) The three digits of a certain number are in arith. prog. : if the number be divided by the sum of the digits, the quotient will be 26 ; but if 1 98 be added to the number, the digits will occur in reverse order : what is the number ? (15) There are three infinite series, namely : — ^x=l+;+^+...., 5,=!-^^+^-...., ^3=1+1+^+.... Prove that S^xS^^^zS^. 88. Piling of Balls and Shells. — Cannon shot and shells are deposited in arsenals in piles of three different forms, called respectively triangular, square, and rectangular or oblong piles, according to the figure of the base of the pile. Each side of a pile presents an inclined face of rows one above another, each row diminishing by a single shot. If the pile be triangular, or square, it terminates in a single shot at top ; if it be rectangular, it terminates in a ridge, or row of shot. The three different forms of a pile are here represented. 4^ ^ Triangular Pile. Square Pile. Rectangular Pile. Every horizontal layer of shot is called a course. 89. Triangular Pile.— In this pile each layer or course of shot, till the top shot is reached, is an equilateral triangle : the side of the first of these triangles, proceeding from the top, is 2 . * . the number of shot in that course is 2 + 1 , or 3 ; the number in the next course, the side of which is 3, is 3 + 2 + 1, or 6 ; the number in the next, the side of which is 4, is 4 + 3 + 2 + 1, or 10; and so on .-. the number of shot in the nth course, counting from the top is 1+2 + 3 + +?i=in(n + l) (art. 82), so that if the nth be the bottom course, and we represent the sum of all the courses by S, we shall have — n(n-{-l) ,S'=l + 3+6+10+ .... + „ . (n—l)n , ot(?i+1) or, S= 1 + 3+6+10+ . . . +—5— + — n— F 2 68 SQUARE PILE — RECTANGULAR PILE. .-. adding, 25=1+22+32+42+ .... +%2^^:^^+l> that is (p. ee), 2.='^^!±1|!^) = "6 . . S= g .... [I.]. If the pile be reduced loj the removal of any of the top courses, the number of shot in the remaining incomplete pile is found by subtracting the pile removed from the original pile. 90. Square Pile. — As the courses here form the series of squares 1+22^32 + ... +,^2^ the number S of shot is ^^ K^+l)(2/i + l) |. ^^^^ and the number in an incomplete pile is found from subtracting the pile wanting from the complete pile. Rectang^ular Pile. — It is convenient to conceive a rectangular pile to be formed thus. To one of the slant faces of a square pile, each side of whose base is equal to the shorter side of the intended rectangular pile, imagine an equal face of shot to be applied (see third /ig. at p. 67) : we shall thus have a rectangular pile terminating in a ridge of two shot, and the base of the rectangle will have one shot more in its length than in its breadth ; and continuing to add faces of shot in this way, when we have added m faces, there must be m + 1 shot in the top ridge, and m shot more in the length of the base than in the breadth. Calling, therefore, the number of shot in the shorter side of the base n, the entire number in the rectangular pile will be — No. in square pile of base n^, +No. in the m faces added to it. The number of shot in the m faces will, of course, be m (1+2 + 3 + n(n + l) , , , . , .1 . n{n + l)(2?i + l) ...+»)=»i-^— — •, and the number in the square pile is ^ -, .-. S=-^ — ■ ] 3w + 2n + 1 /- . . . [III.], where n is the number of shot in the shorter side of the base, and m is the difference of the numbers in the longer and shorter sides, or it is the number of shot in the top ridge minus 1. If p represent the number of shot in the longer side of the base, then m=p^n; and making this substitution, 8 becomes ^^!!(!!±i)2^i:^L±i). . . [IV.]. As the number of shot in the top row is m + l=p— /i + l, if we call 1 1, 1 CI n(?i + l)(3( + 2.n — 1) ^-Tn this number t, we shall have 0= -^^ —^ ^'...LV.J. 91. Now, it is worthy of notice that the last factor in the num. of this expression being the same as i + 2(i + 7i— 1), and that p=t-^n — 1, the factor referred to is t-r^p, where t is the top row, and 2p the two equal base rows parallel to it. Moreover, as -^r — expresses the number of shot in the triangular or end face of the pile, if we call this number A , we may express 8 thus : 8= A — 5-^, which, being independent of n, o GENERAL RULE. 69 equally applies to a square pile, in which t—l. With slight modification, it also applies to a triangular pile, for in this, the last factor in the num. of the expression for S is the same as 1+n + l, which we may regard as expressing the top shot + the two opposite base rows, one of these rows being n, and the opposite one merely a single shot. Consequently, allow- ing this latitude to the word row, and representing the two opposite rows of the base, which are parallel to the top row, hyp, p\ we have in general, for each pile, 8= A ...[VI.], a formula which may be expressed o in words as follows : — 92. General Rule.— Multiply the number of shot in a triangular face of the pile by one-third of the number of shot in the three parallel edges of the pile : the product will be the number in the pile. In all the cases an incomplete pile is regarded as the difference of two complete piles. (1) How many shot are there in an incomplete triangular pile of 1 5 courses, 6 shot being in a side of the upper course ? As 6 shot are in the side of the upper course, 5 courses have been removed, so that the com- plete pile had 20 courses : hence in the formula [I.] putting first 20 for n^ and then 5, and subtracting, we have iS= — '- — '—^ '—^ =7(10.22—5)= 5.7(2.22— 1)=] 505. The operation would be just the same by using the formula [VI.], or the above rule; for t-{-p' is here=2, so that the number of shot in the three parallel edges is 22 for the complete pile, 20 21 5 6 and 7 for the pile removed ; also A is -^ — for the former, and -^ for the latter. (2) There are 18 courses in a rectangular pile, and 45 shot in the ridge : how many shot are there in the pile ? In the formula [III.], n = 18, and m=45 — 1=44 ... s^l8.19.{3.44+2.18 + l}^3^3^g3^gg33 ^ By the formula, or rule above, ;S'=^^.i^^!^^=3.19.169=9633 : for 2?=/=«+w— 1=62. Examples for Exercise. (1) Find the number of shot in a triangular pile, and also in a square pile, each composed of 12 courses. (2) How many shells are there in a rectangular pile of which one side of the base contains 16 shells and the other 7 ? (3) How many shot are there in a rectangular pile of 15 courses, 21 shot being in the top row ? (4) If the number of courses be 30, and the top row contain 31 shot, how many will there be in the pile ? (5) How many shot are there in an incomplete rectangular pile of 12 courses, the longer and shorter sides of the base containing 40 and 20 shot respectively ? 70 THE BINOMIAL THEOEEM. 93. The Binomial Theorem-— This theorem, which is one of the most important theorems in algebra, was first given in a general form by Newton: its object is to express any power or root of a binomial without actually performing the involution or extraction : in other words, its object is to express the development, as it is called, of (a + a?)", by a formula applicable to every value of n. To a#-rive at this general formula, we must first establish the following preliminary theorem, namely : 94. Preliminary Theorem. — If the series A-\-Bx+Ca!"^ + DaP -{•..., whether the number of terms be finite or infinite, be equal, as a whole, to the series A'-\-B'x-\-C'x~-\-D'x^ + ..., (the coefficients being all finite) whatever value be given to x, then the coefficients of the like powers of x must be equal : that is, A=A\ B=B\ C—C^, D=D\ &c. For the condition being that the series — each taken as a whole — are equal, whatever value be given to x, put for x the extreme value x=0 : then all the terms in each, after the first, must vanish, since the coefficients are finite; so that the two equal series are reduced to A=A'. Having thus ascertained that the first term in one series is equal to the first in the other, we have, by suppressing these terms, Bx-{-Cx'^-\- 'Dx^ + ...=B'x+C'x*'-^I)'x^-\-..., and therefore, dividing by a;, B + Cd;-^- Dx2+...=^'+C"i»+DV+...,which, when^=0,become8simply5=Z^'. Thus the first two coefficients in the one series are equal, each to each, to the first two in the other. And proceeding uniformly in this way, we find that A=A\ B^B\ C=C\ D=D\ &c. It follows from this, that if ^ + ^^-f C^^+ ...=0, or, which is the same thing, that ii A+Bx-\-Cx'^-\-.,.=Qi-\-^x-\-Qx'^-^..,Jox bX\ values of x, the coefficients A, B, C, &c., must each be 0.* 95. The binomial theorem, the proof of which the above will assist us in establishing, is this, namely : A A.6 X or, dividing each side by a*, and for brevity putting z for -, it is It will be sufficient to prove it in this more simple form ; for if this be true, the former development must be true, inasmuch as this is converted X into that by simply putting - for z (which latter symbol may be anything) and then multiplying each side by a". I. No Fractional or Negative Exponent can enter the Develop- ment OF (1 +2;)" WHATEVER BE Ti. Or, to be moro explicit, the development must always be of the form {\^zf=l + Pz-\-Qz'^ + Bz'-\-Sz^+ &c...[l.] 1. When n is a positive Integer. In this case, since the development of (1 -\-zY may be obtained by repeated multiplications of \-\-zhj itself, it is plain that the first term of this development will be 1, and that all the powers of z entering it must be positive integers. In this case, there- fore, [1] is obviously the form of the development. • It may be proved, too, that if A-\-Bx-{-Cx^-^...-\-Lx^:=iO be satisfied for n+1 values of Xj the coefficients must each be 0. THE BINOMIAL THEOREM. 71 2. When n is a positive fraction. Let - be the fraction : we know that p {l-^z)i indicates that after the power ;? of l+« has been found, the ^th root of that power is to be extracted. But the power ;? of l+;s has already been proved to be of the form [1], and if the ^th root of [1] be extracted, this root must be such as, by repeated multiplications by itself, to reproduce [1] : its leading terra must, therefore, be 1, and no fractional exponent of z can occur in the root, for this reason, namely, that whatever fractional exponent enters an expression whose leading term is 1, must also enter every integral power of that expression : the multiplier 1 obviously necessitates this. Hence, a fractional exponent of z cannot p. enter the development of (1 -\-z)'}. Neither can negative exponents enter; for if this be supposed possible, let — k be the greatest negative exponent of z in the development ; then, if every term containing z were to be divided by ;?~*, or, which is the same thing, if each exponent of z were increased by k, all the resulting exponents would be positive, and one of these exponents, namely, that which was originally ~k, would be : hence, calling these increased exponents, when arranged in order of mag- p nitude, 0, a, /?, y, &c., the form of the development of (l+z)i would be 1 '^z^\a-\-hz'^-^cz^-\-dz'^-\- &c.), a being the coef. of «~*, so that multiply- p ing by z^, we should have ( 1 + z^z^-=.z^ •\-a-\-hz'^-\-c7^ •\- dz^ + &c., whatever be z, since the two expressions are identical. But if ;^=0, the equation becomes 0=a, Hence, the coef. a, of the supposed negative power, is ; that is to say, such negative power cannot enter the development of p (I +5;) 9 ; and since the entrance of fractional powers has been shown to be also impossible, the development in this case too is of the form [1]. 3. When n, either integral or Jr actional, is negative. — Here we have to seek the form of the development of (1 +^2)"", or of its equal ■ .„ that is, we have to determine the form of the quotient arising from dividing 1 by an expression of the form [I]. The first term of this quotient must, of course, be 1 ; and since the first remainder cannot con- tain any fractional or negative power oi z, neither can the next term of the quotient, neither, therefore, can the second remainder, nor, in conse- quence, the next term of the quotient, and so on. Hence, generally, whether n be integral or fractional, positive or negative, the development of (l+^r)" will always be of the form [1]. II. Determination of the coefficient P in the general form [1]. 1. When n is a positive integer. — Suppose the first two terms of any positive integer power of l-|-;2 to be \-^mz, then the first two terms of the power a unit higher, got by multiplying again by 1 -f^, will be 1 + [m + \)z) so that by increasing the exponent of the power by a unit, the coefficient of z becomes increased by a unit. In other words, the differ- ence between the exponent and the resulting coef. of z is always the same ; but if the exp. be 2, we know that this difference is 0, for (1+^) = 1 4- 2^ + «^ hence the difference is always 0. Consequently, when n is 72 THE BINOMIAL THEOEEM. a positive integer, (1 -]-zf=l+nz-hQz''-\-Bz''-{-Sz*+ [2], in which, however, nothing at present is known about the coefiBcients Q, R, S, &c. 2. Whe7i n is a negative integer. — In this case (1 +;j;)-"=l-^(l+^)"= 1-^(1 -]-nz-{- ...), and actually performing the division, as far as the second term of the quotient, we have l-\-nz+ ...)1 (l—nz+ .... Hence, whether the integer n be positive or negative, the form of the develop- ment is [2]. 3. When n is a fraction, either positive or negative. — Let n=:g^ p being either positive or negative, and, for brevity, put z' for Pz-\-Qz^-\- &c. in [1], then {l + zy=l-^z' ,: {l+zY=(l-^zy .'.l-\-pz+ &c., =l+qz' + &c.,by [1] and [2] above, that is, replacing the value of ^r', l^pz-{- &c. = i'\-q{Fz-\-Qz'^-^ &c.) + &c. Now, this being an identical equation, is true, whatever be the value of z, therefore (94), p=qP .-. P=-=the proposed exponent. Hence, whatever be w, that n is always the coefficient of z in the development of (1 +zY : in other words, the development is always of the form [2]. If now we replace z by -, and multiply by a", we have [2], (a+a;)"=a"+na''-^^ + ^a'-V + l?a"-V+5'a''-V+ &c. ...[3] whatever be the value of w. III. Deteemination of the Coefficients Q, R, S, &c., in the Ge- NEEAL Forms [2], [3]. — In the identity [2], as z may be anything, put 5(1 -\-a:) for z ; we shall then have [1 + Z>(1 +a;)]"=l +n6(l +;z?)-f Q6-(l +a;f -\-Rb\\-\-a!f + Sb\l+a;y-{- &c. Now we know from [2] that the first two terms of the development of (l+;c)'" are l-^-mx; so that we know every first term and every second term furnished by (l+^)\ (l+a;)^ (l+a:)^ &c., in the right hand member of this last equation; and collect- ing all the first terms together, and then all the second terms, we have \_i-{-h{l-^ai)Y={l-\-nb + Qb''+Rb^-^Sb'-t &c.)-^{nb-\-2Qb'' + dRb'+4.Sb'' H- &c.)^+ &c. Again, since l-\-b{l-\-x) is the same as {l-\-b)+hx, and that by [3] [(l-f-&) + 6a;]"=(l+6)"+??(l+&)"~^&^+ &c., the two series on the right, this and the one above, must be equal, whatever be the value of x ; hence .'. %(l+&)"-'=w+2Q& ^dIlb^+4SP+ &c. Now multiply each side of this by 1+6, and there results w(l+6)'»=w+(2Q+»)5+(3i2+2Q)6»+(4-S+3i2)iH &c. But [2], %(l+&)»=%+w264.nQ&2_j.,j^j3_|_ &c^ The two series on the right are, therefore, equal, whatever be the value of b, consequently (94) THE BINOMIAL THEOREM. 78 iS+ZM=nR .: g^''(''-l)(''-2)(''-3) 2.3.4 &c., &c., where the law of the coefficients, Q, JB, S, &c., is abundantly obvious. Hence the development [3] is vs'hich development is the Binomial Theorem. 96. In the preceding development [A]^ the student will do well to attend not only to the law of the coefficients, but also to that of the ex- ponents ; he will observe that while the exponents of the first term of the binomial, namely a, commencing with n, regularly descend by unity, those of the second term .v as regularly ascend by unity, so that the sum of the exponents is always n ; it thus appears that, omitting the coeffi- cients, each term is equal to that which immediately precedes it, multi- plied by -. Knowing the first term, therefore, we can always write down the subsequent terms (omitting coefficients) without any trouble. And leaving spaces for the coefficients, these may afterwards be supplied from observing that the coefficient of the second term is always the given ex- ponent 71, or -, and that every subsequent coefficient is found by intro- ducing an additional factor into both num. and den., the factors of the num. uniformly decreasing by 1, and those of the den. as uniformly increasing by 1. The additional factor in the num. is no other than the exponent of a in the preceding term, and that in the den. is no other than the exponent of a; in that term increased by 1. Suppose, for example, we had to develope {a-t-xf; then, knowing that the first term is a^ by leaving spaces for the coefficients, we might very rapidly first write ^5+ a'^x-}- aV+ aV+ ax^+ a:^. Then, knowing the coefficient j of the second term, and attending to the simple law just described, we have the following succession of coefficients, namely, 5, 10, 10, 5, 1. Hence, filling in the coefficients, we have (a+a?)' =«»+ 5a*a;+ 1 OtxV+ 1 OaV -f 6ax^+a;\ 97. Whenever the exponent is a positive whole number, the coefficients,, as in this instance, form the same series of numbers, whether taken in order from left to right, or from right to left. It is obvious that such must always be the case ; for, as the coefficients must remain the same, whatever particular letters form the binomial, the development of (x-\-a)" must have its coefficients, taken in order, the same as those of the development of (a -ha?)"; but, omitting the coefficients, the latter develop- ment is «"+ a"-'aj-i- a"-V+ -}-aV-'-|- aaf'-'+ x'' and the former, x'-t aj'-'a-i- aj"-V + -fajV-'^-f- xa''-'+ a"" But either of these, written in reverse order, becomes the other, so 74 THE BINOMIAL THEOREM. that the coefficients must be the same series of numbers, whether they be taken in order from left to right, or from right to left. 98. In the foregoing general development x is preceded by the plus sign ; but as cc is any value whatever, we may replace it by — y, and thus# get the development of {ci — yy\ or, putting the symbol x for y, of (a—xy. But the changing of a; into —x can affect only the signs of the odd powers oi x; so that, in this case, the development [A] will have its signs alternately positive and negative when n is a positive integer ; but if n be a negative integer, then, on account of the coefficients — com- mencing with that of the second term — being alternately negative and positive, the signs will be all plus. 99. As to the number of terms, when n is a positive integer, we may readily determine it by observing that in the first term the power of x is x", and that in the last it is x'^ ; and as the exponents of x regularly in- crease by unity, the entire number of terms must be ?i + l. We say that the term involving x" is the last term, because, when we arrive at this term, the coefficient becomes 1 ; and, by extending the terms beyond this limit, each coefficient becomes 0. But when the exponent is negative or fractional, then there can be no last term. The coefficients never disappear, and the series may be con- tinued indefinitely; for the subtraction of a positive number from —n increases it, and the subtraction of a whole number from a fraction can never destroy that fraction. 100. Confining our attention, then, for the present, to the finite develop- ment, that is, to the case in which n is a positive integer, and for which the terms are limited to the number w-f-l, we see, from the principle (97), that when n is even, we need compute only the first 7: + 1 coefficients of the development of (a-^xY; and that when n is odd, only the first -—— ; for the remaining coefficients will be these written in reverse W -h 1 order, observing that, in the latter case, the ~ — th coefficient is to be repeated. In the former case we compute up to the middle coefficient inclusive ; in the latter, only up to the first of the two middle and equal coefficients. These coefficients are necessarily all integers, since, from the nature of multiplication, (a + a;)", when developed, cannot involve fractions. The same is true when n, instead of a positive, is a negative integer ; for the development of [a-^x)'", that is, the quotient of l-^{d^-{-na"~^x-\- ...), seeing that the coefficients of the divisor are all integral, and the leading one unit, cannot possibly involve a fractional coefficient. We may infer, from what is here shown, that the product of n con- secutive whole numbers is always divisible by 1. 2. 3. 4...ri. 101. Another interesting deduction is, that so long as w is a positive integer, the sum of the coefficients, when they are all positive, is equal to 2" ; and when they are alternately positive and negative, the sum is ; that is, the sum of the positive cofficients is equal to the sum of the negative coefficients. For, making a and x each =1, we have (i+i)«=2»=i+«+2(2=i>+"-<2z2)|i^+ . . . +1 and (l_l)..=0=l-„+i'i2^l^-^-^.i^^>+ . . . (-1)-. THE BINOMIAL THEOREM. 76 102. I. — To develope {a-\-xf when n is either a positive or negative integer. Rule. 1. Write down a" for the first term of the development; then every subsequent term is found thus : — 2. Multiply the coefficient in the term last found by the exponent of a in that term, then divide by the exp. of x in the same term, increased by 1, or, which is the same thing, by the number which marks the place of that term, and the coefficient of the next term will be obtained. As to the letters, the powers of a diminish, and those of x increase by 1, successively. Note. — When n is positive, the coefficients need be computed only up to the middle of the development; they are then to be repeated in reverse order. If one of the two terms of the binomial is negative, all the odd powers of that term will be negative. (!) Required, the development of (^a^xf. By the rule, we have term. 2nd term. 3rd tenn. 4th term. a6 Qa^x 2 We need not proceed further, for we have now arrived at the ( n + ^ )*^' or middle term. Hence the complete development is {a—xf^a^-U^x + 15«V- 20aV + 1 5aV- &ax^-\-x\ w + 1 (2) Required, the development oi{a-^xy. Here we compute — ^7-= til 4 terms. ist. 2nd. 3rd. 4th. of U^x ^a5a;'=21a5a;2 ?^a«:z;3_ 35^4^8, 2 o (3) Required, the development of [^a*'~?>hxf. Here, putting a' for 2a^ and of for 2>hx, we might proceed to develope {a'—xff as above, and then, in the result, replace the values of a' and x'. But the operation will be easier by considering the general development [A] under the form f. « w(7i— l)/a;\2 nin—Vjin— 2/a;\^ . , /^Vl. t-dt (.+.)"=.»{l+»-+^(-) +^^ (-) + . . +(-) Y-.. [B] where - is the second term of the proposed binomial divided by the first; and since, in the present example, this is— --^, we have ,.:3a.,=,.,{i-.|>.o(gy-io(|)% or (<*+^) • The coefl&cients are 1, I, 2 8' 3 16' 4 128 , &c. .'.^/{a+x)=:a^^-\--a ^^x-^-hp-t-—a ^^— jgg* lx*+ kc. wMcli last is the form at once obtained from [B]. (2) Required the development of {b^-^x)h , or (p^+x)—i. The coefficients are 1, -i -iLi=A, i:ii_J=_ __, &c. •••TO 7^17)440-^.+: 3a;2 3.5a:3 3.5 2.4.6 3.5.7a;* -&c.^ (624-a;)4 b\ 262 2.46* 2.4.66^ ' 2.4.6.868 There is an advantage in thus putting the numerical factors in evi- dence, since, after a few terms have heen computed, they may be con- tinued at pleasure. Examples for Exercise. (6) (a+a:)-|=. (7) 3^9=3^(8+1)=. (8) V6=V(8-2)=. (1) V(6H^)=. (2) V(6H^)=. 1 (3) (^a-x)i— (4) (a+x)i=. (5) {a^-x)i=. (9) (10) {a^-^x)i 1 (3-7a;)s' 104. Instead of the entire development of {a+^y, it is sometimes required to express only the single term involving a specified power of a?, as the power ;^;^ From the general form, we know that the den. of the coef. belonging to x"" is 2. 3. 4...r, and that the last of the r factors in the num. is n— (r — 1); the required term is therefore readily found. For instance, let the term involving a>'' in the development of {a-\-x)^t be required. The coefficient would be m)ih)i-i){-l) • • • (i-r+l) ^ 5.3.1.1.3 (r-2r) 2.3.4 . . . . r 2.3 A 2r 5 3 113 . (7 2r) And the term itself would be ' ' *, ^^-»"~''*'"> which is called the 2.3.4 . . . Q\ 2^ general term of the series. THE BINOMIAL THEOREM. 77 It is sometimes, too, required to find, for given numerical values of a and X, which is the greatest term of the series. Now, from [B] we see that each term within the brackets is equal to the constant factor -, a multiplied by the preceding term, and also by — ^^ -, where the num- r ber r marks the place of that preceding r term ; that is, this term is the rth of the series, or that in which the exponent of x is r — 1. So long, then, as the multiplier -. - exceeds unit, will the successive r a terms continue to increase in magnitude, but when at length r becomes 80 great that the multiplier becomes less than unit, we may be sure that the numerically greatest term is about to be passed ; that is, that the term, to which this multiplier must be applied to obtain the next, is the - , . ^ . , /. n—ir—\) greatest term of the series; for, as r increases, the factor decreases, so that when once the terms begin to diminish, they must con- tinue to diminish. This is readily proved thus : take the next term of the series, that is, change r into r+l ; we have to prove that > — -, or that — ^ 1 >— !— — 1,* or that -— - > —- :, which is necessarily r r-\-l r r+1 r r-\-l the case, because ->Tfr-T- For example: What is the greatest term in the development .£ (4+^)'? Here l=\, and >.-(r-l)=5-r .-. ^^^-r)=-^, &c. 2' 2.3' 2.3.4' .-. a-^=l+Ax^^x-+f^x^+^^x*+ &c [3] In which it will be remembered that .4=6— J6^-f J6^— &c. ; or, since b=a—l, A={a^l)-^l(a--lf + i(a^-iy-i{a-lf-{.&e [4]. The development [3] is the Exponential Theorem. 107. This development, unlike that of the binomial theorem, is always THEORY OF LOGARITHMS. 79 interminable, whether x be positive or negative, integral or fractional. Not only so, but even the constant A, which enters each term, is ex- hibited, at present, only under the form of an interminable series, namely, the series [4] ; but A will be computed independently of this series here- after. , ] 08. Theory of Logarithms. — If we take any positive number a other than unity, then n representing any positive number whatever, it is always possible to determine x so that a'' may be equal to that positive number ; that is, it is always possible to solve the exponential equation a'=n, as we shall presently show. For the same proposed number n, x will of course be different for different values of the assumed base a; but this base being fixed once for all, the value of x, which satisfies the equa- tion a''=n, is called the logarithm of the number n, to the base a: we ex- press this by the notation x= log n. Making n= 1, 2, 3, 4, &c., up to any proposed limit, and determining the corresponding values of x, as will be hereafter shown, we shall obtain the logarithms of all the natural numbers in succession up to the prescribed limit: these, arranged in a table, constitute a Table of Logarithms. Such a table is therefore simply a table of exponents. The assumed base a, to which these exponents all belong, is, in modern tables, the number 10 ; any number, therefore, be- ing proposed — the number 256 for instance— the table informs us, by inspection, what exponent {x) must be put over the 10, so as that lO'' may be equal to 256 ; we should find that ic=2-40824, in other words, that log 256=2-40824. Reserving the methods of constructing logarithms for the next section, we shall here enumerate their fundamental properties. 1 . The sum of the logarithms of two or more numbers is the logarithm of the product of those numbers. Let a^=%, and a^'=?i' .'. a^Xa'''=a^+'''=%%'. But by definition, x=z log n, x'=. log w', and x-\-x=. log n'n! .'. log w-f log n'= log nn'. Similarly, putting N for nn\ log N+ log n''=\og Nn'' .-. log n' + log w'4- log n''= log jinV, and so on. 2. The difference of the logarithms of two numbers is the logarithm of the quotient of those numbers Let a^=n, and a^=n .'. a^-^-a* =a^--^=-. n. But a:=log n, a;'=log n', and x —x'=log — .•. log n — log n'= log — . n ^ 3. The logarithm of any power of a number, whether the exponent be integral or fractional, positive or negative, is the logarithm of the number itself multiplied by that exponent. Let the number be n, and let m be any exponent over it, and as before put a*=7i .-. «""= w'" .-. mx= log n'", that is, m log n= log ?i'". 109. These general principles sufficiently show the use that may be made of a table of logarithms in arithmetical operations that would be laborious without such aid. If we wish to find the product of several large numbers, we have only to take the log of each out of the table, and to add them : the sum will be the log of the product ; and against this log in the table the product will be found inserted. So if we have a power of a number to raise, or a root of it to extract, all we have to do is 80 CONSTRUCTION OF LOGARITHMS. to take the log of the number out of the table and to multiply that log by the given exponent, the product is a log against which in the table stands the power or root sought. # 110. Construction of Logarithms. — The base of the pro- posed system of logs being a, and N any positive number, whole or fractional, we have to find x, so as to satisfy the equation a'^^N, By [3], since a'^^N^, the two following series are equal; namely: — . l+Axy+^:»Pf+ &c. =l+^'y+^lV+ &c. The denom. (A) of this fraction is the same for all values of N, since the base a is constant, but the num. differs for different values oi N ; it is unaffected, however, by any change in the number a. Substitute 1 + 71 for Ni and calling unity divided by the den., that is — , M, we have — log{l+n)=M(n—i7i^-{-^n'^—ln'^+ &c.) where, as already observed, If or — is constant : it is called the modulus of the particular system of logs adopted ; as before remarked, the system in general use is the system whose base (a) is 10, so that for this system M=l^(9-i9Hi9'-i9H &c.) But Napier, the original inventor of logarithms, constructed his system on the hypothesis that a was such a number as would render M=\, or, which is the same thing, A=l. Let Napier's base be distinguished from all other bases by being called e, instead of a ; then, since a=e corresponds to ^=1, we have, by the exponential theorem — .^=l+.+-+_ + _-H .... [1] so that, putting a;=l, g=l + i + - + ~ + ^^4. ... [2] If in [1] we put —a? for x, we have .--!-.+-__ +—- .... [3] so that each of the series [I], [3], is the reciprocal of the other, whatever be the value of x. 111. On account of the rapidity with which the series [2] converges — that is, the rapidity with which its terms diminish — the value of e may be found to several decimal places by summing only a few of the leading terms : thus, taking only the first nine fractions which follow 1 + 1, or 2, and then adding this 2, we find 6=2-7182818, as in the margin ; and taking a few more terms, we have e:=2-71 8281828459. This value of the Napierian base e, would be equally got by putting 2-7182818 2 1 8 •5 4 'iQmm7 5 416667 6 83333 7 13889 8 1984 9 248 10 27 3 CONSTRUCTION OF LOGARITHMS. 81 1 1m 2 for X in [3], page 78; hence a'i=e .-. a=e ; and, taking the Na- pierian log of each side, writing that log thus, log,, we have -log,a=l, that is Mlog,a=l .-. M=. , -^ iogga so that the modulus of the common system of logarithms, or that whose base is 10, is j=^=i — Tq", or I divided by the Napierian log of 10. 112. If, in the general series for log (1 +w), p. 80, we make n negative, we have log (l-w)=:ijf(-%_l7i«-l%3_1^4_ ^(5 J .-. Subtracting, log(l+%)- log (1-w), or log ^^=z2M{n+i'n?-{-^n^+ &c.) . . . [1]. But if in the same series we put - for w, we shall have n log (!+«)- log .=m(1-±+±-±+ &C.) . . . . [2] .♦.Subtracting, log w=il!f{(»i-n-i)_i(7i2_^-2)^^(^3_^-3)_ &c.} . . . . [3] 2 3 113. This last series for log n will find its application hereafter ; but for the actual computation of logarithms none of the foregoing series are directly available, on account of their slow convergency, except in par- ticular cases. But as n is quite arbitrary we may, by judicious substi- tutions for this symbol, obtain a variety of series of suitable convergency. Put- =- , from which we eret n= : this converts fl] into l—n p 5 2p4.1 L J the following, when log p is transposed ; namely, log (p+l)=:2M{—- — I- 1 1 1 h&c.i+logo . . . . m by means of which we may compute the logs of all the natural numbers in succession, from 1 up to any prescribed limit; for log 1=0, since a^=}, whatever be a (art. 42), so that putting jp=l, the series gives log2, thence log3, log4, and so on; supposing, that is, that M have been previously computed. It has been seen that M becomes known so soon as the Napierian log of 10 becomes known; we shall first, therefore, show how Napier's logs may be found in succession, as far, at least, as log.lO. Making, then, jo=l, 2, &c., we have from [4], remembering that in Napier's system M=l, ^%K^+o.+5y ) = ''''''''^ H^=,„=»i(7i-l)(7i-2)(u-3) (7i-m+l) [2]. where the notation on the left indicates that n is the total number of things, and m the number in each set taken out of them for permutation. If ')n=-n we return to the case [1], as is plain; for, reversing the factors here written, we should then have 1.2.3.4 n for the right-hand member of [2]. But the product would have been just the same if we had made m=w— 1, instead of m=?i, for, still reversing the factors, it would have been 2.3.4 n: we may, therefore, safely conclude that the number of permutations furnished by n things, by taking n — \ of them at a time, is the same as the number furnished by taking them altogether. Thus, in the case of three things, a, b, c, as seen above, the permutations, when they are all taken together, are 6 in number ; and when taken two and two, they are ab, ba, ac, ca, be, cb, which are also 6 in number. If some of the n things out of which the permutations are to be formed are identical, then certain of these permutations will recur, differing in no respect from one another. In many inquiries these recur- ring groups must be dismissed, and only those retained which really differ from one another. We have, therefore — Problem II. To find the number of different permutations of n things, taken altogether, when certain of these n things are identical. 1. Beginning, as before, with the simplest case, suppose only two of the n things are alike ; then, since these two enter every permutation of all the n things, they enter 1.2.3 n times [1]. But every permutation is accompanied with another, in which the two like things are simply interchanged : hence there can be only half the whole number of per- mutations that are really all different; so that the foregoing product must be divided by 2, or by 1.2. 2. Suppose three of the n things are alike. These three enter every permutation, each being accompanied with five others, formed by merely interchanging the places of the three like things ; so that the whole set may be divided into sets of six, in each of which there will be but one distinct permutation ; hence to obtain the number of distinct permutations we must divide the entire number by 1.2.3, the number of permutations furnished by the three like things. And, continuing this reasonmg, we arrive at the conclusion, that if p of the n things are alike, there must 90 PERMUTATIONS. be as many sets of recurring permutations, each set consisting of the same number of individual permutations, as there are permutations ol tliose p things. Hence the number of different permutations of n things, taken altogether, when p of them are alike, is — ^?^=g=(.+l)(.+2) [3J 3. After thus eliminating all the recurring permutations arising from the p like things, q other like things may be still conceived to enter. In this case, the permutations just determined divide themselves into as many sets of recurring permutations, as there are permutations of those q things. Hence the number of different permutations of n things, taken altogether, of which p of them are alike, and also q others of them are alike, is \.2.Z...n _ Pen) (1.2.3.. .^)(1.2.3...2)-P(p)XP(,)' and this reasoning may be extended to three, four, and, generally, to any number of distinct sets of recurring things. If the factors in the nume- rator be reversed, and the p factors, common to num. and den., p being >g', be suppressed, we shall have Pcn^ jn(n-l)(n-2) (jp+1) P(;,)P(,) 1.2.3 q If p-]~q make up the whole number n of things, then q=n—p, , Pen) ^ n(n-l){n-2) (n-p-^l) "Pa»P^n-p) 1.2.3 .p LJ- It is sometimes required to form permutations of m things out of w, when liberty is given to introduce into the permutations repetitions of any of the n things, or to solve the following problem. Problem III. To find the number of permutations of n things^ taken two and two, three and three, and so on, allowing repetitions. Let the n things be represented by letters connected together by the plus or minus sign, as a + 6 + c + d + .... It is obvious that if this quan- tity be multiplied by itself, the product will exhibit all the possible per- mutations, two and two, admitting the entrance of the repetitions aa, hh, cc, &c. If this product be multiplied by the same multiplier, all the permutations, three and three, will be exhibited, admitting repetitions such as aaa, aab, abb, &c. ; and so on. Hence, allowing repetitions, the number of permutations of n things, taken two and two, is n^ ; taken three and three, n"'; and, generally, taken m and m, the number is n"". (1) How many changes may be rung with a peal of 8 bells, and how many may be rung with 5 out of the 8 ? P(g3=1.2.3.4.5. 6.7.8=40320, and «P5=8.7.6.5.4=6720. (9) How many different permutations may be formed of the letters in the word Mississippi, taken altogether? Here are 11 letters : i and s each occur four times, a.nd.p twice, Examples for Exercise. (1) Nine men stand in a rank, in how many ways can their order be varied? (2) In how many different ways may 1 persons seat themselves at table ? COMBINATIONS. 91 (3") How many different permutations may be formed of the letters in the word Salamanca, taking them altogether ? (4) How many different numbers can be formed of the figures in the number 223334444, taking them altogether ? (5) How many different numbers can be expressed by four different figures, taking them one at a time, two at a time, three at a time, and altogether ; repetitions of the same figure being excluded ? (6) How many can be expressed when repetitions are allowed ? 125. Combinations. — Combinations of things are the different collections — each collection containing the same number — that can be formed out of a given number of things, under the restriction that in no two sets shall the things be all the same. Mere order of arrangement of the individual things in a set is not regarded here as in permutations : a combination, to differ from another, must have in it one thing, at least, "which the other has not : thus, the combinations of a, b, c, taken two and two, are ab, ac, be, and no others : ba, ca, cb, are excluded, because, though different permutations, they are the same combinations as the three above. Problem I. To find the number of combinations that can be formed out of n different things, when p of them are taken at a time, 1. Let the n things be taken two at a time. The number of permuta- tions that may be formed of them will be n{n—l), as already shown. Now each permutation, as ab, is accompanied with another, ba; so that there are only half as many combinations as permutations : hence the - , . . . n(n — 1) number of combinations is — - — \ 2. Let the n things be taken three at a time. Then, p. 89, the number of permutations that may be formed will be n(ri— l)(n— 2). Now every combination, as abc, is only one out of 1.2.3 permutations of the same letters (134): hence the number of combinations is — ^ — rr^" — • ^^^ following out this reasoning, it is evident that when the n things are taken 2? at a time, the number of combinations is n[n-\){n-^). . . .(^~j^ + l) .,-, ^'- 1.2.3. . . .p -L^^- Putting in this general expression 1, 2, 3, &c., successively for p, we have nn-n »r-"("-^^ _ «(»-lX» -2) ^» -lX»-aX»-3) C,-«, (^^--g-. <^3- ^ , 0,- ^jj .&c., •which are the successive coefficients that follow the first term 1, in the development of (1+a;)": hence (101) l+"ei+«C2+«C3+«C4+ +«C„=2" [2], so that the number of combinations of n things, formed by taking them first singly, then two at a time, then three at a time, and so on, is 2"--l. To find how many, out of n different things, must be taken at a time, so that the combinations may be the most numerous possible, we have evidently only to find the place of the term involving the greatest coeffi- cient in the development of (1+^)''. Not counting the first term 1, of the development, the greatest coefficient belongs to the i nth term, if n be even, and to the i(?i-l)th, or i(^+l)th, indifferently, if n be odd 92 COMBINATIONS. (see art. 104). Heuce, in the former case, half the proposed number of, things must be taken : in the latter, half the number, which is a unit, either less or greater. If n—p be put for p in [1], then, observing that [1] is the very same as the expression [4] page 90, we shall have nc,=-l^, and «C7,.-,=— ^ [3], which expressions are identical : consequently the number of combinations of n things, taken n—p at a time, is the same as the number when^ of those things are taken at a time. Hence, whether "(7^, or "C„_^, are com- puted by [1], the result is the same : these two sets of combinations (equal in number) are said to be supplementary to each other. It is plain that "C,="6?„=l. Peoblem II. To find the number of comhinations that may be formed out of Til, ^2> ^3* ^c., different things, by taking one thing out of each of the sets. 1. Let there be two sets, n„ n^. Then, since each of the w, things in the first may be combined with every one of the n^ things in the second, the whole number of combinations will evidently be n{n^. 2. Let now a third set of Wg things be introduced; each of these may be combined with all the n^n^ combinations just deduced : therefore the whole number of combinations furnished by niU/i^, will be n^n.jn^ ; and so on. Hence, generally, the number of combinations that may be formed by taking one thing out of each of m sets of different things will be ^^,^^2^l3...^l;„. If each set contain the same number n of things, this last product will, of course, be n"\ In a similar manner it may be shown, after reference to Prob. L, that if instead of one thing, jt?, things are taken from the first set, p^ from the second set, and so on, then the number of combinations that may be formed from the different selections is (1) How many combinations may be formed with 7 things out of 12? By equation [3] above, ''C7="^(7,= '^^:V'J^'^'^ =6. 11.8.3.2=792. (2) How many combinations may be formed out of 6 things taken 1, 2, 3, &c., at a time, and altogether? Here (art. 125), 2''-l=2'— 1 =63. (3) How often may a different guard of 6 men be selected out of 60, so that no two may mount guard together oftener than once ? Out of the 60 men, ■ ' =30.59 couples may be formed ; and out of 6.5 every 6 men, -^=15 of those couples; but only one of these is to be admitted : hence, taking the 15th part of 30.59, we have 2.59=118. (4) How often may a different guard of 6 men be taken out of 60, how often would the same man be among the six, and how often would the same two men be found on guard together ? 1. 6»(7,=?M?_5|L^I:.^:i5==59.58.57.14.55=50063860 total number 1.2.O.4.5.0 of guards. PROBABILITIES. 93 2. Suppose the specified man were excluded, and that guards of five men only were made up out of the remaining 59 men: the number of the sets of five would be ^^(7,= ^^'^^'^^f f'^^ =^»C6H-10=5006386, and 1.2.3.4.5 into each of these sets must the excluded man be introduced to complete the guard : he will, therefore, be on guard 5006386 times. 3. Suppose the two comrades excluded, and the guard to consist of but f> four men, made up out of the remaining 58 men : then ^^C^—^^C^ X — = 59 5 6006386 X —=424270 ; and into each of these sets must the two ex- oy eluded men be introduced to complete the guard : they will .'.be on guard 424270 times. Examples for Exercise. (1) How many combinations may be formed of 11 things taken 4 at a time? (2) How msiwy fours can be selected out of ten, so that the same indi- vidual may always be one of the four? (3) What is the greatest number of combinations that can be formed out of 10 things ? (4) A person wishes to make up as many different dinner parties, each of the same number of guests, as possible, out of 8 friends : how many should he invite at a time, and how many parties can he give ? (5) How many combinations can be formed of the letters in the word notation, taking three at a time ? (6) Fifteen young ladies at a school walk out daily, three and three : how many walks may they take if arranged so that no two shall walk abreast twice? (7) How often may a different guard of six men be selected out of 60, so that a specified couple of men may never be on guard together ? (8) From 16 privates, 6 corporals, and 4 sergeants, how many different guards can be formed, each consisting of 12 privates, 2 corporals, and one sergeant ? 126. Probabilities.— The doctrine of Probabilities originates in our ignorance of operating causes. If a die with six equal faces be thrown from a dice-box, we say that there is the same chance that any one face will turn up as any other : yet the turning up of the face which presents itself after the throw must be a necessary consequence of all the motions communicated to the die by the thrower. Of the effect of these, however, we are ignorant till after the event : and as the thrower himself has no control over the motions he communicates so as to predetermine the result, we are correct in saying that any one face is just as likely to turn up as any other. If, therefore, six persons were each to predict the turning up of a different face, the six chances would be equal. Suppose a sum of money S to become due to him whose prediction is fulfilled : it is plain that before the die is thrown, each of the six persons has the same interest in the sum S ; and if these interests were all to be purchased at their exact value, the sum paid for them ought to be exactly S, because one of the six predictions must necessarily be successful, and thus the sum S be recovered. Hence the value of each of the six equal chances 94 PROBABILITIES. is ^S, or |th the value of certainty. Representing, then, certainty by* 1, the chance or probability of any specified face turning up is ^. And it is evident that in a case where there are n equal chances instead of 6, the probability of a specified one of the n events happening is -. Peoblem I. If an event may happen in a ways, and fail in b ways, any one of these being equally probable, the chance of its happening is 7, and the chance of its failing 1. For the whole number of occurrences being a-f 6, that any specified one may succeed or fail, the probability is 7, which may be regarded as the value of the chance ; but there are a such chances of success, and b such chances of failure ; therefore the value of all the chances for the event is r, and the value of all the chances against it is 7 : that is, a-\-b ° a-i-b these fractions express the probability of happening and of failing respec- tively ; and the sum of them is 1, or certainty, as it ought to be. (1) A bag contains 12 balls, 7 black and 5 white: what is the proba- bility, in drawing two at once, that they shall both be white ? By the preceding problem, we must find all the possible ways in which two balls can be drawn ; then how many of them are favourable to the proposed result : — the latter number divided by the former will be the probability of that result happening. All the combinations of two out of 12 are ^"(7a= ' =66 i-.A 12.11 1.2 5.4 »C,= ~ =10 Prob>. =|. 7 6 12 11 7 6 7 In like manner, -^ -r —^=z ' =— -, the probability that both shall be black. 33~33' ^^ 22~22 **• 33 is the probability that both shall not be white, that is, that one at least 15 shall be black, and — is the prob^. that both shall not be black, that is, that one at least shall be white. The ratio of the probabilities of an event happening and failing is called the odds : thus, in the above example, 28 6 the odds against drawing two white balls is — to — , or 28 to 5 : the 00 00 odds against drawing two black balls is 15 to 7. The odds is found by merely subtracting the numerator of the probability from the denomi- nator : if the rem. is less than the num. the odds are in favour of the event to which that probability refers ; if the rem. be greater, the odds are against it : if the rem. be equal to the num. the odds are equal, or even. (2) What is the probability of throwing an ace once, and not oftener, in four throws of a die ; or, which amounts to the same thing, in a single throw with four dice ? Only one face of each die can enter into any combination, because two faces of the same die cannot appear at once. Hence, we have first to find the whole number of combinations of four faces, by taking I out of PROBABILITIES. 95 each of the four sets of 6 faces : this number, by Prob. II. (art. 125), is 6^=1296. For the number of favourable combinations, each ace must combine with one out of Jive only of the faces of each of the other dice ; so that each ace furnishes 5^ combinations, and since there are four aces, the whole number of combinations favourable to the event is 5^x4=125 x4 .*. the proy. reqd. is 77-77^=77^-7. 129o 324 Note. Whether one die be thrown m times, or m dice thrown once, can make no difference ; because intervals of time do not enter into con- sideration in determining the combinations. Problem II. If an event may happen in any one of n different ways^ and if the probability of its happening in a specified one of the n ways be jOp the probability of its happening in a second specified way be p,-^, in a third /?3, a7id so on : then the probability that it will happen in one or other of the n ways will be the sum of the individual probabilities^ namely, Suppose a sum of money S to be receivable upon the happening of the event: then the value of all the n expectations will be p^S+p.^S-^ p.^S-\-.., -hp,tS=(j)i-\-p,^-\-p.^-\- ... -\-pn)S' But the quantity that multiplies S must be the fraction of certainty, that is, the probability, that the event will happen in one or other of the n ways : this probability is, therefore, (3) In Ex. 1, what is the probability that the two balls drawn will both be of one colour ? The event may happen in two different ways : the balls may be both white or both black : the probability that both will be 5 7 white has been seen to be — , and that both shall be black, it is — : 00 22 6 7 31 hence that one or other shall happen, the probabiUty is ^+7^.=^. 00 i4),i DO The probability of not drawing either two white or two black balls, that 31 35 is, the probability of drawing one of each, is 1— ^=^* (4) Out of 100 mutineers, two men, drawn by lot, are to be shot : the real leaders of the mutiny are 10 : what is the chance, 1. that one of these, and one only, will be shot ; 2. that one, at least, will be shot ; and 3. that two of them will be shot ? 1. If one and one only is shot, the pair of men must consist of one man from the 10, and one from the remaining 90 ; the number of these combinations is 900; and the whole number of combinations, two and two, is ^^;^=4950 .-. ^=4 is the probability of one and one 1.(4 4950 11 only. 2. If the restriction to one only be removed, then, in addition to the 900 combinations, there will enter the combinations from the 10, taken two and two; these are -— ^ =45, giving the probability -^^ that the 945 21 selection will be made from these 10 : hence, adding the two, 4g5Q=fl0* the probability that one at least will be shot. 96 PROBABILITIES. ^ 3. The probability that two out of the 10 will be shot, is evidently TTrr- , just determined, and which is =t77,' 4950 110 Problem III. If any number of independent events can happen, in con- junction, the probability of their happening together is the ;prodiLCt of the several probabilities of their happening separately. Let the individual probabilities be — , — , — , . . . .— , where a^, a.i,..,a„^, Wj n^ n^ n^ are the numbers denoting the occurrences favourable to the respective events, and w,, n^,...n„^, aU the possible occurrences; then the favourable cases will be expressed by all the combinations of 1 out of a^, with 1 out of a^, with 1 out of a.^, &c. ; and all the possible cases by all the com- binations of 1 out of n^, with 1 out of n^, with 1 out of n3,.&c. But by Prob. 11. p. 92, these several combinations are in number a^ a^ a^ a„„ and nyn^n^...n^. Hence the probability of the compound event is a-^a^a^ a^ n^n^n^ n^ (1) What is the probability of throwing an ace with a single die once at least in two trials? The prob^. of succeeding the first time is -, and of failing, - ; and the same as respects the second time. Hence, by the K K OK above, the prob^. of failing both times is h^a^qa' ^^^ •*• ^^ prob". of o o oo 25 11 not failing both times, that is, of succeeding once at least, is 1— ^ = r^. DO do (2) What is the probability of throwing exactly two aces in four throws, three in four throws, and of ace turning up every throw? 1. The prob^. that any two specified throws may turn up ace is ( ^^ ) , (Prob. III.), and that the other two throws may fail, ( ^ ) » •*• the prob". of the concurrence 52 of these events is ^. But the two specified throws may be any two out 52 25 of the six combinations of two furnished by the four dice, .-. 6—=—- is *' 6^ 216 the prob''. that some two throws will each turn up ace, and the other two fail. 2. The prob^. that any three specified throws will produce ace is { - ) , and the prob^. that the remaining throw will fail is -, .*. the prob'', /1\^ 5 of both happening is ( ^ ) X ^ ; but there are 4 different ways in which (1 \^ 5 20 5 -) X7i x4=— -—=-—, 0/ o 1290 o24 is the probability required. PROBABILITIES. 97 / 1 \^ 1 (3) The probv. of an ace every throw is ( ^ ) =,iwj« ^^^ •*• ^J (J^-) the proby. of throwing either three aces exactly, or four aces, that is, the 20 1 21 7 proby. of throwing three aces at least, is — ^ + _=_g=_ . But the following general problem is better suited to all questions of this kind. Peoblem IV. The prohahility of an event happening exactly m times in n trials is expressed by the {ii—m-\-\)th term of the development of the binomial {p-\-cD'^, where p, q are the respective probabilities of happening and failing in a single trial. And the probability of happening at least m times, in n trials, is expressed by the sum of the first n—m-\-\ terms of tJie same development. Let p be the prob''. of happening in a single specified trial, and q=l—p, the proby. of failing in that trial. Then the prob^. of the specified trial succeeding, and all the remaining n — 1 trials failing, is P5"~' ; and since any one of the n trials may be the trial specified, the prob'. that one or other will succeed and all the rest fail, is npq^~\ Again, the prob''. of the event happening twice in two specified trials, and failing in the others, is p-q'*~'*; and since all the combinations, two and two, out of n things, are — ——^ in number, and that either of these may be the combination specified ; the prob^. that one or other of the combinations will succeed and the remaining trials fail, is \ yV"^- ^^ li^® manner, the proV. of the event happening three times in three specified trials, and faifing in the other n—S trials, is ;?V~^5 ^^^ since all the , . . , , , . , . . n(n—\)(n-^) combmations, three and three, out of n things is -7-5 , and that either of these may be the combination specified ; the prob^. that one or other of them will succeed and the rest fail is -^^ — —-z p^q^'^j and 80 on. Hence, generally, the proby. that one or other of the sets of m trials will succeed and the remaining n—m trials fail, is 1.2.3. ..m ^ ^ ' that is, it is «C«i>"'2'-'«, or (p. 92), ''Cn-n,p'"q^-'"--D-l The prob'. that the event will happen at least m times, in the n trials, is the same as the prob^. that it will happen either n times, or n—l times, or n— 2 times, &c., down to m times, which (II.) is the sum of the pro- babilities of these happenings individually; that is, it is the sum of the first n— m + l terms in the development of {p + qT- Since p + ^=l .-. {p+qf=l, so that the sum of all the n + 1 terms of the development is' 1 ; hence the first n-w + 1 terms will be obtained by subtracting the last m terms from 1 ; but (97) the last m terms ot {p-\-qY are the same as the first m terms of {q+pfj, hence the prob''. is expressed by unity minus the first m terms of (q+pT' Note— The first term p"" of the development of (p+qT expresses the »» PROBABIIJTIES. proby. that all the n ,trials succeed, the last term o", the probv. that they all fail. (1) A shilling is tossed up : what are the odds in favour of four heads at least turning up in ten trials ? In one trial the proby. of head is ^, which is also the proby. of tail ; hence — ^=h =T7v» 9'=T7. (Prob. I.). And for these values we have^ to find what m must be in order that the above expression for the pro-' bability of the combination may be the greatest possible ; in other words, 7 we have to find what the exponent r— ] of — is in the greatest term in (g ijr v6 7 3 7 — +— ) . Now (see art. 104) rr--^:rr=^, and 10 10 / 10 10 3 7 6— (r— 1)=7— r .-. -(7— r)-+^>+^0- And since every numerical equation has a root, there must be some value ^2 of a; which will satisfy the equation a:"~' + +^'2^-+^/^;+ i\r'=0, so that we shall have fiai)={x-a,){x-a,){x--^-{. +^>+^'0- And, in like manner, since every equation has a root, there is some value a, such that the last of the above factors is divisible by x—a.^^ giving a quotient a unit lower in degree that will be divisible by x — a^, and so on. Consequently f{x)={x-a;){x-a.^{x^a.^ (ic-«„). Hence the equation f{x)—0 has n roots. It cannot have any additional root different from either of these, for if any value different from any of the values a^ a^, a.,, a„, be put for x in the above, none of the simple factors will become zero, and neither therefore will the product be- come zero. 136. The last term in the first member of the equation /(:c)=0, being (~'^iX~*2)(—^a) (—«»). is the product of all the roots with their signs changed ; or since the product of an even number of factors is the same, whether the signs of all those factors be changed or not, it follows that, in an equation of an even degree, the last term is the product of all the roots. If one root a of an equation be found, the division of the first member f{x) by x—a will give a polynomial, which, equated to zero, will be the equation involving the remaining roots. For example : — One root of the equation x^-\-^x^-\-W—llx—A^=^0 is 2; the first member is therefore divisible by a?— 2 : and performing j , 9 • 9-41 (2 the division as in the margin (see p. 104), we find the 2+22+62 depressed equation involving the other roots to be ar* + lla;- + 3U+21 =0. A root of this also is found 11+31+21(-1 to be —1 : hence dividing the former result by i»+l, ~ as in the margin, we arrive at the quadratic a;^+10^+ 10+2I 21 =0, of which the two roots are —3 and — 7 ; so that _ the four roots of the proposed equation are 2, — 1,-3, and — 7 ; or that equation, thus decomposed, may be written a:4+9^+9a;2_41a?-42=(a;-2)(^+l)(a;+3)(a;+7)=0. • This is proved a priori, for numerical equations, by Cauchy and others. See the Author's General Theory and Solution of Equations of the Higher Orders, p. 33. IOC THEORY OF EQUATIONS IN GENERAL. 137. Proposition 4. If two numbers, a, /5, when separately sub- stituted for a; in f{a;), give results with opposite signs, then at least one real root of the equation /(:c)=0 must lie between a and ^. Representing the n roots of the equation /(^)=0 by a^, a.^, an...a,^, we have (last prop.), regarding ^„ as 1, as, of course, we always may,* /(a?) = {a}—aj)(a!—a^Xa! - a,). . .(a;— « J. Let the substitution of « for x render this product plus, then, whatever other substitution be made for x, the product can become minus only on account of an odd number of the factors becoming changed in sign for the new value of a^, for an even number of factors changing signs has no effect on the sign of their product. Hence, since the substitution of /S for X produces this change of sign in the product, one factor, at least, as for instance, (x—a^), must take opposite signs when a, and (3 are succes- sively put for X. But when the root a^ is put for x the factor becomes zero ; hence a, which gives a result plus, must be greater than a^ ; and ^i which gives a result minus, must be less than aj. It thus appears that when two numbers, substituted successively for ^ in the first member of an equation, give results with opposite signs, one root at least necessarily lies between those numbers; but when the results have like signs an odd number of roots cannot possibly lie between them; there must be either an even number or none at all. For instance, taking the equation in last page, if and 10 be successively substituted for x, the results are successively — and + ; hence we infer that either one root or three are positive and less than 10. Also, if and —10 be substituted, the results are likewise — and +, we conclude .'. that one or three roots are negative, and that e^ch of them is numeri- cally less than —10. 138. If the roots of an equation are all imaginary, then, substitute whatever numbers we may for x, we shall never get results with different signs, for such results would imply the existence of at least one real root. And conversely, if we perceive that, substitute whatever numbers we may, the results must all have the same sign, we may be sure that all the roots are imaginary. We shall see presently that this unvarying sign will always be plus. 139. Propositions. Imaginary roots always enter an equation, with real coefficients, in pairs; that is, if a+iS^/ — 1 be one root, then a — 0^ — 1 must be another root. [These roots, from their form, are called conjugate roots.] Putting i for the imaginary a/ — 1, as at (61), if a+^i be a root of f(x)=0, then /(«+^i)=(a+^i)«+ +A,{c+^if-^AJ,c.+^if+A,(cc+^i) + N=0; and if these powers be developed by the binomial theorem we know (61) that the result will be of the form A-^Bi=:0, .-. (61) ^=0, B=0. Let now « — /3i be written above, instead of a.-\-^i, the developed powers will differ only in the signs of the terms involving odd powers of jSi (98), these signs will all be opposite to the signs of the like terms in the former development, but the terms themselves will be the same. The terms involving the even powers of ^i are the same, sign and all, in each . * Hereafter, the coefficient of tlie first term of f{x) wiU always be regarded as unit, unless otherwise stated. THEORY OF EQUATIONS IN GENERAL. 107 development. Hence the result is f(a.-'^i)—A—Bi; and since ^ = 0, B=0 .-. /(a— iSi)=0 ; so that if a+iSi be a root of/(a;) = 0, then also a— /3i must be a root. And the same conclusion follows if « be rational, and /Si be a quadratic surd, and not an imaginary, provided the coefiBcients are rational, since the inference ^=0, 5=0, still holds good (p. 44). Since the product of the factors x^{cc-\-^i) and x—{ce,—^i) is real, namely, ar—^ax+ac'^-{-^\ it follows that every polynomial, f{a)), with real coefficients, is composed of real factors of the first and second degrees. If the roots of an equation be all imaginary, that equation must be of an even degree. Moreover, since the last term N of an equation of even degree is the product of all the roots, and since the product of every pair of imaginary roots is plus, inasmuch as a^+/3^ the sum of two squares, can never be negative, it follows that, when all the roots are imaginary, the last term of the equation must always be plus. By putting x=0 in such an equation the result therefore is plus ; but the signs of the results never change, whatever be substituted for x (138); hence, when all the roots are imaginary, the results are invariably + for every substitution. Note. — It is not only the imaginary and surd roots of equations that always enter in conjugate pairs, for a conjugate form may be given to any pair of roots whatever : thus, taking any two of the roots of an equation, ajL and ^g, we may always express them thus : aj=a + |S, a2=oc—l3, where a=^( + (aia2+aia3+a2a3+ .. .)a;«-* -(a,a,a3-l-...):ir«-3+...+iV...[l] where N is the product of all the roots a^, a^, a3,...with their signs changed. Hence, if the second term of an equation be absent, then ay-\-a..^-^a.^^ ...=0; that is, the sum of the positive roots must = the sum of ithe negative roots, when all are real. If the roots be all real and negative, the signs of the terms must be all positive ; and if they be all real and positive, the signs of the terms must be alternately positive and negative. Since — Ai is the sum of all the combinations of the n factors of A'', taken n—i at a time, it foUows that _^=i+i+i+ +1. iV Ui a, a^ On Similarly, ^^=_L4-1+ J_+ , -^=J-.^ , &c. If the coefficient Au of a;**, in what is said above, be other than unity, then — must An N N be written for iVin [1], that is, --r=(—a^{—aM—a^*... .*. An=^-, r, r, ^ — £n (— rA^h'^-'^y-\-Nh''=:Q. The n values of y in this equation are ka^, ka^,...ka„, because y=kx. Hence we see that an equation is transformed into another, each of whose roots shall be k times as great, by multiplying the terms in order by 1, k, k^,...k''. A ready method is thus suggested of removing fractions from the coefl&cients of an equation : we have only to find a common multiple k of the denominators, and then to transform the equation into another whose roots shall be k times as great. This method may be employed with advantage whenever we wish to remove the leading coeffi- cient A,„ or to render it unity, and at the same time to exclude from the other coefficients the fractions which division by A^ usually introduces. If k= — l, the multipliers 1, k, k\ &c., become 1,-1, + 1, &c. ; so that an equation is converted into another whose roots are —1 times those of the former, that is, into one whose roots are the same in value, but contrary in sign, by multiplying the terms in order by I,— 1,-fl, &c., that is, by simply changing the alternate signs commencing with the second ; a truth already established otherwise at (141). Note. — Before using the multipliers, the equation must have its absent terms, if any, supplied by zeros. (1) x^-\--ax''-—-bx-\-c=0. Clear fractions, and keep the 1st coef. =1. The L.C.M. of the denominators is 6 .*. the multipliers are 1, 6, 6^, 6\ and the transformed equation is a;^-\-dax^—V2ba!-{-'l{Qc=0, the roots of which are each 6 times those of the proposed equation. (2) as^-\-^x'^—^x"— - A' -4- 10 = 0. Here also k=Q, and the transformed 2 8 6 equation is, therefore,a;H3^^— 12^^— 36^ + 12960=0, the roots of which are each 6 times those of the proposed equation. 144. Prop. 2.— To transform an equation into another whose roots shall be those of the original increased or diminished by a given quantity. For greater simplicity, suppose the equation to be of the fourth degree only, namely, A,x'-i-A.^^x''-\-A.X' + A^x-^N=0...[l] and that it is re- quired to transform it' into another, of the same degree, whose roots shall each of them differ from those of the proposed equation by r. In order to this, we have only to substitute x' + r for ii?, when the result- ing equation, ^y^+^>'^'-|-^'^:?/H^'i^'+^'=^-L''^] will have each of its roots greater or less than the corresponding roots of the proposed equation by r, according as r is negative or positive ; for x'=x—r. Ine following is an easy way of arriving at the coefficients of the transformed 110 TRANSFORMATION OF EQUATIONS. equation. Eeturn to the original by replacing ad by x—r\ we then have If we divide the first member of this identity by a;— -r, the remainder must be W : the division, therefore, of the second member must also give W for remainder, the quotient furnished by each being In like manner, dividing this quotient by {x—r\ we have A^ or re- mainder, and for a second quotient A,{x-TY^A!lx-r)-\-A'^ Again dividing, the third remainder is A\, and the corresponding quotient, Aix—r)-\-A'y And lastly, dividing this by a?— r, we have for the final remainder A\, and for the final quotient, A^. We see, therefore, that the several remainders, arising from these suc- cessive divisions of the original polynomial [1], by x—r, furnish, one after another, all the coefficients of the required transformed equation [2]. A very expeditious method of executing these several divisions is explained at (138) : we shall give a few applications of it to the present problem. (I) Transform a^-2x^-2Zx-^eO=0 into an equation whose roots shall be the roots of this increased by 2. l-2-23+60(-2=r -2+ 8+30 _4-15+90.-.jy'= 90 -2+12 -6- -2 .-. A\=-Z -8 .-. A\=-S Hence the transformed equation is a;'3_8a/2_3a/+90=0. ('2) Transform the equation 2a*- 15:^3+ 40x2- 45a;+ 18=0 into one whose roots shall be the roots of this diminished by 4. 2_15+40-45+18(4=r 8-28+48+12 -7+12+ 3+30 .-. N' =30 8+ 4+64 1+16+67 8+36 9+52 8 17 /. ^'i=67 .-. ^' =17 Hence the transformed equation is 2a/*+17a;'3+52x'2+67a/+30=0. Note. — The operation may be slightly modified as follows : — (3) Transform the equation 6x^-Bx^-]-4x-l=0 into one whose roots shall ea!ceed the roots of this by 3. 6 _ 3 + 4 - l(-3 -18 63 -201 -21 -18 -39 -18 67 117 [formed eq. .-. 6a;'3-57a;'2+184a/-202=0, the trans- (4) Transform 2a;4-13a;2+10a;-19=0 into an equation whose roots shall be less than the roots of this by 3. 2 -13 + 10 -19(3 6 18 15 75 6 6 12 6 18 6 5 36 41 54 25 123 .-. 2a;'H24a;'3+95a;'2 + 148:r'+56=0, the transformed equation required. TRANSFOEMATION OF EQUATIONS. JH 145. By the preceding process, it is easy to transform an equation into another in which the second term shall be absent, for we see, in diminish- ing the roots of an equation of the nth degree by r, that r times the first coef , added n times to the second coef , gives the second coef in the transformed equation. In order, therefore, that this may be 0, we have only to satisfy the condition wr^„-}-^„_i=0.-.r= ~. Take, for in- stance, the equation ar^—9x^^ + 20^— 12=0. The transforming multiplier 9 that will effect the removal of the second term, is r=-=3, which, em- o ployed as in the preceding examples, conducts to the equation x'^-\-Oa/^-~ 7^'— 6=0, or ^'•*— 7a:'— 6=0, of which the roots are those of the pre- ceding equation, each diminished by 3. Again: take 2^^-|-24a7^ + 95;c' + 148^ + 56=0. (See ex.4 above.) The transforming multiplier is r= 24 — —=—3, which, employed as above, furnishes the transformed equa- tion ^a/^—lSa^'^-\-10a/—19=0, the roots of which are those of the former equation, each increased by 3. 146. Pkop. 3. To transform an equation into one whose roots shall be the reciprocals of the roots of the former. In order to this, it is neces- sary merely to put - for a in the proposed equation : thus, writing as follows, N+A^x +A^^ +A^ + ■\-AnX^=0 , ,, , ^1 . -^2 ■ ^3 , .An - wehaveiVr+- +^ +^ + + - =0 .•.iV2r+^.r-'+^22/"~'+^32/"~'+ +^n =0 [2] the equations whose roots are the reciprocals of those of [I]. It appears, therefore, that, by simply reversing the order of the coefficients in [1], we obtain an equation [2], the roots of which are the reciprocals of those of [1]. 147. Should the coefficients in [1] be the same series of numbers when taken in reverse order, as when taken in direct order, as, for instance, in the equations jl^a^+A,x'-\-A^^+A^^+A,x+A,=0, A^+A,x'+A^x^-{-A^^-\-A,x^-\-A,x+A^=0, then the reciprocals of the roots are themselves roots ; that is, if a^ be a root, then also — must be a root. Such equations are called reciprocal equations, in reference to this property of the several pairs of roots : they are also called recurring equations, in consequence of the recurrence of the coefficients. 148. When the equation is of an odd degree, like the first of the pre- ceding, it is still a reciprocal equation, though the equal coefficients have opposite signs, for in this case the coefficients of [2] become the same, upon changing all their signs, as those of [1]. Unity, with a sign oppo- site to that of the last term, must always be one root of a reciprocal 113 TRANSFORMATION OF EQUATIONS. equation of an odd degree ; because if this value be put for x, the terms with equal coefficients will have opposite signs, and therefore the poly- nomial will vanish. 149. When the equation is of an even degree, and complete in all its terms, it can be a reciprocal equation only so long as the equal co- efficients have the same signs; for here there is a middle term, so that if the equal coefficients have opposite signs, it is impossible, on account of this middle term, that the row of signs, taken in reverse order, can be the opposites of those taken in direct order ; and therefore the two rows cannot become alike by changing all the signs in one row. 150. But if the middle term be absent, then the obstacle is removed; and the equation is reciprocal, whether the equal coefficients have the same or opposite signs. And in this case, + 1 and — 1 are both of them roots, if the last term be negative ; but neither is a root if the last term be positive ; for, in the former case, the signs of the equal coefficients being opposite, and the powers of a; connected with them being both of an even or both of an odd degree, the terms must vanish equally for x=l, and x= — l. But for neither of these values can they vanish if the last term be positive, and therefore the signs of the equal coefficients like, as is obvious. Examples for Exercise. (I) Transform x^—^x^—x+^=^0 into an equa. whose roots are 3 times as great. . (2) Transform ^^— 28^-f48=0 into an equa. whose roots are only half as great. (3) Transform the equa. Qx^—^x^'+Ax—l into one with roots each less by 3. (4) Transform the equa. 12ar^-i-24a;'— 58a;+25 = into one with roots each less by -5. (5) Transform the equa. 19^*— 22;»^— 35a;-— 16;c— 2=0 into one with roots less by 3. (6) Change a:'^ + 6x'^-\-6a-\-l=0 into an equa. wanting the second term. (7) Remove the second term from the equa. £c'''-\-A^a;-\-N=0. (8) Remove the second term from 2^*— 16:c"^H-43a;'— 41a;-|-5=0. (9) Find the equa. whose roots are reciprocals of those of 56a:'^ + 34^Ha;— 1=0. (10) What equation is that whose roots are 1, 2 and 3 ? (II) One root of aj"*— 7^-|-6=0 is 1 : find the remaining roots. (12) One root of a;^— 5a;^— 18;b + 72=0 is —4: find the remaining roots. (13) Three roots of x^ + 6x^-\-x'^—lQx^—^0x—ld=^0 are 2, —2, —4: prove that the remaining roots are imaginary. (14) Find the equa. whose roots are the reciprocals of the roots of a?*— 12:c' + 12;»— 3=0 diminished by 1. (15) Find the three cube roots of unity, that is, solve the equation a?^-l=0. (16) Find the four values of iy !> that is, solve the equation a;*— 1=0. (17) Given the equation a;^— 3a?' -f 2a;- -|- 7a;— 5 = 0: write down another whose roots shall be the positive roots of the former changed into negative, and the negative into positive. LIMITS OF THE REAL ROOTS OF AN EQUATION. 113 (18) If four roots of the equation A,x''-i-A^a;^-{- ..,=0 be a^, a.^, a^, a,, what will the fifth root a^ be equal to ? -5r2 -1 151. Limits of the Real Roots of an Equation.— If in an equation in which the second member is zero, a positive number r be substituted for a?, such that it, and every greater number, gives a positive result, it is plain that r will be greater than the greatest root of that equation ; that is to say, r will be a superior limit to the positive roots. If, therefore, all the terms of an equation are plus, all positive numbers from a?=0, to a;= oo must give a positive result; and therefore, as indeed was already seen at (137), the equation cannot have a positive root. On this fact is founded 152. Newton's method of finding a superior limit. For x, in the proposed equation, he substituted a;'+r, that is, he diminished the roots each by r, and then sought for the smallest positive integer value of r that would render the signs of the transformed equation all positive, and consequently its real roots all negative. Such a value of r would ob- viously be a superior limit to the positive roots of the proposed equation, seeing that if each be diminished by r, the remainders are all negative. As an example of this method, let it be re- quired to find a superior limit to the positive 0=a/^+Sr\x'^-hBi^x'-{- r^ roots of the equation ;c^—5;»'' 4- 7a;— 1=0. Sub- ~ ' X 7^ 8tituting;i/4-^for^» the transformed equation is that in the margin ; and after a few trials, we find that 3 is the smallest integer which, put for r, causes all the co- eflficients to become positive; .-.3 exceeds the greatest positive root of the equation. This method of Newton readily leads to that of Maclaurin, which is free from trials. 153. Maclaurin's Limit. — When the leading coef. is unity, the greatest negative coef., taken positively, and increased by unity, will be a superior limit to the positive roots. For let r be equal to the greatest negative coef., taken positively, and increased by 1 : then r is > 1, and diminishing the roots by r, as already explained, 1 ^„_, ^„_2 ^1 ■ N {r it is evident that, whether J„_, be the greatest negative coef. or not, the number r, added to J„_i, will give a positive result not less than unity : this result, multiplied by r, will therefore give a quantity, not less than r, to be added to A„_^; the result, therefore, whether An-^ be the greatest negative coef. or not, is positive, and not less than unity ; and so on. Hence all the results in the first row of operations are positive. Of necessity, therefore, all the results of the second row, and of the follow- ing rows, must be positive, since only positive quantities are henceforth added. The coefficients of the transformed equation are, therefore, all positive, so that r is a superior limit to the roots of the proposed equation. . , 154. When the second is the greatest negative coefficient, Maclaurms, as a general method of finding a superior limit, is the most convenient that has been proposed. But when the third coef. is negative, whatever the preceding may be, and all that follow positive, a much closer superior 114 LIMITS OF THE REAL EOOTS OF AN EQUATION. limit may be discovered as follows : Writing the three leading coefficients only, as in the margin, An±An^i —An-i{r all we have to do is to find the stflallest positive Anv {Anr+An~^)r value of r that will render the second result Anr±An~i positive positive ; the first result, for such a value of r, will necessarily be positive, as likewise all the following results, inasmuch as all the original coefficients after the third are positive. Hence, to find a superior limit r, we have only to determine this value so as to satisfy the condition A,y^±.A,^^{r=. or >-4„_2. Take for ex. 3;c"^— 2a;'— llaj+4=0 : then we must have 3r^— 2r>ll .*. r=3 is a superior limit. 155. But even should the coefficients following the third be not all positive, this method will still be available : for ex., take the equation Proceeding as above, we have to determine r from the condition r^+ r>15 .*. r= an integer >3. Trying 4, and proceeding with the subse- quent coefficients as in the margin, observing that for r=4, r-+r— 15=5, we see that the results which follow will all —19—3(4 become positive : hence 4 is a superior limit. The superior ^ ^ limit, determined by Maclaurin's method, would be 20. j ^ Lastly, let the equation be 2;»^ + 11:b^— 10;»^— 26;^;^-f 31;^;H'72^'^— 230a;-348=0. The condition 2rHllr> 10, gives r=l, and 2r^ + llr=13, and since 13—10, or 3, multiplied by 1, does not amount to 26, r is too small. For r=2, 2r- + llr— 10=20; trying, there- fore, this 2, in reference to the coefficients after the third, we have -26+31-1- 72-230-348(2 ' 40 28 118 380 300 14 69 190 150 -48 From these results, it is plain that 3 is a superior limit, and that the first figure of the greatest positive root of the equation is 2 (art. 137). It is always easy to see at a glance, when the left-hand member of the inequality is computed, whether the result, under the negative coef. with which we are dealing, is sufficiently great for its product by r to exceed the next coefficient, should this be negative. The foregoing remarks and illustrations will sufficiently prepare the student for the following general rule for finding a close superior limit to the positive roots of a numerical equation. 156. General Kule. 1. Take the ^rsit negative coef. — that in the term —A^x^ suppose, and subtract k from the exponent, and from all the preceding exponents ; then, disregarding all the following terms, find the. least integer value r oi x that will satisfy the condition .4„aj"~*-f ....>^a, which it will not be difficult to do, since all the terms on the left are positive. 2. Use the value of x thus found for a transforming multiplier, and get the first row of results : if these are all positive, the number used will be a superior limit ; but if not, complete the transformation : if the results of this are all positive, the number will be a superior limit. 3. If negative signs occur in these last results — which, however, can happen only in the more advanced of them, proceed exactly as before — taking the first of the negative coefficients, and determining the least integer / for of -, and so on, till a row of positive results is obtained ; then the number r +/-]-..., which is the sum of all the transforming multipliers, will be a close superior limit. Note. 1. As exemplified above, it will always be best, when the tliird LIMITS OF THE EEAL ROOTS OF AN EQUATION. 1J5 coef. is negative, even should the second be negative also, still to deter- mine the transforming multiplier from the condition /4„a;"^±^„_,a'> J^„_2. The equation y4„aT'±^„_,a;— ^„_2=0 has necessarily but one positive root (68), so that if this condition be satisfied for a;=r, r must be such as to render the first two results positive. 2. When the second coef. is negative, and the third positive, it will still be better to deduce r from the like condition ^^a?-— ^„_,;c+.4„_2>0. But the value r of x must not be taken less than i(-^„_i-H^„), otherwise the first result in the second row will be negative. Should, however, the roots of the quadratic be imaginary, that is, should 4:A„A„_.> A''„_i, then we may disregard the above condition, using at once the transforming multiplier r=^(An-i-7-A„), since, for this value of r, the first term in the second row of results is ; and the second in the first row is + for all values of y (68). It may be noticed too here, that in the foregoing conditions for determining r, the symbol > may always be replaced by =. (1) Let the equation be a:*+lla;2-25a;-67=0. Here the condition is 1+0 +11 -25 -67(2 30 2 15 5 2 8 46 4 23 61 2 12 10 ■57 35 as we had reached the positive results 4+ 23+51, for these alone exceed 57. (2) Let the equation be a;5+ 7a;* - 12a^ -49a;2+ 52a; - 13=0. Here the condition is a;2+7a;>12 .-. a;=2 1+7-12-49+52-13(2 2 18 12-74-44 9 6-37-22-57 2 22 56 38 11 28 19 16 Now, without proceeding further, we see that, as the sum of the positive re- sults already arrived at exceeds 57, the completed results would give a/=l : hence 3 is a superior limit, and the greatest root of the equation must lie between 2 and 3, inas- much as for a;=2, the polynomial is nega- tive, and for a;=3, it is positive (137). 8 .-. a/*-^Sx'^+S5x'^+51x'>57 .'. a/=l. Hence 2+1=3 is a superior limit, and the greatest root lies between 2 and 3. And this we might have inferred as soon (3) Let the equation be ajM-16;?r^— Sa;*^— 12^— 6=0 .*. «Hl6a;>2 ,\ x=^ ; and proceeding as in the margin, we see that all the results are positive, except the last, and that their sum exceeds that last. It is unnecessary therefore to complete the transformation, which would necessarily give xf=l : we conclude, therefore, that the greatest root lies between 1 and 2. (4) Let the equation be ;b*--8;b^ + 14;2;- + 4^— 8=0. Here the condition is a;'^—8a}-\-'\4:>0, and takings? not less than 4, (156) .-. x=Q, and proceeding as in the margin, we soon have evidence that 6 is abund- antly large. Transforming again by 5, the final result of the first row is —-13; we conclude, therefore, that the greatest root lies between 5 and 6. As the smallest integer fulfilling the first condition here is 6, the restriction " x not less than 4" is superfluous. I 2 1^-16-2-12-6(1 1 17 16 3 17 16 3-3 1-8+14+ 4- 8(6 6-12 12 96 -2 6 2 16 88 l_8+14+4- 5_16_5_ 8(5 6 ■ 3 -1-1-13 116 LIMITS TO THE NEGATIVE ROOTS. (5) Take the equation ;»*- 8^*^ + 20^*^ -14^- 9=0. 1-8+20-14-9(4 Here 4x20>8^ therefore (156) r=4. Proceeding _^~^ ^ _^ as in the margin, we see that as the positive results _^ ^ 2— i 4 + 18 exceed 1, 5 will be a superior limit: hence the 4 16 greatest root lies between 4 and 5. And this we might - - — have inferred even from the first row of results. 4 18 On account of the importance of the present problem in the solution of numerical equations, we shall give one example more. Required the first figure of the greatest positive root of the equation Now, whether we take the condition x=10 by the i_io+4— 20— 78(10 rule (for > may always be changed for =), or the 10 40 condition in the Note, the same value is suggested, — - — and the operations are as in the margin, where it is ^ ^ ^^ seen that for a; =10, the last result is +, and that i_io+4— 20— 78(9 for ^=9, it is — : the greatest root lies therefore 9—9 between 9 and 10, that is, the first figure of it is 9. -1-5 157. Inferior Limit. — To find an inferior limit to the positive roots of /(a;)=0, all we have to do is to take the reciprocal equation, that is, to reverse the coefficients (146), and to proceed as above to find the superior limit, the reciprocal of which will of course be the inferior limit to the positive roots of the proposed equation. And in this way we may always discover the extreme limits within which all the real positive roots of an equation lie. 158. Limits to the Negative Roots.— As to the negative roots, we know (141) that they are converted into positive roots by simply changing the alternate signs of the terms : if therefore we make this change, and then determine the limits of the positive roots, as above ex- plained — these limits, with the negative sign prefixed, will be the limits of the real negative roots, or the extremes within which they all lie. If a superior limit of the positive roots of /(a;)=0 be 0, that is, if for a}=0, and for all higher positive values, f{x) be positive, we infer that there are no positive roots. In like manner, if be a (negatively) superior limit to the negative roots, that is, if for x=0, and for all higher negative values, /(^) be positive, the equation can have no negative roots, so that if both conditions have place, we conclude that all the roots are imaginary. But more precise information about the number of real roots may be ob- tained from arts. (163), (164), &c., following. 159. Peop.— Let be any equation, and let the following, of inferior degree, be derived from it by the uniform process of multiplying each term by the exponent of a?, and then diminishing such exponent by unity, nAnX^-'i-[-(n-l)An-^x''~^^\-{n-2)An-2X^-^-^..•+ZArjX^-{-2A^-\-Al=0...[2l Then the real roots of this latter equation will all lie between the real roots of the former. [The latter is called the limiting equation to the former : — it is also called the derived equation.] The polynomial forming the first member of [1] is f{x)={x~aMx-a^{x-a.^{x-a,)...{P)...[d] THE LIMITING EQUATION. 1 1 7 where P represents the product of the quadratic factors involving the imaginary roots, should any enter, which product we know (139) to remain positive whatever value be given to x; we shall also regard the real roots ttj, a.,, &c., to be arranged in the order of their magnitude, a^ being the greatest root. Now we have seen (pa. 103), that if any polynomial /(^c) be divided by a— a, and if a be put for x in the quotient, Q, we shall have Q=n^„a"-'+(7i-l)^„^,a«-2-f (7i-2)A^£a"-3+...+3^3aH2^2a+^i---[4] whatever be a*. Let a=a^, then [3], since, when a^ is put for x, Q is also Q=.{a^—a^{a^—a^{ai—a^...{P), in whicli, by hypothesis, all the factors are positive, it follows that when a^ is substituted for a in [4], or which is the same thing, for x in [2], the result is positive. Let az=zac,, then [8], for 0;=^^, Q is Q=^{a^—a^{a2—a^{a2—a^...{P)y in whicli, by hypothesis, the first factor is negative, and all the others positive : hence, when a.^ is substituted for a in [4], or which is the same thing, for x in [2], the result is negative: hence a real root of [2] must lie between a^, and a.,. Again \Qta=a^, then [3], Q=:{a.^—a^)[a^—a.^){a.^—a^) (P), in which the first two factors are negative and all the others positive : the product is .*. positive: hence when a.^ is substituted for a in [4], or for x in [2J, the result is 'positive, .'.a real root of [2] must lie between a.,, and a.^. And so on : therefore the derived or limiting equation [2] has at least as many real roots, wanting one, as the primitive equation [1], so that if all the roots of [1] be real, all the roots of the derived equation [2] must be real. It follows, moreover, that if imaginary roots enter [2J, the same number, at least, must enter [1]. 160. If a positive number greater than a^ be substituted in [1], the result will be positive; and if a root of [2], lying between a^ and a^, be substituted, the result will be negative; if again a root of [2], lying be- tween a^ and a.^, be substituted, the result will be positive ; and so on. Consequently, if the real roots of [2] be substituted, one after another, in [1], the changes of sign in the results will make known, not only the exact number of real roots in [1], but also between what two numbers each root lies. 161. We may add here that the function of x derived from f{x\ as above, is usually called the first derived function, and is marked /X^'). ov f\x) ; the second derived function, deduced in like manner from this, is marked/2(a:), orf'\x); and so on. The function [2] above iaf^{x), which, if we put x=0, reduces simply to A^ ; the function fj[x) (see [3] p. 105), for x^O, reduces to ^A^y the next function /^(a;), to ^.SA^; and so on: consequently, the polynomial f(x)=N+A^x+A^x'^'\-A^x''+A,x^-\- -f-A*" * We may arrive at this conclusion a little differently, thus :— Taking for convenience a polynomial of the fourth degree only, namely, A^x*-\-A.iOi^-{-A2i^-^A^x-[-^f and di- viding by a;— a, after the manner already explained, we have A, -{- A, -{-A, + AM A^a A^o?-\-A:fk Aifi^-\-A^(^-\rA^a q=A^s^-^{A^a^-A>ix^-\-(^A^a?^-A^a^-A^x-\-A^a?-{-A^a^-\-A^-\-A^. Putting a for x, this becomes iA^Q?-\-^A^o?-\-1A^{n-l)A\^i [1'] or, indeed, provided they satisfy it when = is put for > . If the order of the coefficients be reversed, we shall have an equation the roots of which will be the reciprocals of those of the original : the existence of a pair of imaginary roots in either implies, therefore, the existence of a corresponding pair in the other. The criterion [I'J may, therefore, be applied to the last three coefficients of [1], as well as to the first three ; so that a pair of imaginary roots will be also indicated, pro- vided that 2nAoA^>{n-l)A^^ [2']. 172. Now we know (p. 117), that if a limiting or derived equation have ima- 2nAoA2> {n—l)A^i [1] ginary roots, the primitive equation S{n—l)A^Ar^>2{n—2)A% [2] must also have imaginary roots : taking 4(?i— 2)^2^4>3(7i— 3)^% [3] therefore the several limiting equations 5{n-d)A^A^>i{n-4)A\ [4] derived one after another, each from the „ . j '< /^ in ^2 u n preceding, [1] being the primitive, and 122 Newton's bule for imaginary roots. applying the criterion [2'] to the last three coefficients of each, we shall arrive at the series of conditions in the margin, and the existence of any one of which will imply the existence of at least one pair of imaginary roots in the equation [1]. The criteria here established pave the way for the proof of an im- portant rule, first given by Newton, but given without demonstration ; it may be established from the following considerations. 173. Newton's Rule for Imaginary Roots.— -From the general equation ^„a;"+ +A^a^+A^3i?-{-A^'^+AiX-\-Ao=i0y the following series of derived equations are deduced, namely, nAnX^'-^-i- +4^4x3+3^3^^4-2^^+^1=0 w(u-l)^„a;»-2+ +4.5^5CB3+3.4^4a;2+2.3^3X+2^2=0 w(7i-l){^-2)^„a;«-3+ -\-i.5.6As3^+ZA.5A^z^-\-2.dAA^z+ZZA3=:0. &c. &c. &c. And if any of these have imaginary roots, we know that as many, at least, must enter the primitive : the same remark applies if we reverse the coefficients of each, as also if we take the limiting equations derived from them when the coefficients are thus reversed. Reversing, then, the coefficients of each equation, commencing with the primitive, we readily see that the derived cubic equations, or equations of the third degree, are as follows : — 4.5 nAo^-{-SA n-lA^x^-}-2.Z {n-2)A^-}-2.Z (71-8)^3=0 4.5. ..(71-1) ^ix3+3.4 (71- 2) 2^33;'^+ 2. 3 {n-B)SA^z-\-2.d (71-4)4^^=0 4.5. ..(7i-2)2^2a;3^3.4... (71-3)2.3^3X2+2.3.. .(7i-4)3.4^4a:+ 2.3.. .(71-6)4. 5^5=0. &c. &c. Or, expunging the common numerical factors, they are n{n-l){n-2)A^a^+S{n-l){'n-2)A^x^+2.Z{n-2)A^-\-2.SAs=0 (n,-l){n-2){7i-Z)AiX^-{-Z{n-2){n-2,)2J^+2.d{n-d)SAsX+2.dAA^=0 (7i-2)(7i-3)(7i-4)24^+3(7i-3)(7i-4)2. 3^3^2+2. 3(71-4)3. 4^ 4X+2.3.4.5^5=0. &c. &c. Now, if any of these limiting cubics indicate imaginary roots, when tested by the criteria [1'], [2'], at p. 121, we may be sure that imaginary roots exist also in the primitive equation. But since only one pair of imaginary roots can enter a cubic equation, it follows that whether the criterion of imaginary roots be satisfied by the three leading terms of any of the above cubics, or by the three final terms, or by both sets of three — one pair of imaginary roots, and one pair only, is implied. Now, upon examining the coefficients of the above cubics, we see that the first three terms of each have a common factor, so that in applying the criterion to these, this common factor may be suppressed. Let us suppress the common factor, namely (n— 3), in the second cubic : then the product of the first and third coefficients will be 2.3^(7i— l)(n— 2) A^A.^, and the square of the middle coef. will be ^~.S\n—2f A^. But these are the same as we should get by employing, in like manner, the last three coefficients of the preceding cubic : and we see the same to be true of the following cubics, that is, if the criterion of imaginary roots is satisfied by the last three terms of one cubic, it must be satisfied by the first three of the next, and vice versa', so that the fulfilment of the condition by two newton's eule for imaginary roots. ]S3 consecutive sets of three terms, implies but a single pair of imaginary roots. We thus arrive at the following conclusions, namely : — 1. If the first three terms in the first cubic satisfy the criterion, we infer the existence of one pair of imaginary roots. 2. If the next set, the three final terms of the same cubic, also satisfy it, the preceding condition merely recurs, and supplies no additional information. In this case the following set of three — the leading terms of the next cubic — must of necessity furnish the same concurring con- dition, and so on, till we arrive at a set of three terms for which the condition /(di'/s, thus putting a stop to the series of concurring indications, and preparing the way for new and distinct conditions. 3. So soon as the criterion is again satisfied, the condition being entirely independent of, and non-concurring with, the former, must imply another and distinct pair of imaginary roots. And so on, to the end of the series of cubics. The criterion which we have here supposed to bj applied to the terms, taken three at a time, of the successive cubics — namely, either [!'] or [9/], at p. 12], supplies, one after another, the entire series of criteria there given, as we shall presently see. Bat as the three final terms of any cubic always furnish the same condition as the three leading terms of the next, the repetitions may be omitted. Attending to this, and applying the criterion to each of the foregoing cubics in succession, we have the following conditions for imaginary roots, namely : — or, suppressing com. factors, 2n,i4(,^ 2^ (^~l)^i'* [!]• Slid. 22.33(7l-l)(7l-2)^,^3>23.3>-l)2^a2. or, suppressing com. factors, S{n—l)AiA3>2{n—2)A\ [2]. 3rd. 2KSM{n-2)in-S)A^A^>2\^\n-d)^A\; or, suppressing com. factors, i{n—2)A2A^'>B{n—S)A^3 [3]. 4tli. 23.3*.4.5(;^-3)(»l-4)^3^5>2^3^4V-4)M24; or, suppressing com. factors, 5{n—d)A^A^>i{n—4)A^^ [4]. &c. &c. And thus, as stated above, are we led to the series of criteria at page 121 ; and which we now know to be such that if, when proceeding from one set of three terms in an equation to the three next in order, the consecutive criteria both have place, the concurrence is to be regarded merely as a second indication of the same thing — the existence of a single pair of imaginary roots ; but that so soon as the condition fails, preparation is made for a new and independent indication, and so on, till all the sets of three have been submitted to the tests. And this is substantially the rule of Newton.* 174. In employing the criteria [1], [2], &c., at page 121, we may consider the symbols A^, A^, A^, &c., to represent the coefficients taken in order either from the beginning or from the end. As an example, let the equation x*— 4=0}^ + 8x^—1^0! -\- ^0—0 be submitted to examination. Taking —iai^ for the middle term, x* and 8x- being the adjacent terms, the condition [1] is satisfied. Taking 8^- for the middle term, the condition [2] fails. And lastly, taking —IQx for the middle term, the condition * The above investigation of Newton's rule, here slightly modified, was first published by the author of this work in 1844 in his " Researches on Imaginary Roots," forming an Appendix to the " Theory and Solution of Equations of the Higher Orders." ]24 IMAGINARY ROOTS OF A BrQUADRATIC. [3], or which is the same [l], is satisfied: hence all the four roots are imaginary. Note. — The student must bear in mind that although we may always be certain of the existence of imaginary roots when any of the criteria are satisfied, yet that we cannot be sure of the non-existence of such roots when the criteria are not satisfied. If imaginary roots are indicated, we may safely infer that so many, at least, enter the equation : — if the criteria are all unsatisfied, we cannot afiirm anything as to the character of the roots : — the quadratic equation alone excepted. The series of criteria at (172) may be very readily written down, without reference, if the first only be remembered, and this will be suggested by the well-known criterion for the quadratic. In those which follow the first, the numerical factors regularly increase by unity, and the literal factors as regularly decrease. It is plain, too, that any set of three coefficients may be deprived of any factor common to them. 175. Theorem.— If in the biquadratic, or equation of the fourth degree, the condition {A:A^A2—A'^.^N>A^A'^^ is satisfied, then all the roots of the equation are imaginary. Multiply by .4^, and extract the square root : — " A^A^a?-^A^A^x^ 2A,x^^\a^^ A^A^-\- - A\x^ Qi,A,-^A''^x^+A,A,x^A,N. Consequently the first member of [1] is made up of the two compound terms (^^4;rH^^3^y+{(^4^2-^^'3)^+^4^1^+^4^}- Now if that within the braces be always positive, whatever real value be given to x, the whole expression must be always positive, because the first compound term is a square. But the second is always positive for real values of x, if a(a,A^-\a'^A^N>A\A\ or if {iA,A^-A^^N>A,A\...[^l Hence, when this condition is satisfied, N being positive, [1] cannot become zero for any real value of x; that is to say, all its roots are imaginary. If .^4=1, the condition is {iA^—A'^^N.>A'^^, and if, in addition, ^3=0, it is simply AA^N>A\..m. 176. It is well worthy of note that, whether A^ be absent or present, and whatever be the degree of the equation, that equation cannot have a real root within any of the intervals throughout which the portion of the equation preceding A^"^ is positive, whenever the condition [3] is satisfied ; and this condition, for N positive, may always be brought about by in- creasing A^jxi' by some multiple of itself, and then diminishing the pre- ceding terms by the same quantity. If N is negative, the condition may SOLUTION OF NUMERICAL EQUATIONS IN GENERAL. 125 be made to have place by diminishing A.^x^, instead of increasing it ; but the second portion of the polynomial will then be negative for all real values of x, so that real values will be excluded from the intervals within which the first portion is always negative also. Should the two portions of the polynomial, when separated by the negative sign, be both always posi- tive, it will in general be easy to see within what limits one always exceeds the other : — there can be no real roots in the interval. (See the examples at page 133.) Note. — Although in the above we have considered the condition to be uniformly secured for the last three coefficients, yet it may be equally brought about for the first three; and this will sometimes be found to be the more effective. (See ex. 10, p. 133.) 177. Solution of Numerical Equations in General.— Although the process about to be explained applies equally to all nume- rical equations, as will readily be perceived, yet for simplicity sake we shall at present confine ourselves to equations of the fourth degree. Let the successive figures of one of the roots of the equation be r, /, r", &c. When the first figure r is discovered (see art. 156), we may readily obtain the transformed equation involving the remaining portion of the root, that is, the root of which the figures are /, /', &c. When the first figure / of this is found, we can arrive at a second transformed equation A^x"*-\-A"^"^+A"^"^+A\x"-\-N"=z(i...\Z] in which the first figure of the root we are tracing is /' ; and so on. Now the leading figure r of the root having been found, and the trans- formation [2] obtained, we may avail ourselves of r to discover /, the next figure, much in the same way as, in the extraction of the square and cube roots of numbers, the leading figure of the root supplies a trial divisor for the discovery of the next figure. Thus, transposing N' to the right, and regarding it as a dividend, the first figure / of the root of [2] must be such that when the dividend is divided by A/^-\-A'./^-]rA\r^-\-A\, the leading figure of the quotient may be /, for the finding of which we are assisted by the known value A\ — the partial or trial divisor. That this, in general, forms an important part of the true or complete divisor above, will appear from reflecting that the composition of A\ is A\=f,{r)=.iA^^+^A^r^-\-2A^r+A, (p. 118), and that r is a place higher in the arithmetical scale than /. The trial divisors A'\, A"\, &c., for finding the subsequent figures /', r"\ &c., of the root, as in the case of the operation for the cube root already al- luded to, will become more and more efficient as the work proceeds. An example or two will make this apparent. (1) Required a root of the equation a;^ -f 8;»- -f- 6;??— 75-9 =0. This equa- tion has a positive root, because the last term is negative (140), and it is easily seen that this root lies between 2 and 3 (156). The first trans- forming factor, or the first figure of the root, is therefore r=2 ; and avail- 126 SOLUTION OF NUMERICAL EQUATIONS IN GENERAL. iiig ourselves of the several trial divisors A\, A'\, &c., the successive steps of the work, for finding /, /', &c., are as follows :— 43 A, 1 8 2 10 2 6 20 26 24 60=^', 5-76 N -76-9(2-4257 62 - 23-900=iVr' 22-304 r=2 /=-4 12 2 l-596000=ir 1-239688 r''='02 \4:=A\ •4 65-76 6-92 •356312000=iV" •311827625 r"' = -005 14-4 •4 61-68=^''i •3044 44484375000=iV"" 43716797593 r""=-0007 14-8 •4 61-9844 •3048 767577407=iV""' 16-2=^% •2 '62-2892=^" 76325 15-22 2 62-365525 76350 15-24 2 15^26=il'% 5 15-265 5 16-270 5 15-275=^' 7 15-2757 7 15-2764 7 62-441876=^'"', 1069299 62-45256799 1069348 62-46326147=^' The transformed equation at wMch we have now arrived is a" "'3+ 15 -2771a;" "'2_i_62 -46326147a;'""= '000767 And by regarding the small decimal •000767... as 0, we have the depressed quadratic equation a;'""3+ 15 •2771a;"""2+ 62 •463=0 involving the remaining roots diminished by 2 •4257. But since 4x62'463>16^2771^ these two remaining roots are imaginary. 15-2771=^"'"2 The depressed quadratic, obtained as above, ought not be regarded as involving any practical error because we have replaced the small value W" by 0. We have stopped the development of the root of the cubic at the fourth place of decimals, considering Q^dSST to be an approximation to that root sufficiently close to be regarded as the root : it is, however, in strictness, the complete root of ;c"' + 8a;^-|-6:c=75-9 — -000767... Conse- quently, in thus putting a termination to the development, we have tacitly assumed -000767,.. =0; we are, therefore, quite consistent in adhering SOLUTION OF NUMERICAL EQUATIONS IN GENERAL. 127 to this assumption ; but the roots of the depressed quadratic, had they been real, could not have been obtained true beyond the fourth decimal place. For the extent to which the root of the cubic has been carried, it is plain that several of the right-hand figures in each column of the work are superfluous : we may preclude the entrance of these useless figures thus : Having arrived as above at that absolute number N\ or N'\ &c., which contains as many figures plus one as there are root-figures still to be determined, work, as in contracted division, so as to provide against the entrance of figures in the last column beyond the limit thus fixed for the absolute number. It is scarcely necessary to mention that the carry- ings from rejected figures are not to be neglected. Taking the preceding example, limiting the number of root-figures to five, the contracted form of the work will stand thus : — 8 6 75-9(2-4257 2 20 52 — — [1] 10 26 23-900 2 24 22-304 — -[1] [2] 12 60 1-596 2 6-76 1-240 -[1] [3] 14 55-76 •356 •4 6-92 [2] 61-68 •312 14-4 44 •4 •30 43 14-8 61-98 1 •4 \ [2] •3 15-2 [3] \ \ 62-3 •1 62-4 The numbers [1], [2], &c., mark Ike successive complete transformations. The three figures 15-2, at the bottom of the first column, and 62 -4 at the bottom of the second, remain constant for all subse- quent transformations : therefore the approximate quadratic a:2+16-2a;+62-4=0 would guide us sufficiently to the leading figure of each of the remaining roots, diminished by 2 -4257, of the proposed cubic, if those remaining roots were real. (2) Find a negative root of the equation a;^— 3a?+6=0 to six or seven decimals. This equation in a complete form is x^-\-Oa)^—Sx-\-G=0; and changing alternate signs (141), we have to find a i o — 3 —6(2 positive root of x^—Ox'—hx—Q=0. For the superior 2 4 2 limit r we have (156), aP—S>0 .'. r=2 ; and trans- - - — forming by this 2, as in the margin, we see that ^ 8 ~^ as 4 + 9 exceeds 4, the first figure of the root is 2. 4 9 Hence, arranging the given coefficients as in the preceding example, and checking the unnecessary accumulation of decimals as above, we arrive at the successive figures of the root, and the corresponding transformed equations, as follows : — 128 SOLUTION OF NUMERICAL EQUATIONS IN GENERAL. -3 6(2-3553014 2 4 2 _ - -[1] 2 1 4 2 8 3-267 _ -[1] [2] 4 9 •733 2 1-89 •660875 -[1] [3] 6 10-89 72125 •3 1-98 68014 [2] [4] 6-3 12-87 4111 •3 •3475 4092 6-6 13-2175 19 •3 •3500 14 —[2] [3] — 6-9 13-5675 5 5 353 5 6-95 13-6028 5 * 35 7-00 [4] 5 13^638 [3] \ 7-05 2 13-640 \ \ \ \ Hence the negative root of the proposed equation is — 2-3553014. The approximate quadratic is 0;- + 7*05^ + 13-64=0, the roots of which are imaginary. 178. In each of the preceding examples, it is the greatest root that has been sought for ; but it sometimes happens that a number greater than the greatest positive root of an equation may, nevertheless, be much too small to render all the terms of the equation which results from trans- forming by it positive, for variations may still exist which no longer imply real roots, but imaginary pairs. On account of such imaginary roots entering, the superior limit to the real positive roots, as determined by the rules already given — and which limit must overstep all the imaginary, as well as all the real indications — may be a number considerably greater than the greatest real root. The following is an example : — (3) Required a positive root of the equation a;^— 5:c^H-7a;— 1=0. A glance at the terms here is sufficient to show that if be put for a the result is —1, and that if 1 be put, the result is +9, so that a root lies between and 1. But the lowest 1—5 7 —1(2 transforming number that would give results all posi- _? ^ ? tive is r=3, as appears by the work in the margin, _3 1 1 which shows, however, that a root lies between and 2 —2 2, on account of the change of sign in the final term. — — Now the two variations still presented by the trans- —1 —1 formed results a/-^ -f sd'^—af + 1, in the last three terms, _ imply a pair of imaginary roots, as Newton's rule, 1 (p. 123), at once shows. The real root of the equation has been shown to lie between and I : its leading figure is, therefore, a decimal ; we may consequently use the SOLUTION OF NUMERICAL EQUATIONS IN GENERAL. 129 7 (=^i) as a trial divisor, and proceed to develope the root as fol- lows : — 1-5 7 •1 --49 -4-9 6-51 •1 --48 [1] -4-8 6-03 •1 --2784 -[1] -47 57516 6 --2748 ■4-64 5-4768 6 -32 -[2] -4-58 5-4736 6 -[2] -4-62 [3] 5-47 The approximate quadratic is a;^— 4-520? -f- 5*47=0, which verifies the rule of Newton as to the remaining roots being imaginary. But whenever the approximate quadratic is such that 4 times the product of the extreme terms is very nearly equal to the square of the middle term, it would be unsafe to pronounce confidently on the character of the remaining roots, seeing that, on account of the contractions introduced, the quadratic is only an approximation to the true quadratic. Take, for instance, the following example : — (4) Find all the roots — if real — of the equation ar'— 7a?+7=0. Sup- plying the absent term Ox', and changing the alternate signs, in order to convert the negative into a positive ^ ^ —7 —'^(3 root (141), we have a:H Oar— 7a;— 7=0, and since for ^ ? _^ a?*^— 7>0, we have r=3, we proceed as in the margin. 3 2 —1 And as l-j-3 + 2>l, we conclude that the positive root lies between 3 and 4, and we then develope that root as in next page. [The student is recommended to pay special attention to the remarks subjoined to the developments at pages 130, 131 : to notice first, how the determination of one root of an equation, by the process exhibited, con- ducts to an equation, a unit lower in degree, from which the character of the remaining roots, or the first figure of each of them, when they are real, may be ascertained ; and second, when any doubt exists, as to whether a pair of these roots is real or imaginary, to observe that the true character of the doubtful roots may always be discovered by proceeding to approximate to one of them as if it were real, by aid of the first figure, which, on the hypothesis of its reality, that root must have. There is a theorem — Sturm's Theorem — which enables us to dis- cover the character of roots indicated in any interval, with unerring cer- tainty ; but the numerical operations necessary for this purpose, although very simple in themselves, sometimes become laborious on account of their length. The author of the present work has endeavoured to reduce this numerical labour to its smallest amount ; and the inquiring student is recommended to make himself acquainted with this improved form of Sturm's Theorem from the chapters on that subject in the " Analysis E 130 SOLUTION OP NUMERICAL EQUATIONS IN GENERAL. and Solution of Cubic and Biquadratic Equations." All discussion of the theorem has been excluded from this work, because — not only of the additional space which such discussion would require, but because the character of the roots of an equation may be determined, as hinted at above, without its aid, and, usually, with less labour.] 3 6 3 •04 9-04 4 9-08 4 1 9-12 -[2] 9-128 8 9-136 * 8 9-14 -[3] -7 2 18 -[1] 20 •3616 20-3616 •3632 [2] 20-7248 73024 20-797824 73 20-871 8 20-879 \ \ \ [3] 7(3-048917 6 -[1] •814464 1 •185536 •166383 -[2] 19153 18791 362 153 146 [3] Hence thie negative root of x'^— 7a; -{-'7=0 is —3-048917; and the ap- proximate quadratic is aj'^ + 9* 14a; + 20-879=0. Four times the prod, of the extreme coefl&cients is 83^51... and the square of the middle one is 83-56..., results which imply a pair of real roots differing but little from each other. But, on account of the above curtailments of the decimals, it is possible that the roots, which thus appear to be nearly equal, may be imaginary. We proceed, therefore, to develope the remaining roots on the presumption that they are real and nearly equal. * Since the sura of the nearly equal positive roots of x^-\-0x^~7ic-\-7=O must be equal to the negative root taken positively, seeing that the sum of all three is 0, (142), half of 3-048... must be an approximation to each positive root ; so that both roots must lie between 1 and 2. A like in- ference is deducible from the approximate quadratic; for the half of —9-14 is —4-57, which increased by 3-04..., the quantity by which each root has been diminished, gives —1*5 3... as an approximation to each root when thus rendered negative. The developments are as follows : — * As observed in the foot-note at the bottom of next page, we could settle the doubt at once by anticipating the deductions at page 138, since the present equation is only of the third degree : for as ( - ) >^-^ , we conclude with certainty, from the page referred to, that the roots are all real and unequal. SOLUTION OF NUMERICAL EQUATIONS IN GENERAL. 131 1 1 1 2 •3 •3 3-6 •3 —[2] 3-9 5 3-95 5 4-00 5 1 4-05 [3] 4-066 4-062 ^ 6 1 4-068 -w -7 1 2 -[1] -4 -3-01 1-08 [2] -1-93 •1975 -1-7325 •2000 -7(1-356895 -6 -[1] -1 -•903 [2] -97 -86625 -1-6325 24336 -1-508164 24 -1-484 2 -1-482 -w -10375 -9049 [ -1326 -1186 -140 -133 .[3] -w -[3] — - -7 -7 1 1 1 2 1 -[1] 3 •6 3-6 4-2 4-8 -[2] -[3] 5^07 6-072 2 6-074 * 2 6-076 -M 2 -[1] -4 2-16 -1-84 2-52 [2] •68 •4401 -7(1-692021 -6 -[1] -1 -1-104 [2] -104 •100809 1-1201 •4482 -[3] 1^5683 10144 1^578444 10 [4] 1^588 3191 3157 -[3] 34 32 2 2 -[4] In proceeding from the transformation [1] to [2], in the second develop- ment, we see that one variation disappears, and that a real root is passed ; showing that this root lies between 1 and 1*6: — it is the root 1^356..., previously developed. If the roots assumed to be real had been imagi- nary, we should have discovered the fact after a few steps : the several remainders, in the last column, instead of continually tending towards zero, or evanescence, would have approached towards some constant finite num- ber, as a limit, which the diminishing remainders could never pass — a decided indication of imaginary roots. If the roots had been equal, the remainders, in the second and third columns, would have simultaneously converged to zero.* The cubic equation here discussed is justly regarded by Lagrange as one of more than ordinary difficulty. (5) Required a positive root of S^b*— 3a?^+6a;— 8=0. To find a supe- rior limit, we have to make 2a;— 3>0, .*. r=2; and proceeding as in the margin, we see that the greatest root lies between 1 and 3, for the sum of the given coefficients is —3. 1 2 10 12 ■3 4 2 - 8(2 20 * As far as cvhica are concerned, criteria for determining the character of the roots with the same certainty that the character of the roots of a quadratic is determined, k2 132 SOLUTION OF NUMERICAL EQUATIONS IN GENERAL. -3 2 -1 2 1 2 8 2 -[1] 5 •8 5-8 •8 6-6 •8 tT •8 —[2] 8-2 8-22 2 8-24 -1 -1 1 3 2-32 6-32 2-64 -[2] 10-92 822 11-0022 824 11-0846 826 [3] 11-2 6 -1 5 -[1] 5 2-128 7-128 3-184 -[2] 10-312 110022 10-422022 111 10-533 45 10-578 4 -[3] 10-62 \ \ \ -[4] 8(1-414214 5 -[1] 2-8512 [2] •1488 •10422 4458 4231 [3] 227 212 15 10 5 4 [4] 8-26 2 8-3 [3] Applying the second criterion at (172) to the three coefficients marked [1 ] above, we see that two of the roots are imaginary : and since the last term of the proposed equation is negative, we conclude (140) that the other real root is negative. In order to find the leading figure of it, we change the alternate signs, writing the equation thus, Sa^^ + S^^^+Oa?"'— 6^—8=0, and seek the value r of ^ which satisfies the condition 2x^ + 3^^— 6>0.'.r=2i; and since for r=l a change of sign takes place in the final term, we infer that the positive root of the changed equation lies between 1 and 2, .*. the leading figure of the negative root of the proposed is —1. The foregoing process for finding the real roots of numerical equa- tions was first proposed by Mr. Horner, and is known by the name of "Horner's Method." But for a more ample discussion of the general theory and solution of Equations, the inquiring student is referred to the Author's octavo work on that subject, as also to his smaller volume on " The Analysis and Solution of Cubic and Biquadratic Equations." (6) Required the character of the roots of x'^—Ax^-\-Sx^—lQa; + 20—0. The rule of Newton applied to this equation detects the existence of two imaginary roots, and two only: but employing the criterion at (175), we find, since (4 x 8— 4^)20 > 16', that all the roots are imaginary. •will be given hereafter. (See the subsequent article, 183.) But the remarks and di- rections above apply generally. -2 3 1 3 1 -1 -1 2 -1 -1 2 3 1 -1 -1 1 1 1 1 1 SOLUTION OF NUMERICAL EQUATIONS IN GENERAL. 133 (7) Required the nature of the roots of ;c"'— 3:c* + 2:^;H3^---2;^; + 2 = 0. This equation being of an odd degree, with its last term positive, has one negative root (140) ; the remaining roots, if real, are positive (1 64). Newton's rule, applied to the last three co- efficients, detects the existence of two imagi- 1—3 2 3 —2 2(1 nary roots. But writing the equation thus 1—2 31 (x''—Sx-h2)a^+{Sa!'^ — ^x + ^) = 0, we know (176) that no real positive roots can exist in the intervals within which x^—dx-\-Q, is plus; and if a^, a^, be the roots of ar—Sx-{- 2=0, the first member will be plus through- out the intervals [0, aj and [a^, oo]. These roots are «!=!, ttg^'^ ' ^^^ trans- forming by 1, as in the margin, we see that the indications of positive roots all lie in the interval [0, 1] : hence the equation has one real negative root and four imaginary roots. (8) Required the nature of the roots of x^-^Oar^—Ax+QzszO. This equation has one real negative root (140) : the other roots, if real, are positive (164). Adding and i ~i _^ subtracting x-, to bring about the condition (176), we _ _ _ have {x—i)x'-\-{x^—4.x-{-Q)=0; and since the first 1-3 3 member is positive throughout the interval [1, oo], and ^ ^ from the results in the margin, it is in this interval o -Ii that the indications of positive roots lie, seeing that i none lie in the interval [0, 1], we at once conclude the roots indicated in the positive region to be imaginary. 3 (9) Required the nature of the roots of 2;»*— 3a?* + 6a;— 8=0. Here we know (140) that there are two real roots with opposite signs, and that (164) the remaining roots, if real, are positive. Subtracting and adding 2x\ we have (2a;'— 3^-i-2)c"^— (2^?-— 6;» + 8)=0, where each term is positive for every value of :c ( 138). Now froma;=0 toic=l, it is plain that (2;??^— ^ ~"n ? ? "k^"^ Gx-\-S)>{^x''-dx+2), ov (S-6x)>{2-^x\ _ _ "II _ .-. within this interval, (2^*^— 6^-1-8) >(2«''—3;» _1 _1 5 ^ -f 2);^'^; hence no roots exist in the interval 2 10 [0, I], (176). But proceeding as in the margin, ~ ~ " we see that two variations are lost in this inter- val. Consequently the two remaining roots are imaginary. (Ex. 5, p. 131.) (10) Required the nature of the roots of a?^-j-8^- + 6a;— 75-9=0. Changing alternate signs, the equation may be written (adding and sub- tracting IO4 (a;2-8;ir-f-16)^ +(75-9 - 10a;)=0, which is positive for every value of x from x=0 >. h __.^^ to ^=7, as we know from regarding the second term only. Hence no root exists between these —1—1 -t- liraits. But transforming by 7, as in the margin, 7 42 we see that the two variations are lost, .*. the equa- ~ ~ tion has two imaginary roots. (Ex. 1, p. 125.) "*" 134 Newton's method of approximation. 179. Examples for Exercise. Required a positive root of each of the following equations by Horner's method: — (1) 3a;H22a;-102=0. (2) 17a^-2dx-i5=0. (3) a^-21x+21=0. (4) a^+x-lB=0. (6) a^+5x^-\-Sx-i8=0. (6) x*-^x^-8x-15=0. (7) 2x^-16a^-^i0x^-S0x+l=0, (8) a^-67337309n25*=0. (9) The cube root of 9 to nine decimals. (10) x*-Bx^-\-75x-10000=0. (11) x*-ia^-ix'^-llx+i=zO. (12) 25a;*-298a;3^576a;2_281a;-26=0. Required a negative root of each of the following equations : — (13) a:3-5a;2+3a:+48=0. (14) a;3_2la;+21=0. (16) cc<-12a;2+12ic-3=0. (16) x^-2a^+dx-20=0. (17) a;*+a;3_8a;_i5=:0. (18) 4a^+240.z;'+3996a;+14937=0. (19) Find aU the roots of a:3_49a;2+658a;-1379=0. (20) Find aU the roots of a;*-12a;2+12a;-3=:0. (21) Required the character of the roots of x*—4:a^-\-7x^—ix-\-l=zO. (22) Required the nature of the roots of x*—ixi^—Zx-i-2S=0. (23) Required the nature of the roots of x^—2x^-\-2c(P—x-{-3=0. (24) Required the nature of the roots of a^-{-x*-^xi^—2x^-[-2x—l=0. Note. — The student must be careful to remember that the non-detec- tion of imaginary roots, by any of the methods now given, does not necessarily imply their non-existence : whenever the character of the roots still remains doubtful, we have only to proceed to develope them on the supposition that they are real, when all doubt will be speedily removed, as noticed at page 131. 180. Newton's Method of Approximation.— The method of Newton, like that which has just been exolained, requires that the leading figure of the root to be approached to, be previously found ; and not only this, but also that the interval within which it lies be so nar- rowed, by preparatory transformations, that the extreme limits of that interval may not differ from one another by more than the small fraction J^ or -1, in order that each limit may be within this small amount of the value of the root. Calling the partial value of x thus obtained r, and the remaining portion of the root x\ we shall have w=r-\-a/, and there- fore (161) Ax)=f{r+x')==f{rnAir)x'+'^^x'^4^x'^+ where, since a/ is less than -1, a/^ must be less than -01, x^^ less than •001, &c. ; hence, rejecting the terms into which these small factors enter, we have for a first approximation to the correct value of a/ the * This is simply to find the cube root of the ahove number. We imagine the wanting terms Oaj'^+Oa; to be supplied, write the top row 1 673373097125, pointing off \ \ \ every three figures — as in the common method — and then proceed with the leading figure 8, of the root, as in Homer's process. APPEOXIMATION BY DOUBLE POSITION. 135 f(r) expression af ■=. — — . which will usually give the value true to two places of decimals. Adding, therefore, this correction to r, we obtain a nearer value r' to the root x, the error being usually below -01. We now have a;=/ + a;", and proceeding as before, ar"=— ^.-— ^, which will usually give of' true to four decimal places. And continually repeating the operation, the approximation to the true value of x may be pushed to any extent. Newton illustrated his method by only one example, certainly a most favourable one for the purpose: this is it, namely, a;'*— 2;»— 5=0 ; so that we have y(;c)=a?«~2:c-5 .•./i(^)=3x2_2...[l]. A root of ^;c)=0 lies between 2 and 3 : to narrow these limits, trans- form successively by 2, 2*1, 2*2, &c. : the root is thus found to lie between 2 and 2"1, so that 2*0 is the preparatory approximation, .*. ic=204-a;'. ... .=.o.a+." ... -=-i^=-S^=— .•.a;=2 -094551 48. Even in this, one of the most simple examples that could be chosen, the process of Newton, requiring the successive substitutions of r, /, r", &c. in [1], becomes very irksome. It is entirely superseded by the elegant and effective method of Horner. It will be seen that each of Newton's divisors /^r), /,(/), &c., is Horner's trial divisor, which, by the con- tinuous series of uniform operations peculiar to his mode of approxima- tion, becomes converted into the true divisor, by aid of previously-computed results. 181. Approximation by Double Position.— Let a be one of the real roots of an equation, and let a -fa/, a-^x'^ be two values near to that root, either both less, both greater, or one less and the other greater. Then, transposing the absolute number N to one side, and writing the equation thus, f(x)=N, we shall have /(a)=iVr,and(161),/(a-fa/)=/(a)-f/,(ay-H^^a;'^-f... Aa-]-x")=:f{anf,{a)x"-/^x"'+... Now supposing that the errors x^, x" are so small that the squares and higher powers of them may be disregarded, we have X — X Bnt fi{a)=:— ~ , 0T=— ;, ■ therefore X X J{a+x')-f{a-]-x") ^ f{a^x')-f{a) ^ f{a+x")-f{a) x'—xf' «f ic" ' where, although a/ and a/' are unknown, yet x'^af' is known, being the difference of the suppositions, namely, {a-\-af)—{a-\-xf'). Hence this Rule. Transpose the absolute number iV" to one side of the equation, and find by trials two numbers, which, when separately substituted for x, 136 cardan's method for cubics. give results each very near to N. Then, as the difference of these results is to the difference of the suppositions, so is the difference of the true result IV and either of the former, to the error involved in the cor- responding supposition. Correcting for this error we shall have a first approximation, which sub- stitute for one of the suppositions, and with this and the other supposition, or with a new supposition still nearer the truth, proceed as before ; and so on till an approximation sufficiently near is obtained. Ex. : A value of o! in the equation a)^-\-Sar =600 is found by trials to lie between 7 and 7-1. Making, then, these substitutions, we find the approximate errors thus : — x=7 .'. ar^=343 a;=7-l .'. a^=357-911 3x2=147 3x2=151-23 490 Results 509-141 509-141 iV=500 Diff. of results 19 '141 9-141 .-. 19-141 : -1 : : 9-141 : -047= error of 7*1 .•.x=7-l- -047=7 -053. Again: x=7-05 .-. ar'»=350-4026, x=7-06 .'. x3=351-8958 3x2=149-1075 3x2=149-5308 499-5101 Results 501 -4266 i\r=600 499-5101 -4899 Diff. 1-9165 .-. 1-9165 : -01 : : -4899 : -00255=error of 7*05 .-. x=7-05+ -00255=7-05255. As an example for exercise, take the equation cc^ +x^-^a=lOO, the positive root of which lies between 4 and 5. By the above rule it will be found that ic=:4-26443 very nearly. Note. — The preceding method may be applied with success to the approximate solution of any equation, whether its terms be rational or irrational, or even exponential, provided only that we can find two near values of the unknown to begin with. 183. Cardan's Method for Cubics.— The foregoing methods apply exclusively to numerical equations, and we see that they are per- fectly general. But it is far otherwise with equations whose coefficients are letters instead of numbers; not only is the general solution of such equations beyond the powers of algebra, but even literal equations of so low a degree as the third or fourth have hitherto proved unmanageable, except under certain restrictions. The quadratic, or equation of the second degree, is the only literal equation, after a simple equation, of which algebra, in its present state, can furnish a perfectly general and satisfactory solution. We shall here investigate the formula at which Cardan arrived for the general solution of a cubic. Let the general cubic equation A.^ar^ -\- A.jc^ -\- A^x -\-N=0 be deprived of its second term by (145),-and reduced to the form x^ -\-px+q=0 ....[l]. For x substitute y-rz .-. y'^-^z'^ + ^yz{y + z)-\-p(y-\-z)-^q=0....['il]. Assume Syz=—p, which we are at liberty to do, since two unknowns (y, z) may always be cardan's method for cubics. 137 made to satisfy two conditions; the two conditions here are y-\-z=x, dyz=—p, and therefore [2], y'^-\-z^= — q. From the second of these we have i/V=--^; hence, taking this in conjunction with the third, we have the sum of two quantities, y^-\-z^, and their product y^z"^, given to determine those quantities. Now we know (68) that the quantities sought ©■* -will be the roots of the quadratic v^+2V—^=0.... [3]. Solving, there- fore, this equation, we have therefore since x=y+z=y—^, we have, by taking the cube root, which is called " Cardan's formula," although, strictly speaking, it is due to Tartalea. This formula, like the general formula for the solution of a quadratic, has the appearance of presenting the three roots (for every cube root has three values, see p. 105) as a finite function of the coefficients of the equation. But if it should happen that those coefficients are so related that j4-^<0...[4], each of the two expressions which make up the value of a? will be imaginary ; so that, for such values of p and q, it will be impossible to compute a real value for x — and one real root a cubic must always have — , except by developing each expression by the binomial theorem in an infinite series, and expunging the imagi- naries, which, entering both with opposite signs, destroy one another. That all the three roots are red when the condition [4] is satisfied, may be proved thus : — Let a be one real root, and let the other roots be h-\- s/c,h — ^/c) these will be real if c is positive, and imaginary if c is negative. The quadratic involving the two latter roots will be a;^— 26:c + 6'^— c=0, and since the third root a is such as to cause the second term in the resulting cubic to vanish, the factor by which the quadratic must be multiplied to produce that cubic will be a* + 26, for a must be equal to —26. The product being x'^—{W-\-c)x-\-U.h^—'ilhc=0, we must have (|)=(.M-..c+0X-|=(3.^-|yx-|, which is necessarily negative when c is positive; that is, when all the roots of the cubic are reaU and positive when c is negative ; that is, when two of the roots are imaginary. When j + q;^=0, then evidently c must be 0, and conversely; hence, in this case, two of the roots must be real and equal, and the formula of Cardan would determine them ; but when the roots are real and unequal the formula is said to be in the irreducible case, and for practical purposes is useless. 138 cardan's method for cubics. 183. Nevertheless, the investigation just given is not barren of note- worthy results, for we learn from it the following interesting Theorem. In the cubic equation aj'^-^px + q^O; whatever be the signs of p and q, the following are sure and complete tests of the character of the roots : — 1. If ( — ^ ) '^v I ) ' *^® ^°°*^ ^^^ ^^ ^^^^ ^^^ unequal. 2. If ( — ^ ) "^^V 9 ) » *^° °^ *^® ^°°*^ ^^® imaginary. 3. ^^("q/^Vo)} t^® roots are real, and two of them are equal. The student is recommended to apply these tests to the cubic equations already discussed, after freeing each from its second term, and dividing by the leading coefficient. Whenever p is positive, there is no occasion to refer to these tests, since, by the rule of De Gua (170), two roots are then necessarily imaginary. As an example, let us take the equation - j < ( ^ ) » two roots are imaginary : applying, therefore, Cardan's formula, we have = — '655...H — — — =—2 '355..., which is the real root. — '000... The two imaginary cube roots of unity are , and , (ex. 15, p. 112), and each is the reciprocal of the other (147);* therefore, introducing each of these separately, as factors of — '5 5 5..., the two ima- ginary roots are 184. From what has now been shown in reference to Cardan's formula, it is evident that it cannot be considered as a general solution of an algebraic cubic. Every general solution of an algebraic, or literal equa- tion — to deserve that designation — while it exhibits the several roots, as functions of the literal coefficients, should also be competent to supply, in a finite form, the numerical values of those roots when the letters are replaced by numbers. We have brought it under the notice of the stu- dent, partly on account of its celebrity, and because a more complete solution of a general cubic has never been discovered ; but chiefly because of the criteria given above, to which the investigation has conducted us, and which criteria preceding writers have usually passed over without directing that special attention to them which they certainly deserve. Up to the present point, the only numerical equation, the character of whose * Moreover, either one of these imaginaries is the square of the other, so that if we call the real cube root of any number iV^, and put a for either of the two imaginary 1 ill values of ^1, all three of the roots of ^iVwill be expressed by either N^, aN^,~^^, 1 1 I or JV^, ecN^, a^N^. This is obvious, because if a^=l, then the third power of either of these expressions must be N. DECOMPOSITION OF A BIQUADRATIC INTO A PAIR OF QUADRATICS. 139 roots may always be ascertained, without any tentative operations, is the quadratic : we may now add the cubic. Formulae for the general solution of a biquadratic have also been pro- posed by several of the older algebraists, Ferrari, Descartes, Euler, Simpson, and others; but as they all presuppose the general solution of the cubic, they cannot be regarded as fully effecting the object sought. It would be little better than waste of time to enter upon these early attempts to accomplish for literal, what Horner has so satisfactorily effected for numerical equations ; since the roots of the former class of equations are available in practice only when numerical values are given to the letters.* It remains for us now to present to the student a brief view of a method of dealing with certain biquadratic equations, which will often be found of successful application. It differs from all the foregoing pro- cesses in this peculiarity : — the object sought is, not at once to develope the roots, but to decompose the equation into two quadratics, without the aid of any subsidiary equation. 185. Decomposition of a Biquadratic into a Pair of Quadratics. — The way of doing this, when practicable, is to extract the square root of the polynomial, conducting the operation with this aim, namely, to secure a remainder which shall be either a monomial of the form mx^, or a mere number, or else a trinomial square. If the quadratic remainder, instead of being a trinomial square, should have four times the product of its extreme terms — these being positive — greater than the square of the middle term, we may conclude that all the roots are imaginary. The following are examples of this mode of decomposition : — (1) x*-\-ix^-Zx^-8x-\-i=0 4a^-f4a^^ 2x^+ix-2) _7a;2_8a;+4 -Ax^-Sx+i -3a;2 The number —2 is suggested for the final term of the root, as it is seen to he the num- ber which, when applied to the divisor, causes the terms —8a; 4- 4 to disappear from the subsequent remainder. The two component quadratic factors of the polynomial are thus f ovind to be {(ic2+2a;-2)^-^/3.ic} {x^-\-2x-2)-s/B.x} and consequently the two quadratics themselves are a^+i^+s/S)x-2=0 and aP-^{2-;^S)x-2=0. * Able and interesting investigations, respecting literal equations of the higher de- grees, will be found ia Mr. Jerrard's " Researches," and in various papers by Mr. Cockle, and Mr. Harley, in the ** Philosophical Magazine," and in the "Manchester Memoirs." The advanced student is also referred to the writings on these subjects of Sir W. R. Hamilton and M. Abel : those of the former distinguished analyst, in the " Transactions of the Eoyal Irish Acadamy," and those of the latter, in his " (Euvres Completes." 140 DECOMPOSITION OF A BIQUADRATIC INTO A PAIR OF QUADRATICS. (2) x*-\-23t?+2x^-6x-15=0 x*+2a^+2x^-6x-15{x^-^x-{-l 2aP+x) 2o^-\-2x^ 2«2+2«+l) x^-Qx-U 2a^+2a;+ 1 -tB2_8a;-16=-(a;+4)2 The polynomial is therefore {x^+x-\-Vf—{x-\-iy, so that the equation is {(a)2+x+l)+(x+4)}{(;r2+a;+l)-(ic+4)}=0 and the two component quadratics are a^+2«+6=0 and gj2— 3=0. (3) a;<+2a:?+3a^2+»+3=0(ai2+a; 2si?+x) 2a?^-3a^» 2x3+ 352 2x24-*+ 3 As fonr times 6 exceeds 1^, the roots of the equa. are all imaginary, for the equa, is (x2+x)2+ (2x2+a;+ 3)=0, where each of the terms forming the first member is always positive, and the sum of two real ^positive quantities cannot be zero. (4) x3+7x2+llx-4=0 Multiply by x : — / 7 1 a;*+7»3+lla^5-4xrx2+-x-- _x* 2x2+|x^ 7xHlla^^ 4Q 7a^+^2 4 2a;2+7x-^^ -5x2-4x 1 . 1 1 /I IV Hence the polynomial of the fourth degree is that is, (x2+4x)(x2-f 3x— 1), and therefore the proposed cubic is (x+4)(x24-3x— 1)=0. In the foregoing example, the proposed equation is multiplied by x, in order to render the leading term a complete square ; and for a similar purpose, the equation in the next example following is multiplied by ^x. When the factors of the first member, thus multiplied, are discovered, the extraneous multiplier is expunged from their product. RECURRING EQUATIONS. 141 (5) 2xH8x2+9a;+9=0 Multiply by 2x : — 4x* Aa?+ix\ 16r»+18x2 160^34-16x2 4a?+8a;+?^ 2x2+18x Q 6x2+12x+- 4 3\2 _4a52+ 6x--=-(^2x--^ /. the polynomial of the fourth degree is {2.H4.+?+(2x-|)}{2a=H4.+?-(2.-|)} that is, (2x2+6a;)(2x2+2x+3), and dividing by 2a;, the cubic is (ic+3)(2x24-2a;+3)=0. Examples for Exercise. (1) Find the four roots of x4+4x3-3x2-8x+4=0. (2) Find all the roots of x^-^y?-\-lb3(^-\-ix-^—0. (3) Show that the roots of 6x*4-4x3-f Sx^— 4x+2=0 are imaginary. (4) Find the three roots of the cubic x^— 6x2+6x+9=0. (5) Find aU the roots of x4-4x3+7x2-4x+l=0. (6) Resolve x^-\-a^-\-hx^-\-cx-\-( - ^ into its component quadratic factors. 186. Recurring Equations. — ^Whenever the biquadratic is a recurring equation (147), the above method of decomposition always applies, as the following general example shows: — aJ*+ax'+&x?+a«+l=0^x2+-ax+l X* 2x2+-ax^ ax3+6x2 2x2+aa:+lV6-ia2')x2+ax+l 2x2+ax+l Consequently the first member of the equation is (^x2+iax+iy-(^a2_j+2^x2 and therefore the component quadratics are *'+{^^-\/G»'-*+2)}^+l=0, and x2+|ia+V^(ia2-J+2)|x+l=0. 142 EECURRTNG EQUATIONS. 187. But the usual way of treating recurring equations in general is the following. Since (^'^+^.)(^+-)=^'^+H^^i+^'-H^i, by putting « for 1 + - we have a^+'^-i — = { ^»4 — U—fx'^-^-i -V from which general relation we see that from any two consecutive forms iB»-i_j a;" -I — . all that follow may be derived one after the other, thus:— &c. &c. &c. which shows that every recurring equation in x may be reduced to another in z of only half its degree if that degree be even. If the degree be odd, we know that one of the roots will be +1 or — -1, according as the last term is — or + (140); so that eliminating the root a:=l, or = — 1, the depressed equation will be a recurring one of an even degree, and this may always be reduced by the above conditions to an equation of half that degree— as in the following example: — (l)aj5— ll^*4-17^ + 17a;'-ll^i-l=0. 1 -H 17 17 -11(-1 ^ ' ' 1 12 29 12 Sinceoneroot of thisisii;=--l, wedivide by _ _ x-\-\, and get the coefficients of the depressed —12 29—12 1 biquadratic, as in the margin. This biquadratic is therefore o;^— 12ii?^H-29a;-— 12a;+l=0, or dividing by x"', and bringing the equidistant terms together, it is ix'^-\ — ^ j — l'2ra;4-- j -1-29 = 0; so that putting z for x-\ — , we have {z^^^)~ 12;2;-|-29 = 0, or ;2;^— 12^^+27=0, which is only half the degree of the proposed. The roots of it are «=9, and z=.^y that is, x-\ — =9, and X aj-f--=3 ; and thus the two quadratics for determining the four remaining X roots of the proposed equation, are ^-— 9^= — 1, and a?^— 3d?= — 1, of 9±v/77 , 3±5 ^ ^ , . . wmcn the roots are , and — ^r-, where one of each pair of roots is the reciprocal of the other of that pair, for the four roots may be ^ntten thus: -^, ^-^-^ ; -^, ^^j-^, as will at once be seen by rationalizing the denominators. The preceding quadratics are BINOMIAL EQUATIONS. 143 of course the same as those at (186), when —6 is put for -a, and 29 for h. 188. It may be here noticed that the recurring biquadratic, when put in the form f x^-\ — ^ j-faf a;+- j+&=0, is lowered in degree by one- half by simply adding 2 to the first term, and then subtracting it from the last, that is, by writing the equation thus, («- + 2 + — 3)-fa(;i? + -)-f fc— 2=0, which is (a?-i--] +ara;-f-j + 6—2=0, or ;s'4-a5; + 6-2=0. And if the form be ix^ — :\-\-a{x — ]4-6=0, we effect a similar reduction by subtracting 2 from the first term and adding it to the last ; for we shall then get ix — J -\-aix — j +6 + 2=0; but here the original is not a reciprocal equation, since it does not remain the same when - is put for x. In the general forms at (187), it is assumed that the signs of the terms paired together are like; but they still answer every requisite purpose : for when the equation is of an odd degree, with its last term negative, we may divide by a?— 1, and thus get a reduced reciprocal equation of an even degree with its last term positive : and if the equa- tion be of an even degree, with its last term negative, we may divide by a?—\ (150), and thus also get a reduced reciprocal equation of even degree with its last term positive. The extreme terms will thus have like signs ; and in a reciprocal equation, if the extremes have like signs, all the intermediate pairs must have like signs (149). 189. Binomial Equations. — A binomial equation is an equation of two terms of the form 2/"±a"=0, where a" is a known quantity. By putting ax for y, it becomes a";»"±«"=0, or, more simply, a;"±l=0, to which form, therefore, every binomial equation may be reduced. And it is plain that, after this reduction, the equation is a reciprocal equation in which all the terms between the first and last are so many zeros. Theorem 1. If « be one of the imaginary roots of ic"— 1 = 0, then every power of a. will also be a root. For since a is a root, therefore a"=l/.a2"=l, a^"=l, &c. ; also a-^=\, a-2"=l, a-3"=l, &c. Consequently «, a^, a^ , oT^, a,-^, a■"^ , all satisfy the conditions of the equation, and are therefore roots of it. The roots may, therefore, be exhibited under an infinite variety of forms, but of course only n of them can be essentially dif- ferent. No two of the roots can be equal, because the limiting equation na?"~^=0 has no divisor in common with a;"±l = 0. Theorem 2. If « be one of the imaginary roots of /»" + l = 0, then every odd power of a will also be a root. For since a is a root,therefore »"= — 1, and every odd power of —1, is also —l. Consequently a, a;\ a^ ..., «~S a.-'\ «~% ..., all satisfy the 144 VAKISHING FEACTIONS. equation, and are therefore roots of it, though no more than ?i of them can be essentially different. It is plain that if n be even, the equation a;'*-- 1=0, or a;"=l, has but two real roots, namely, + 1 and — 1 ; since of no other real values can an even power be 1. If n be odd, the equation a?"— 1=0 has but one real root, namely, + 1, this being the only real value the odd powers of which produce 1. If n be even, the equation ^'^ + 1=0, or aj"=— I has no real root, since no even power of a real value can produce — 1. If w be odd, the equation a*" + 1=0, or ;??"=— ] has one real root, and one only, namely, — 1 ; this being the only real value of which an odd power is — 1. The actual determination of the imaginary I'oots of a binomial equation is best effected by the formulaa of Teigonometry. Examples fob Exercise. (1) Required the four roots of the reciprocal equation cic^-\-a^—iay^-\-x-{-l=zO. (2) Required the four roots of the recurring equation 26a;*— 85x'+102ic2— 85a;-f- 26=0. (3) Required the six roots of the recurring equation 4:X^—2ix^-^57x*—7^x^-{-57x^— 24a;+4=0. (4) Required the three roots of the binomial equation ic3-fl=0. (5) Required the four roots of jc*+l=0, or x^-\-—=0, or x^-\-2-\-—z=.2. (6) Required the five roots of the equation x^-^x*—x—l=0. 190. Vanishing Fractions. — It sometimes happens, upon putting a particular value for a? in a fraction involving this general symbol, that both numerator and denominator vanish, the fraction, in this particular case, assuming the form -. This is called a vanishing fraction, and can arise only from num. and denom. of the general fraction having a common factor which becomes zero for the particular value of a? mentioned. We shall here show how such common factors may be eliminated, and thence the value of the vanishing fraction determined. Let the general fraction be ■— r, the numerator and denominator being rational polynomials, or capable of being developed into such. If for the same value a of x, we have F(x)=0, and f(x)=0, we shall know that a is a root common to both of these equations, and therefore that each of the polynomials must be divisible hy x—a. If the division of num. and denom. by this were actually performed, the result would be an equivalent fraction free from the vanishing factor x—a. There might, however, still remain a second factor i»—rt, equally evanescent for x=a: a second di- vision hj x—a would then be necessary to free the fraction from vanish- ing factors, and so on. But when all are removed, the substitution of a for X in the reduced fraction — the general equivalent of the original — would be a particular fraction in the ordinary form. Now, the successive divisions here alluded to may be avoided thus : we VANISHING FRACTIONS. 145 have proved (134), that if a be put for x in the quotient F{x)-^{x—a), the result will be F^(a); and that if a be put for x in the quotient f{x)~^(x— a), the result will be f^(a). Similarly, for the same substitution of a for x, F-^(x)-^(x—a) becomes F^ia), and/^(ic)-r(a?— a) becomes /^(a), and so on. Consequently we have only to replace the fraction ■-— by -r^-, -7~r, &c., till we arrive at a fraction which, upon putting a for the x, assumes the ordinary intelligible form. The student will remember that, in the following examples, the notation F^{a), F.j(a), &c., means what the derived functions F^(x), F.Jx), &c., become when a is put for x: — F,(.x) _2x , I\(a)_2a /i(^)""i •'•7>)~T Consequently, in this case, T=2a. (2) Eequired the value of — 7-— — --- when x=2. Here -r7^= 5 — =- sc3-7ic+6 /,(^) 3x2-7 .*. ; ' Y=~T» the value required. (3) Required the value of ,, J — -^ when a;=l. Here (l-x)2 /'/x) w(%+l)a;»-»(7i+l)ic'— 1 , 7-7^= :tp, ^^ > which for 05=1 is still -. /,(x) -2(1-^) F^(x)_ 'nP{n-^l)x«-^-n{n-\-l)(n-l)x»~^ ^ F^(l) _ n{n-\-l) /a(a')"" 2 •'• /a(l)- 2 .* Examples for Exercise. x2_i (1) Required the value of (2) (3) (4) (5) (6) a;H2a;-3 x3— 2x2— a;+2 a;:*-7x+6 a^-5x^+Sx-\-9 ar*— 05^— 21a:+45 h{a—x^a) a—X' a;3^_2x2-4x-8 when x=l. when a!;=2. when 05=3. -^ — when x=>Ja. —or? ^ ar*+3x2-4 2x3-5x2- 4a;+ 12 when x=— 2. when x=2. a;3_i2x+16 When the general symbol x is under a radical sign, the following method may be employed to get the value of the vanishing fraction for x-—a. Instead of a, substitute a-^h for x in numerator and denomi- nator; develope the terms containing « + /fc, by the binomial theorem; expunge from num. and denom. the highest power of h common to all the terms, and then put ^=0, as in the following examples : — (1) Required the value of v^^^~^)~v2 ^j^^^ ^__i^ Putting l-\-h for x, the frac- X — 1 (A k\\ k 4*+i.4''^5A-&c. -2^ J.4~*5A-&c. tion is changed into ii±£^LzL= i =! . Divid- 146 SCHOLIUM. ing num. and denotn. by h, and then making A=0, we have -.4~4 5=_— — - for the 4 4^04 value of the fraction when x=zl. (2) Required the value of ^, — — — ^, when x=a. Putting a-\-h for jc, we ^(/ U imaginary, u must have reached a maximum state, and if a diminution, however small, of u renders ^ U imaginary, u must have arrived at a minimum state. (3) For what value of x is x-\- s/[a?—x-) a maximum or a minimum? Here x ■\- ^{a^ — x") — u = .-. 2x' — ^ux -j-u^ — a^^O/. x—-u. Also ^U=^{u~—2u^i-^a)=0.\u'"=^a\\u=a\/^.\x=-aK/Q.. And it is plain that if any value greater than 'Ha^ be put for u^, then v U will be imaginary: hence for a;=-a-s/2, the value of the expression, namely, u= POLYNOMIALS IN GENERAL. 149 «v/2, is a maximum ; and since for every greater value v/U" is imaginary, the value a\/2 is the greatest possible the expression can have. / s ^ , , /. . iJ?'— 14^4-44 . . . „ (4) For what values of x is — ^ a maximum or a minimum ? ^'" 14:P-f-44 ^ o^ «N iir^ ^ « Al Here u=0 .-. a;- + 2(i^--7):B4-4:4— 8m=0.-. ^=7— m. Also 8—2^ ^U='^{(u-7f-\-Su-U}=0.\ v/K-6tt + 5)=0.-.'M=l,or5. Now for w=0, C/=5 : hence for u=\ +^\ 17 must be negative, since the root «=1 lies between and 1 + J, .'. u = l is a maximum, the corresponding value of x being a?=7— w=6. Again : for it=a large number (100 say), U is positive ; hence for 5 — ^, it is negative, since 5 lies between 100 and 5 — S; .'. w=5 is a minimum, the corresponding value of a being 7— ^=3. 2^-— 10^ + 8 (5) For what values of x is — — ; — r- a max. or a min. ? Here 2x2— 10^ + 8 — 2wa;2 + 2Maj~M = _.^ (^i ^ u)x'^ + (u — 6)x + —-—=0 lit .•.a!= J'~\ . Also s/C7=^/{t*--10w + 25-2(l-w)(8-t*)} = n/(— w- + 8m-|-9)=0 .-. w = 9, or —1. Now for u=lO, U'is — ,and there- fore for t*=9 — ^, it must be -t-,and must continue so down to u = — l, so that for It = — 1 — ^, U is — . Hence u=9 is a maximum, and u= — l is a minimum, the corresponding values of a being x=- - = -,and -. 2(m— 1) -4 2 Note. — If the two roots of U=0 be equal, there cannot be either a maximum or a minimum, for U will then preserve the same sign whatever be put for u; and for the same reason neither a maximum nor a minimum can exist if the two roots are imaginary. It is scarcely necessary to observe that when u^ is absent from U, as in examples (1) and (2) above, the max. or min. may be found in the same way. 194. Polynomials in General. — ^We have seen (161) that if /(a?) be any rational and integral polynomial, and that if x be replaced by x-\-l we shall haYef{x + S)=f{x)-\-f^{x)^-/-^^^+'^^P-i- [A], where li /«.0 3 may be either positive or negative. Now if r be a value of x such as to render /(a;) a maximum, then a value for ^ exists, so small, that for it, and for all values of ^ still smaller, we must hB.ye f[x)>fix + S), both when ^ is positive, and when it is negative. In other words, we must h3i\ef(x-^r.) — /(a?)=a negative quantity. But (see Note below) a value so small may be given to ^ as to cause /,(^)^ to become numerically greater than the sum of all the terms that follow it; for this small value therefore the entire series, after/(a;), must amount to a positive quantity for one sign ±of ^ and to a negative quantity for the other. In the case of /(^) a maximum how- ever, we must have always fix-^^)—f(x):=a, negative quantity: hence for such to have place, for any value r of x, f^{x) must vanish for that value, that is, we must have f^[x)~0, and moreover for the value r of x, deter- /. (x) mined by this condition, we must further have —— =a negative quantity. By similar reasoning, if r be a value of x such as to render /(^) a 150 POLYNOMIALS TN GENERAL. minimum, we must have/(^ + ^)— /(^)=a positive quantity ; also, as in the former case, we must hix\ef^{x)=0; and ~-^ must be a positive quantity. Hence — If /(;2;) = maximum, then /((^) = 0, SLnd f.,{a;)= ^ . If /(^)=min., then /^(^)=0, and fj^a;)= -\- . When therefore /(^) is a rational and integral polynomial, the following is the mode of proceeding. Rule 1. Find the real roots of /^(;z?)=0. 2. Substitute each of these infj^x) : those of them which give a minus result will make f{x) a maximum : those which give a pins result, will make it a minimum.^ [Note. — That a value so small may be given to ^, as to cause the first term /1(^, of the polynomial A^-\-B^~-^C^^-\-D^^-\- , to exceed the sum of all the terms after the first, may be proved thus. Divide the poly- nomial by y. the quotient is ^+(B + C^ + D^' + )^, and however great may be the numerical values of B, C, D, &c., it is plain that the multiplier ^, without the brackets, may be made so small, that the product, by the quantity within, may become less than any assigned value A: so that for this small value of ^, we must have A^>(B^'--]-CP-]-D^^+ )]. (l)'LGtf{a;)=x-—4x-^S .•./^(;k)=2:c-4 = .-. :c=2; a]so/,(a;)= + 2. As this is^^Ms, x=2 makes /(^) a minimum, namely, /(a?) =4, which is the least possible, and there is no maximum. (2) Let f{x)=4:X-a!^^—Q .•./,(^)=4-2x=0 .-. ^=2; also/2(^-)=-2. which being minus, x=Q, makes /(^) a maximum, which is the greatest possible, and there is no minimum. (3) Let f[x)=a'—}Sx--\-9Q.v-^0 .•./,(^)=3^^— 36^+96=0, or x"— Ux-tS2=0 .'.f.^xy^^x-U. The roots of /,(.r)=0 are x=4 and ^=8. The first makes f.J^x) negative, the second makes it positive; hence for £c=4:,f(x) is a maximum, namely, 140, and for ;z;=8, it is a minimmn, namely, 108. But these are not the greatest and least values possible, for the equation /(a:)=0 has all its roots real (183), and each of them makes f(x) zero ; also, by giving a sufficiently great value to x, we may make /(a?) as great as we please. Examples for Exercise. (1) Divide a given number a into two parts such tliat their product may be the greatest possible. (2) Divide a given number a into two parts such that the sum of the squares of those parts shall be the greatest possible. (3) What two factors of the number a are such that the sum of their squares are the least possible ? (4) What fraction is that which exceeds its square by the greatest number possible ? (5) When is Zx^--5x-\-Q the least, and Q-\-5x—Zx^ the greatest possible ? ^2 5a;-f-4r (6) For what value of x is -— ; — — a max. or min ? 4x^—32^;+ 9 4a;2_j_4_^_3 (7) Does — :; — admit of a maximum or a minimum ? 2x-\-\ (8) For what values of x is 2sc^—9x^-\-12x a maximum and a minimum ? (9) Divide 12 into two parts such that the less multiplied by the square of the greater may be the greatest possible. * It would be out of place to enter more deeply into this subject here. See the Differential Calculus. INDETERMINATE EQUATIONS. 151 (10) Find a fraction such that its square diminished by its cube may be the greatest possible. (11) Given the equation y^—xy-^x^=a'^ : determine x so that y may be a maximum. (12) Solve the equations aj;^—a;*= max., and -^ = min. X (For further applications of the theory of Maxima and Minima, see Mensuration.) 195. Indeterminate Equations. — An equation is said to be indeterminate when it admits of an unlimited number of solutions, which it will always do when it involves more than one unknown quantity ; but if the restriction be imposed that only integer values of the unknowns are to be admitted, the number of solutions often becomes limited. We shall here show how the integer solutions of a simple equation with two unknowns may be found whenever they are limited in number. Prime Number. A prime number is an integer that cannot be decora- posed into factors, the only divisor of it being the integer itself and unity. Such are the numbers 3, 5, 7, 11, 13, &c. Two or more integers are said to be prime to each other when they do not admit of a common divisor; thus 5, 8, and 9 are all prime to each other. Numbers not prime are called composite numbers. Theorem I. If a, b be any two numbers prime to each other, then if each of the terms of the series h, S6, 3i, 46 (a— 1)6, be divided by «, the resulting remainders will all be different. For let it be supposed that some two of the above terms, as mh, nb, leave the same remainder, r. Let the respective quotients be q, q\ then we must have qa-^r=mb, and q'a-{-r=nb .-. a{q — q') = b{m~7i), so that b{m—n)-^a is an integer. But neither b nor m—n is divisible by a since the b is prime to a, and m — n less than a, for m, n are both less; hence b{m—n)-i-a cannot be an integer, and .*. the supposition of two equal remainders is inadmissible, and the following inferences are established : — 1. Since the remainders are all different, and are a— I in number, they must include all integer numbers from 1 to a— I. 2. Therefore some one of the remainders must be 1, so that some integer x, less than a, may be found that will make bx—l exactly divisible hy a; in other words, the equation ba!—ay=l is always possible in positive integers, when a, b are prime to each other. But if a, b be not prime to each other the equation will be impossible in integers ; for, in this case, a, b, having a common measure, one side of the equation bx—ay^l would be divisible by it, and the other not. 3. Since bx—ay=l is always possible in positive integers, it follows ih&t ay— ba!=~ I is equally possible .*. a;r—%=±l is always possible in positive integers when a, b are prime to each other. Theorem II. If a, b be prime to each other, the equation ax—by=±:c will admit of an unlimited number of solutions in positive integers. For since (Cor. 3) aa/—by'=±l is possible .*. aca/—bcy'=±:ci9 also pos- sible; or, putting x for cx", and y for cy\ the equation ax—by=:±c is possible. Suppose one solution to be x=p, y=q, then ap^bq=ax—by, . , x—p b mb J , or ax—ap=by—bq .-. — -=-= — , or x—p—mb, and y—q—ma .-. x^ y—q a ma p-rmb, and y—q-\-ma. And since m may be any integer whatever, the number of positive integer values of x and y is unlimited. 162 INDETEBMINATE EQUATIONS. If a, h have a common divisor, which is not also a divisor of c, there can- not be any solution in integers ; for, dividing by this common measure, the second member of the equation would be a fraction and the first an integer, which is impossible. PROBLEM I. To find all the positive integer values of x and y in the equation ax±.hy=^c. Since x={by-\-G)-T-a is a positive integer, it follows that, if the division by a be actually performed, the remainder, jt?!/ + ^/, must also be divisible by a; hence, if the difference between ay and the nearest multiple to it of py + d be taken, the remainder, qy-\-e, must be divisible by a, and q will be less than p. Again, if the difference between py-\-d, and the nearest multiple to it of qy + e, be taken, the remainder, ry-^f, must be divisible by a, and r will be less than q. Proceeding in this way, we shall evidently at length arrive at a remainder of the iform y-{-k, which will be divisible by a. Now the leasjt positive integer value that y can have, in order that (2/-\-k)-^a may be a positive integer, when k is negative, will evidently be equal to the remainder arising from the division of k by a, but when k is positive, the least integer value will be a minus this remainder ; hence the following rule ; — hy "4~ c EuLE. Having reduced the proposed equation to the form x=— , perform the division. Take the difference of ay and the nearest multiple to it of the remainder, py-\-d, then the difference of py-\-d and the nearest multiple to it of this last remainder, and so on, till we get a remainder of the form y—kovy-\-k. In the former case the least positive integer value of y will be the remainder B, arising from dividing khy a: in the latter case it will be a minus R. (1) Given 2lx-\-l7y—'^000, to find all the positive integer values of 2000-17^ ^, 172/-5 ^ a; and y. Here x= ^=95 ^- — , and ly the operation, by the rule, will be as in the margin. — — : Now 25-f-21 gives a remainder i2=4, the least ^^"^ integer value of y : this substituted in the expres- sion for x, gives ^=92, the greatest value of a. 16y+20...[2] And by adding 21, the coef. of x, to each succes- sive value of y, and subtracting 17, the coef. of y, [1]— [2]=y— 25=y— ^ from each successive value of x, we shall have all the possible integer solutions, as in the margin below. (2) Show the number of diff"erent ways in which '"^Z^^l^f |5«l«l|«f II a it is possible to pay £20 in half-guineas and half- ^z- 4|25|46|b7188|100 crowns only. Let x be the number of half-guineas, and y the number of half-crowns: then reducing to sixpences, we have ^lx + by= 5y_2 800 — 5v „^ 5?/ — 2 :, 1 r . , 4 800 .'. x= — — — ^=38 — — , we proceed therefore as m the ^1 "^^ 20y-8 margin, and thus find that R=8, and .-. 21—8 = 13, the least 21y value of ^; and .•, 35 is the greatest value of x. Consequently if we add 21 to the least value of y, and subtract 5 from the 2^+8 greatest value of x, and continue thus to add and subtract, we shall have all the possible integer values of x and y. It thus appears that the payment may be made ^~?.^I5?|?5|??I1SIJ?„|?„^ ^^ ^^cc 4. v=13 34 55 76 97 118 139 m seven different ways. ^ i i i* i m i INDETEEMTNATE EQUATIONS. 153 Problem II. To determine the numher of solutions that the equation ax-\-by=c will admit of in positive integers, zero being excluded. Find a/, 2/' so as to satisfy the condition ax'—hy^=l (Theo. 1 Cor. 2), then acx' — bcy^=c,'.ax-{-by = acx^ — bcy\ Consequently, putting x=^caf—mb, and y—ma—cy\ m may be any in- teger, taken at pleasure, that will make these values of x and y positive integers : but if no such value of m exist, it will be a proof that the proposed equation is impossible in positive integers. On the contrary, as many suitable values of m as can be found, so many solutions will the equation admit of, and no more. Hence, because we must have ca/>mb, and cy' <,ma, the entire number of solutions will be expressed by the difference between the integral parts of — and — . For as m must be less than the first of these fractions, and greater than the second, the difference of their integral parts mtist express the number of different integer values of m, except when — is itself an integer, in which case, since m<-— , the difference of the integral parts would be one more than the number of different values of m : therefore, when — is an in- teger, we must treat r as a fraction, and reject it therefrom ; but this CXI Cu must not be done with the other quantity — , because m>—. a a Ex. How many solutions in positive integers will the equation 9a; + ]3!/=2000 admit of? From the equation 9^?/— 13/=!, we have , ISw'+l , 4?/' + l , ^ ,. , V+l a = — jj — ~^+~q — ' therefore, proceedmg as in the mar- 2 gin, we see that 2/'=3j and .•.a;'=3; hence the number of solu- 8y'+2 2000 X 3 ^y' tions is the integral part of — — minus the integral part of 2000x2 .... ^^ , which IS 17. y'-2 Examples for Exercise. (1) Given 5^ -f 11?/ =254 to find all the integer values of x and y. (2) Given \'dx—lMy = \\ to find the least integer values of x and y. (3) In how many ways can 12 guineas be paid with half-guineas and half-crowns ? (4) Is it possible to pay ^650 by means of guineas and three-shilling pieces only ? (5) In how many ways can £1000 be paid in crowns and guineas? (6) How many ounces of gold, of 17 and 22 carats fine, must be mixed with 5 ounces of 18 carats fine, to produce a composition of 20 carats fine ? Problem III. To find the positive integer values of x, y, z in ax-\-by-^ cz=d. Take the greatest coefiicient : — suppose c, then since x and y can neither of them be less than 1, z cannot be greater than {d—a—b)~'C. 154 INDETERMINATE EQUATIONS. If, therefore, we ascertain this limit, and then proceed as in Prob. I., we shall at length arrive at a remainder of the form y±z-±ik, in which, if I, 2, 3, &c., up to the limit, be successively substituted for z, all the integer values of x and y corresponding to these values of z will be found. Ex. Given 3a? + 52/ + 7;^=100, to find all the integer positive values of X, y, z, zero being excluded. Here z cannot exceed (100 — 3 — 5)-r-7 = 13J-, so that 13 is the integer limit of z ; or, rather, 12 is the limit, since the above division by 7 cannot give 18 without a fraction. 100— 5y— 7z „^ ^ 2v+2— 1 , , . ^~ 3 =33-3/-22--^-^^ ; and 32/-(2i/+0— l=y-2+l. Now, by taking z=\, y becomes 0, and a:=31 : but the value 2/=0 is inadmissible. Adding therefore 3, the coef. of x, to this value of y, and subtracting 5, the coef. of y, from the corresponding value of x^ we ob- tain the solution z=^i, y=3, ^ = 26. And continuing thus to add and subtract, we shall find the number of solutions in positive integers, for ;^=l to be six. In like manner for z=^, the number of solutions will be also six: for ;^=3, the number will be Jive; and so on up to 5;= 12, for which there will be but a single solution, namely ^=12, 2/ =2, ^=2. The entire number of solutions will he forty -one. The number of solutions due to any particular value of z is found by dividing the greatest corresponding value of a; by 5, the coef. of y ; the integral part of the quotient, increased by 1, will be the number sought. If we have two simple equations involving x, y, z, we may eliminate one of these, and thus get a single equation involving the other un- knowns only. And this may be treated as in Prob. I. For example — ( ^xA-5vA-^z:^ 511 Giren \ , „ ■ r> - r>r. f *o ^^^ 8,11 the positive integer values of x, y, 2. (^10a;+oy+2z=j.20 I Multiplying the first by 5, and subtracting the second, 222/ + 13^;= 135.-..=i^=^=10-y-^. And proceed- ing as in the margin, we find ?/=2.-.«=7.*.a?=10. These are the only positive integral values of x, y, z : but when z and y admit of several values, some of the corresponding values of x may be fractional : it is necessary, therefore, to ascertain these by actual substitution, and to reject them. Examples for Exercise. (1) How many solutions in positive integers does the equa. 17a;+19y+21z=400 admit of ? (2) How many solutions does 5ar-f7y+ 112=144 admit of in positive integers ? (3) Exhibit all the solutions of 17:r+l%+21z=200 in positive integers. (4) Give all the solutions of 2a;+72/+ 52=27 in positive integers. (5) How must gold of 14, 11, and 9 carats fine be combined so that the mixture may make 20 ounces of 12 carats fine ? (6) Given -I ■ or , ^q — ooon \ *° ^^^ ^^ *^® positive integer values of x, y, 2. 196. Problem IV. To find the least positive integer which, being divided by given numbers, shall leave given remaindei-s. %-5. ..[1] 4y+6 2 8y+10., ..[2] [1]- -m =2,-15 INDETERMINATE COEFFICIENTS. 155 Let a^y rto' ^'•' ^^ ^^® given divisors, 6^ b.,, b.^... the respective re- mainders, and N the required number : then putting x, t/, t/, &c., for the respective quotients, we have N=a^x-\-b^^=a.,y-\-b2=.a.i/-^b^, &c., .-.a^x — a^^^b^—by Find the least values of x and y that satisfy this condition, then will a^x-\-b^ be the least integer that fulfils the first two of the required conditions. Call this integer c ; then it is plain that this, and every other integer fulfilling the same conditions will be furnished by the expression a^a^' -\-c, xf being 0, 1, 2, &c., successively. [See Note below.] We have .'.a^acpc' -^-c^^a^y' -^b^ to find the least values of of andi/, for which values, a^a^ -^c, or a'^ -{-b^, will be the least integer fulfilling the first three con- ditions. Call this integer d, and deduce in like manner the least integer values of xf' and /', which will satisfy a^a.,a.^x" -{-d^^^a^y" -^-b^^ for which values either of these expressions will fulfil the first four conditions; and so on. Note. — If a^, a,, &c., have a common factor, it should be expunged from a^a.,, a^a.,a^, &c, ; that is, these products should each be only the least common multiple of the given divisors here represented as forming them. Required the least positive integer, which, when 11^*:— 2 divided by 11, 19, and 29, shall leave for remainders ^ 3, 5, and 10 respectively. Here JV=lla;+3= 22a;— 4 ril 19i/ + 5=29/H-10.-.2/=— ^^, and from the an- ^ ^^ 3x- 4 nexed work it appears that a?=14; and since c= 7 11a; -I- 3 = 157 .-. 11.19;c' + 157 = 29^/'+ 10 .'. 2/= 209^4- U 7_,^ 6a/H-2 21x-28...[2] 29 -^^ + ^+ 29 • [l]-[2]=__£+24 And the operation annexed shows that ^=19, and 6x'-\- 2 consequently that 209*'-^ 157=4128 is the number £ required. 30:r'+10 29a/ aZ+lO Examples for Exercise. (1) Required the least whole number which, when divided by 14 and 5, will leave 1 and 3 respectively for remainders. (2) Required the smallest integer which, when divided by 7, 8, 9, re- spectively, shall leave 6, 7, 8 for remainders. (3) Find the least integer which, when divided by 3, 5, 7, 2, respectively, shall leave 2, 4, 6, for remainders. (4) Find the least integer which, when divided by 28, 19, 15, re- spectively, shall leave 19, 15, 11 for remainders. [For more extensive information on the Indeterminate Analysis, see " Barlow's Theory of Numbers."] 197. Indeterminate Coefficients.— The principle of indeter- minate coefficients has already been apfjlied in the investigation of the Binomial and Exponential Theorems (94) : we shall here show how the 166 INDETERMINATE COEFFICIENTS. same method may be employed to develope other algebraic forms, as also to effect useful changes in such forms. (1) Required the development of - — 77-. [It is plain here that the " &c.," in the assumption below, vanishes for a;=0]. Assume , .^„ =A-^Bx-\-Cx^-^Dx'^-\-k(i. a -^ox then multiplying each side by a'-\-h'x, and transposing, we have As this is true whatever be the value of x, the several coefficients must each be =0 ; hence, we have the following conditions for determining A, By C, &c., namely, ^a'-a=0, Ba'-\-AV=0, Ca'-\-Bb'=0, Da'+C6'=0, &c. .\A=z-„ B=--,A, C=--,B, Dz=-~C, &c. a a a a a'-^b'x a' a' a' a' a'\ a' \a'/ \aV j It is desirable, in the summation of certain series, that the general term of the series should be such that the factors involving a; may be in arithmetical progression. Whenever the general term is a rational poly- nomial, it may always be changed into the desired form by the method of indeterminate coefficients, as in the following examples : — (1) Let the expression be ax'^-^bx-\-c. Assume ax^-\-bx-\-c=:Ax{x-\-l)-\-Bx-\-C=^ Ax^'-\-{A-\-B)x+C .'. A=a, B—b-a, C=c .: ax^-\-bx+c=ax{x-\-l)-\-{b-a)x^c. (2) Let x^ be proposed, and put x^=zAx{x-\-V){x-^2)-\-Bx{x-\-l)-\-Cx-tD=Ax^-\- {ZA-\-B)x^+{2A-\-B-yC)x-^D.'.A=1, 3^+-B=0, 2^+5+C^=0.-.^=1, ^=-3, (7=1, D—Q.:x^=x{x-{-l){x-\-2) -'6x{x-\-l)+x=x{x-\-l){x-^2--Z)+x=:. {x-l)x{x+\)+x. (3) As another application, let it be required to find the two partial fractions by the addition of which the fraction — results. x^-\-x—2 2a:+3 _ 2^+ 3 _ A B ^ 2x-\-Z_ ^ , B{x-1) ^^\-2+;,_2-(^-l)(;.-|-2)-^3i+^= *^^^ ■^+2-^ + "^+2~' For x=l, the second member of this becomes simply A, and the first becomes 6 5 2^+3 ^ , A(x-\-2) -.'.A=-. Also ::=B-\ ^ -. 3 3 x—1 x—1 For a;=— 2, the second member becomes simply B, and the first, 1^1 2:r+3 5 1 3 3 x^+x-2 d{x-l) ' Z{x+2y Or thus : In the proposed fraction put 1st, x=0, 2nd, x=2 ; Zx+2 , _ ^ .^ -4 , X 3:r+2 ^ , Xx (4) Let ~— -rp-g be proposed. Put it = — [- . , ^^. .'. , =J-\- x{x+ly ^ ^ X {x+lf {x+iy {x-\-iy Zx-\-2 2 2ar^+6a;+3 Puta;=0.-.2=^.- (a;+l)3 x{x^\f J? («+l> INDETERMINATE COEFFICIENTS. 157 Put this, disregardmg the minus sign, =_+^_^^+^_--^^, then _^__= C D jB+— — +— — — [1]. Multiply by {x-\-lY, then we have x-\-\ \x-\-\) .:B=2, 25+0=6.-. C=2, B+C-\-D=d.:D=-l. XT ' . A • .X. , . ^ • . 3a:+2 2 2 2,1 Hence, introducing the neglected minus sign, — — r^rrs= — — - . + x{x+lf X x-^1 {x+iy^ {x-\-lY Or thus : In [1] put ar=0, 1 and 2, in succession, we shall then have the three conditions 11 1 1 2S 11 3 = 7?+ C+Z), _.=5+_e+-Z), —=B-\--C-\- D, from which we readily get 5=2, (7=2, Z)=— 1, as before. Note. — Tn thus decomposiug a rational fraction into its component partial fractions, the numerator should be rendered of lower degree than the denominator by reducing the proposed to a mixed quantity, when such is not the case. And it must be observed that when, as in the last example above, the denominator is a power, that power, and all the in- ferior powers, must each form a denominator of a partial fraction. There is a difference therefore in the assumptions when the factors of the denominators are unequal, and when they are equal: p A B Thus, - — ; — - — — - is assumed = 1 -, but if a=&, then the assumption must {x-\-a){x-\-h) x-\-a x-\-h P A B be - — ; — -5= — ; — )-- ; for, clearing fractions, P=iAx-\-Aa-^B, which supplies {x-\-ay x-\-a (x+a)2 ' ' * ' . • > tftr two conditions — and two are necessary — for determining the unknowns A and B. But for the above form of decomposition, we should have only one condition : and similarly in all such cases. If the denom. of the proposed fraction involve a quadratic factor X, of which the roots r, / are imaginary, then, instead of decomposing in the P Q form j-- f- &c., it is better to unite the two fractions, with x—r x—r imaginaries in their denominators, into one, which will be of the form Ax + B X ' X . Ax+B Thus, suppose we have to decompose r-:-, — : t-t; r:> we assume it :=-r-—- — 7~z-\- ' '^^ (xH4a;+5)(x— 4)' x^-\-ix-\-6 C * -. Now put 0, 1, —1, suecessively for x ; then we have the following equations of x—i , B C ^ A-\-B C 1 B-A 1^.. condition, namely 5—^=0, -io~ 3=-30' "i 6=1^' *^'''^°'' 45—5(7=0, 3^1+35—10(7=— 1, 55—5^—2(7=1, from which we readily find _ 4 _5 _4 ^—37' ^~d1' 37' X 4 4a;— 5 " (x^^ix+b){x-4) 37(x-4) 37(a;2+4a;+5) 158 summation of series. Examples for Exercise. . (1) Write (2a;— 1)2 as an expression in whicli tlie factors involving x shall be in arithmetical progression. iC+1 (2) Decompose ■ into its partial fractions. X -|-0.^-pO x>X-\-\ iC^-f-l (3) Decompose -r- -, and -r — - into partial fractions. x^-\-x—1 oc^—\ 4/;c3_|_g^2 3^_j_4 (4) Decompose ^ — — -^ — into partial fractions. 198. Summation of Series. — The general terra of a numerical series is such a function of ri, that when n is put =1, 2, 3, ..., n, the expj'ession becomes the first, second, third ... , nth term of that series. Suppose the general term of a series to consist of factors in arithmetical progression, that is, let G eneral term = n'(n' + j?)(w' + 2;}) . . . (?i' + mi^ . . . [A] . Where n' is either equal to n, or differs from it, or a multiple of it, only by some constant number. It is easily seen that the expression [A] may be got thus : — From n'(n'-\-2)){n'-^2p){>i'-{-Sp) {n'-\-mp){n'-\-[m-{-l]p) .... [5] Subtract {n'—p)n'(n'-\-v)(n'+2p) {n'-\-m,p) [C] There remains {n'-^[m+l]p—{n'—p)} times [A], that is, {m+2)p times [A]. And from this it follows that any series, of which the general term is [.41, is equal to -. — - x the difference between two series, each of the same extent as the former series, and of which the general terms are respectively [B] and [jC]. (1) Let the sum S ot w terms of 1. 2+2.3+3.4+. ..+7i.(n.+l) be required, [B]= 1.2.3+2.3.4+3. 4.6+. .. + (n-l)/i(w+l)+w(w+l)(/i+2) [CJ=0+1.2.3+2.3.4+3.4.5+... + (7i-l)«(7i+l) .*. Taking the difference (and observing that jp=l, m=l), we have 3 The proposed series is double the series in the example at (89), so that the number of shot in a triangular pile may be investigated as above. Note. — The series \_C] is written so that the leading term of [B] may have its equal immediately under it, and as the number of terms is always the same in each series, the lower series extends towards the left beyond the upper, to as many terms as the upper extends beyond the lower, to the right. It is plain, therefore, that the intermediate terms, common to both series, may be suppressed, and account be taken only of those terms thus projecting to the left and right, since these are the only terms which actually appear in the resulting remainder. The projecting terms here spoken of are always p in number, because p units must be added to n to increase (n—p) till it reaches n. (2) Find the sum of n terms of 1.3+2.4 4-3.5+.. -\-n{n-\-2). Here p=2, and 9»=1, so that the series [5], [(7|, are [B]= 1.3.5 + (»i-l)(?t+l)(ri+3)+«(ri+2)(n+4) [C]=— 1.1.3+0+1.3.5+...(w— 2)(?i)(w+2). The difference of these is 8+(„-l)(„+l)(„+8)+„(„+2)(»+4) ... g^ (2r-'+9.+7). ^ .(.+l)(2.+ 7) _ THE DIFFERENTIAL METHOD. 159 (3) Find the sum of n terms of 124-22+32+ ...+7i2^ Here n^={n—l)n-\-n, so that the series is composed of two others, whose general terms are {n—l)n and n respectively. And by the foregoing principle, the sums of these latter are i(n-l)n(+l), andin(ri+l) In this way, therefore, may the formula for the number of shot in a square pile be readily found. (4) Let it be required to find the sum of n terms of 1^+23+ 33+... +7i^ Here since n^=.(n—l)n{n-\-l) +n (ex. 2, p. 156), we have by the preceding principle, S=^{n-l)n{n-\-l){n+2)-]-'^n{n+l)=ln{n-\-l) . iu(w+l)=|ln(7i+l)| .-. 13+2'+33+...+»l3=:(l+2+3 + ...+7l)2. Series such as those here considered are, however, usually summed by the Method of Differences, or as it is sometimes called, — ] 99. The Differential Method. — Let a, b, c, d, e, &c. be any series of numbers following a regular law of increase or decrease, then if each term be subtracted from the next following, the series of remainders is called the first order of differences; and if, in like manner, the suc- cessive differences of the terms of this new series be taken, the results will form the second order of differences, and so on. Problem I. To find a general expression for the first term of the nth order of differences. Let a, b, c, d, e, &c. be the series, and let A^, Ag, A3, &c., stand for the first term of the first, second, third, &c., orders of differences respectively ; then Ai=&— a, c — 5, d — c, e— d, &c. h—a c — h d— c A.^=:c-26+a, d—2c + 6, c— 2rf+ c, &c. c— 26+0 d—2c-{- h Ay=cZ— 3c+36-a, e-3(;+3c— h, &c. d— 3c+36— a li^=ze—id■\^Qc—ih^\-a, &c. And it is easy to see that, by continuing the operation, the numerical coefficients in A„ are the same as those in the development of the bi- nomial {a^xf or (1 — 1)" : hence, reversing the order of the terms, — If n IS even, A„=a— 716+ ^ ' c ^^ -'-^ 'd+.. „ . ,, %(w-l) nln-l){n-2) '^^ If n is odd,A« =^-a-\-nb- ^ , ^ c+-^ -^- -t A A. 6 Problem II. To find the nth term of the series a, b, c, d, e, &c. In the general expressions (1), put 1, 2, 3, &c. for n in succession, then — b= o+A, ' J=a+Aj c=— a+2&+A2, And by substitution ■ c=a+2A,+A2 [1] dz= a-36+3c+A3 " rf=a+3A,+3A2+A3 (2) e=-a+46-6c+4cZ+A4 c=a+4A, + 6Ao+4A3+A4 /= a-66+10c-10ci+5e+A. „ . /=a+5A,+10A2+10A3+5A4+A5 &c. &c. &c. &c. -r 160 CONSTRUCTION OF TABLES. Consequently the (n + l)tli term of the series is n(n — l) n(n~-l)(n—2) {n+l)t}xterm=:a-^nA,+-^-^A,-\- ^ J>^ ^A,-}- [A] .'. nth texm=a-\-{n-l)A,+^ '^ ^A,+ ^ L^J^ 'a,-{- [B]. Problem III. To find the sum of n terms of the series a, b, c, d, e, &c. Of the series 0, a, a + b, a-\-b-\-c, a-rb-{-c + d, &c. ...[1], the first order of differences is a, b, c, d, &c. ...[2] ; and the sum of n terms of this is the (n + l)th term of the preceding series ; all we have to do, therefore, is to find the (?z + l)th term of [1]. As this series commences with zero, the (Ai+l)th term will be found by putting in [A], for a, a for A^, Aj for Ag, &c., so that the sum S of the proposed series is /oy.ti" »Sf=»ia+^— 2 — A,+ ^-^ A^+ .... (1) Required the sum of n terms of 1.2 + 2.3 + 3.4+4.5 + 2, 6, 12, 20 Taking the differences as in the annexed operation, we see ^ 4 6 8 that A, =4, A2=2, and A.=0 : hence ^>. 2 . 2 Sz=2n-\-2n{n-l)-\-:^{n-l){n-2)=^{{n-l){n-2)+6n}=-n{n,-\-l){n+2). O o o (2) Find the sum of n terms of 1.3 + 2.4 + 3.5 + 4.6+ 3, 8, 15, 24 Taking the differences, we see that Ai=5, ^2=2, and 5 7 9 A.,= 0: hence 2 2 „ „ 5n{n-l) , n{n-l)(n-2) {2n^-\-9n-\-7)n S=Zn-h ___+ ^ = ^ . These two examples have been worked by a different method at p. 158 ; the student is recommended to solve the following examples by both pro- cesses : — Examples for Exercise, (1) 13+ 22+32+. •.+«'^. (2) 13+23+33+...+#. (3) 1+3+6+10 + .. .+i^(w+l). (4) 1+4+8+13+.. . to 12 terms. (5) 1 + 3+5+7+.. .+2(w-l). (6) 1.2.3 +2.3.4 + 3.4.5+...+ n{n-\-l){n-\-2). (7) 1+4+10+20+35+.. . to n terms. (8) 13+33+5'+. ..+(2/1-1)3. 200. Construction of Tables. — One of the most important purposes to which the Method of Differences is applied is the construction of Mathematical Tables. Suppose, for instance, it were required to con- struct tables of squares and cubes, we might proceed thus. By differencing a few of the leading squares 1, 4, 9, 16, &c. 1 ^^^-^^ as in the margin, we find that the second differences are all 22 constant. Adding, therefore, 2 to 7, a new term 9 will be introduced into the series of first differences, and adding this INTERPOLATION. 161 9 to the 16, the next square 25 is obtained. The row of first differ- ences, therefore, may be prolonged to any extent by simply continuing to add 2, and the row of squares, by adding to the last-found square the last- found first difference; and the arrangement may stand as below, where 16 is added to 9, 25 to 11, 36 to 13, and so on. The second difference . 2 Row of first differences . 7 9 11 13 15 17 19 21 23 &c. Row of squares ... 16 25 36 49 64 81 100 121 144 &c. 1 8 27 64 125 In like manner for the cubes. Differencing as in 7 19 37 61 the margin, and computing the several rows as above 12 18 24 the work stands thus : — ^ ^ The third diff. ... 6 Kow of second diffs. . 24 30 36 42 48 64 60 66 &c. „ first diffs. . . 61 91 127 169 217 271 331 397 &c. „ cubes ... 125 216 343 512 729 1000 1331 1728 &c. And it is plain that whenever the differences, found as above, at length become constant, the series of numbers may be extended to any limit by easy successive additions only. But in the more advanced class of tables— Logarithmic and Trigono- metrical Tables — where the numbers tabulated are given, not accurately, but only to a limited extent of decimals, the differences never become con- stant ; yet as the successive differences continually diminish, we may carry on the operation till they at length become so small as to have no in- fluence within the range of decimals to which the tabulated numbers are confined ; the preceding differences may therefore be regarded as constant without practical error, and many additional numbers for tabulation may be computed as above. The small error, however, consequent upon sup- posing any order of differences to be constant, will by its repeated intro- duction have at length a sensible effect, so that the extent of numbers thus deduced must not exceed a certain limit ; when this is reached, a new set of numbers, arrived at independently of differences, are to be differenced ; and another series deduced from them, and so on. But for very ample information on this mode of constructing a table of logarithms the student is referred to the author's "Essay on the Computation of Logarithms." 201. Interpolation. — The process alluded to in last article is in strictness one of interpolation ; its object being to insert a set of num- bers, computed by differences, between two numbers computed indepen- dently, and separated by an interval more or less wide, and such that the entire series of numbers may follow a uniform law. The formulae [A], [B'] at p. 160, express how any remote term [the [n + l)i\ or the ?ith] may be computed by help of only the first term (a), and the first of the several orders of differences. But when a single term only is to be interpolated among a set of previously-computed terms, one or other of the forms marked [1], at the preceding page are to be employed : for " example ; — 162 INFINITE SERIES. Given the logs of 101, 102, 104, and 105, to interpolate log 103. Here of the five terms a, b, f=J°S J^J= I'^^af^^at 7 ^ 11 1 1 ^ ^1 xi • J -n J- 6=log 102= 2-0086002 c, «, e, all are known but the third c. Kegardmg (^— iogi04= 2-0170333 then, as we may safely do, ^^=0, we have by [1], e=logl05= 2-0211893 Mh+d)-{a-\-e) , .-.4(6+60=16-1025340 e='--a+4.b^Qc + Ad .-. c= ^ ^ a > a^^d (a+e)= 4-0255107 D this we calculate as in the margin, and find that 6) 12-0770233 c=log 103=2-0128372. And in this way may =:i (t103= 2-0128372 the logs of prime numbers be found from those of ^~ °^ composite numbers. 202. But if instead of it being required to supply an absent term, it is wanted to interpose a new term, between two consecutive terms of a regular series, we must work by the formula [A] or [B]. If, for instance, it were required to find, not log 103, but log 103-55, from having the logs of 102, 103, 104, 105, given, then, regarding a as log 103=2-0128372, we must make 7i=-55 : and since by differencing the logs of 103, 104, 105, given above in the margin, we find (omitting leading zeros), Aj=41961, and ^2=— 401, by neglecting A3, on account of its smallness, we have log 103-55=log 103-|-%A,+^^^^^^A2=2-0128372+-65(4196+^401)=2'0151500. The small error arising from neglecting A3, will be still further diminished, if instead of taking A^ for the first term of the second order of differences, we take for it the arithmetical mean of the first two terms, as obtained from differencing the four given logs : this mean is —405, instead of —401, which will give 20151501 for log 103-55. Note. — What is done in this last article is not a strictly-legitimate deduction from the foregoing theory, in which n is assumed to be, not a fraction, but a whole number. That the theorems [A,] [B] comprehend the case of n=a fraction, will, however, be satisfactorily shown Hereafter. At present we can enter no further into the subject of Finite Differences than to prove the following interesting theorem : — Theorem. If in a rational and integral polynomial of degree n we sub- stitute for a; a series of numbers in arith. prog., the nth differences of the results will be constant. In the polynomial ^„aj"+^„_;^a;'*"^+...-}-^2^^+^^iC+^o» let aj+Ti be put for x: the difference will evidently be of the form A^n-]^~^-\- A\_c^x''~^-\-..,-\-A\x^-{-A\x+A\, a polynomial a unit lower in degree, from which the constant A^ has disappeared. In like manner putting x-\-h for X in this, and differencing, the result will be of the form ^''„_2a;™-2 4. ^''^_^a:"-3 + ... + ^ V' + ^'> + ^''o» and so on. Hence after thus differencing n times, the leading power of X will be ^"-"=1, and therefore the nth dif- 6 30 92 210 402 ferences constant. As an example, take 3a?* + ^^ ^s ^^'ifi ^^^ 74 ^^^ a?^4-2 ; and put 1, 2, 3, &c., successively for x: 18 18 then differencing, as in the margin, we find that A3=18, and A4=0. 203. Infinite Series.—Imitating the method explained at (198), let the nth term of an infinite series be -j—: -,— — =-r ; r, where n\n' + p){n' + 2/?) . . . [vf + mp) INFINITE SERIES. 163 n' dififiers from a multiple of n only by a constant number ; then it is easily seen that this expression is equal to 2f I 2 mpl7i'(»'+i')K+2j?)...»'+(m— l)j) (7i'4-jp)(n,'+2p)...(?i'+»M^) }...[!>]. (1) Required the sum S of the infinite series 7-5+^^+?"^+ where ^3=2, 1.0 0.0 0./ and m=i. -^ V^- -QA^h-^ 1+1+1 (2) Bequired the sum 5 of m terms of the above series : 1 1+ |+H+-+2iZi (3) Required the sum S of the infinite series ^=3, and w=2 1 \8.11^11.14 /J =1-- =:r^.-.^= 2»+l 2/1+1 2/1+1 ; 9 15 i.+ -L+JL. 5.8 8.11 11.14 5.8.11 ■ 8.11.U ' 11.14.17 ^ 2.5.8"^8.1l"^11.14"''*" + .., where « 1 1.1 ^^* 8"Tl+lTli+-=3 24" 80"^24 240' (4) Required the sum S of 1 2 1.3.5.7 ' 3.5.7.9 ' 5.7.9.11 +: ■+... where ^=2, and m=3. 1.3.5^ 3.5.7^5.7.9^ ^ \3.5.7^5.7.9^ / _1/ 1 1 1 \ '~6\1.3.5"^3.6.7"^5.7.9'^"V* ^^* 076+347+ -4 1.3^ 3.6^ I -(f.+-)i 12 72 Examples fob Exeecise. (1) ^=r3+2-4+3-5+-=- (2> ^=r3-2-4+3:5- •=• <^> ^=r5+^+9T3+-=- (^> ^=il+6T2+9T6+-*'^^*'^'°^''- (6) x=_l.+_5_+_^ + ...=. ^ ' 1.2.3^2.3.4^3.4.5^ ^"^ ^~r3i+3:5:7+5?ri+"*~' _ 1 _ 4 7 ___ ^^^ ^-L3r5~3"X7+577:9+" — _ 62 7'» 82 _ ^^ ^""1.2.3.4+2.3.4.5+3.4.5.6+""* ' 2 3 4 5 S' of ^+^+g'g'y+ g'^'y'^ +-- to n terms. 204. On account of the great variety of forms which infinite series may assume, the subject of their summation is an inexhaustible one, Tiie formulse given in the present section comprehend, as we have seen, some extensive and interesting classes of series : they were first pub- lished by the author of the present work in his " Treatise on Algebra " (1823) : a further extension of them may be seen in his " Mathematical Dissertations," pp. 120-134. We shall now give a few examples of the summation of particular series by aid of known algebraic developments. (1) Find S when :s=-^—+—-^..., y;=-^—^—^,„^ &c. By division, l-^{l—x)=:l-{-x-\-x^-{-a?-{- &c. : hence putting -, -, &c., for a?, we have 1,1,1, .11,1, 1 , 2+2^+2^+-=^' 3+^+33+-=2' ^^d ^^ °^- It must be observed that the infinite series in a; cannot have its sum expressed by 1-^(1—^) when x>l. The remainder arising from this division, and which is regarded as concealed under the "&c.," must not, in such cases, be neglected. This remainder when divided by I— a: must, in fact, be so great as to exceed the sum of the series, for the right-hand member of the identity must be negative, whereas the sum of the series is positive. (2) Find the invelopment of J'(.r)=^-f 2 V+ 32^^+4 V+ &c. By the Binomial Theorem, {l—x)~^•*=l-]-Zx-\-6x^-\-l0x^+ &c. .'.X (l—x)~3z=x-\-SaP-}-6x^-\-10x*+ &c. Hence, by subtraction, F{x)—x{l—x)~^=x^-]-Zx^-\-6x*-{- &c. =zx^{l—x)-^ RECURRING SERIES. ]65 (3) Find ^= — + ^j-^^— £----+,., By the Exponential Theorem (page 80), X^ CC^ X* 2 ^2.3^2.3.4.5^ 2 (4) Find the invelopment of ^+^+^+^+ &c. By (112), - log, (l_:t)=a:+^+y+^+ &c. = log, ^ .,;,log,-l_=:^2+^4.|:+&c. 1 a;' a:^ ic* Subtracting, (a;-l)log,- =_«;++_-+_+ &c. 1— a; 2 2.0 o.4 . - , a;-l , 1 ar . a:2 , a^ . a;* . „ ••'+—'°^' 1=^=1-2+273+0+0+ "'■ 205. Recurring Series. — There is one class of series which has not as yet been adverted to — Eecurring Series. These arise from de- veloping rational fractions of the form ^ — f. ./^, ''' — r-rzi- Thus /^, '^ ., =A+Baf-{-Ca!^+Dar^~\-Ea*+ &c., where, by the prin- ciple of indeterminate coefficients,* A=a, B+a'A — h, C-\-a'B-\-b'A = 0, D-^aC-^b'B=0, E+a'D-^-b'C—O, &c. In these equations of condition we see that the coefficients of A, B, C, of B, C, D, of C, D, E, &c., are those of the denominator of the generating fraction taken in reverse order. The values deduced from these conditions are A-=a, B=b—a^A, C=z—WA—a'B, D=—b'B~a'C, E=—b'C—a'D, &c. Consequently each term of the development, after the first two terms A + Bx, is got by multiplying the two terms preceding it by —b^x^t —a'x, and adding the results : — Z>V, —a'x is called the scale of relation of the recurring series. It consists of the terms of the denom. of the fraction (omitting the unit) taken in reverse order, and with changed signs. To take a 3 ^ g^2 particular case : - — — — -= ^^5x4- 7x^+1 3a;'' + 23a-* + &c., in which the scale of relation is — 2^^ i»^ 2^. (1) Required the invelopment of the recurring series 14-2a; + 8^- + 28j;* + 100a;*+ &c. Assume the scale of relation to be p:p\ qx; then ^-1-2^=8, and 2j3 + 83'=28 .-. ^=3 .-. 2?=2. Upon trial, these are found to be correct, so that the denom. of the invelopment is \ — ^x-\- Slx", Assume then il.f_=l+2a;+8a;2+ &c. .-.^=1, J5=2-3=-l 1— 3a;+2a;2 ■^~* •=l+2a:+8a^+28a;'+ &c. l-3a;+2a;2 * It is plain that liere the ** &c." vanishes when a?=0. 166 REVERSION OF SERIES. (Q) Required the invelopment of l-j-Sx-\-6z'^-\-7x^ + 9x^+ &g. As- sume the scale of relation to be px', qx; then ^ + 3gr=5, and 3p + 5g=7 .\q=2.-.p=—l, which, upon trial, are found to answer: hence the denom. of the invelopment is 1— -2a?-f ^'^ Assume therefore --—^—--^=l-\-Sx+5x^+ &c. .-.^=1, ^=3-2=1 {i-xY Note. — Should p and q, determined as in these examples, ever be found upon trial to fail, in reference to the advanced terms of the series, we should then assume the scale to be px^, qx^^ rx, and determine p, q, r from three conditions : and so on. If the series be really recurring, the scale must in this way be at length determined. If the sum of n terms only of a recurring series be required, we must in general find the in- velopment, as above, of the entire series, then the invelopment of the series which follows the first n terms — the scale of relation being uniform — and subtract the latter fraction from the former. By invelopment (a term very judiciously introduced by De Morgan) we mean the fraction or function which generates the series, and whatever besides may be included in the " &c." In all interminable algebraic series, the " &c." stands for the invelopment, minus the series itself: when the invelopment is, as above, an algebraic fraction, the " &c." re- presents the remainder with the divisor underneath. We have not, therefore, as is customary, called this invelopment the sum of the series to infinity, inasmuch as it is this and something more. Examples for Exercise. Find the invelopments of the following recurring series : — (1) l+3a;-i-5:c2+7a;3+ &c. (2) l+2a;+3a;H5a;3+ &c. (3) H-4a;+9a;2-{-16a^+ &c. (4) Find the sum of n terms of l-|-2a;+ 3a;2+4.z;3^ &c. 206. Reversion of Series.— To revert a series such as y=ax-\- hx'^-\-cx'^-\-.., is to express ^ in a series proceeding according to the powers of y. The principle of indeterminate coefficients enables us to effect this. Assume x=Ay-\-By'^-\-Cif-\-..., which call Y: then we have ar+trHcr^+^^H-..— 2/=0; that is, putting for Y its value, "IV isf^V +'^^AB\f +hB^ + 3c^2^ + dA*^ ■ '+.... =0. Hence the coefficients of y and its powers are each =0, so that aA — l =0, aB-\-bA^=^0, aC+^bAB-^cA^=0, aD-h^bAC-\-bB'-\-ScA'B-^dA'=0, &c. And from these equations of condition we get , 1 ^ b ^ 2P-ac ^ 5P-5aic-\-aH „ ^=? -^=-^3. 0=^^, !>= -, , &c. cc^ x^ x^ For ex. Let it be required to revert the series y— l=^+ir+j7^+ir-5--,+... OONVERGENCT OF INFINITE SERIES. 167 „ 11 1,61 2&2_ac 1 5b^-5abc-\-a^d 1 Herea=l, b=:~,c=-, &c. ... -=1, -_=_-, -_^=-, ^^ =_ &«. Hence ar=(y—l) — | ^ \- &c. From (110), we know that this expression for x is the Napierian log of y ; but if x=logye.: e^=y .'. e-=l+^+|4._+2^+... which is the exponential theorem, otherwise established at (110). Note. — The series to be reverted is, of course, considered to be a series, and nothing more : — it may be finite or infinite. The reversion of it is, however, in all cases an infinite series, and will involve in general a supplementary &c. ; we must again, therefore, caution the student against treating what may be implied in this vague symbol as of no moment, and thus deluding himself by the notion that it may be regarded as valueless. On the contrary, it may turn out to be the most important part of the entire expression, and serious error may therefore arise from neglecting it. This may be easily illustrated in the matter before us. Let us take 2/=a;-f a;^ + 2, or ?/— 2=ir+a;'^ : then by reverting, as above, we find for x, ^=(y~2)-(2/-2)H%-2)^-5(y-2)^-t- &c. or, fory=l, a;=— 1 — 1— 2— 5—&C. But the values of i??, in— l=a;+a;-, or in a:--\-x + l=0, are both hnaginary ; the "&c," therefore conceals the most important part of the foregoing expression for x. Such expressions are of course practically useless ; we can never estimate the value of a series, or of any expression, which contains in it a quantity about which we know nothing. Such series or expressions can be available in compu- tation only in those cases in which we know that what is hidden under the " &c." must disappear. 207. Convergency of Infinite Series. — An infinite series is convergent, as already observed at (111), when the more of its terms we add together, the nearer do we approach to some fixed numerical limit, the remote terms diminishing continually down to zero. The fixed numerical limit here spoken of, could never be actually reached by adding term after term, on account of the terms being infinite in number, though the greater the number of terms we include in such an approximate summation, the more nearly does the rejected part of the series approach to zero. The limit, or exact sum of the series, is attained only when nothing is rejected; how this sum may be calculated with the most rigid accuracy — in a great variety of instances — has been suflS- ciently explained in the preceding sections. A diverging series, on the contrary, is such that, take as many terms of it as we may, their sum does not approach to any fixed limit : there is no limit which, by including more and more of the terms, the sum may not exceed. That the series is diverging when the terms continually increase is obvious ; but there are many series which have one of the two distinguishing features of converging series, namely, that have their terms continually diminishing down to zero, and are nevertheless di- verging; the other peculiarity, namely, the existence of a numerical limit, which the sum of the terms, however numerous, cannot exceed, being absent. Such a series, for instance, is 1+77+0+7 + ^+ +-, 2 o 4 5 « 168 CONVERGENCY OF INFINITE SERIES. for however remote the nth term may be, n more terms of the series would be -H -H 7: + "-+7r* ^^6 sum of which evidently ex- n + 1^ + 3 n + 3 2n ^ ceeds — x w, or -, however great n may be. It is of importance, there- fore, to have some means of ascertaining whether a series, such as this, is convergent, and therefore summable — at least approximately — or whether it is divergent, and therefore not summable. A series such as 1 — 1+1 — 1 + 1 — 1 + ---. in which the terms neither increase nor di- minish, has been called a neutral series. Its sum, if we take an even number of terms, is ; if we take an odd number it is 1 ; its sum to ivjinity, since oo is as much even as odd, is ambiguously or 1. Some writers say it is J, but that is impossible, for no fractions enter the terms. The test of convergency for a series, whose terms are all positive, is the following : — Theorem I. An infinite series, of which the terms are all positive is convergent, when, commencing at any term however remote, the quo- tient, arising from dividing that term by the immediately preceding one, is less than unit. Let the series be ... +e+/+5r + A + ..., and such that/-+e, g-r-f, h-r-g,&c. are each less than a certain number r, then/(a— &), and (a— ft)+(c— fl?) + (e— /), and . For an obtuse angle the tangent like the cosine is negative. •215. Cotangent, Cosecant. — If a line Bt be drawn from D, the extremity of the first quadrant, touching the arc at that point, and con- tinued till it meet the secant of BC, or that secant prolonged, in t, the touching line Bt is called the cotangent, and At, the cosecant of that arc. In like manner, Bt! is the cotangent and At' the cosecant of the arc BC . The ratio of each of these lines to the radius, or the numerical values of the lines themselves, to the scale radius=l, are the cotangent and cosecant of the angle BAG, or BAG'. 216. The ratios or abstract numbers to which the above six names are given, comprehend all that are peculiar to trigonometry. There is, how- ever, another term sometimes used : the terra versed sine ; it is employed to denote the excess of unity above the cosine of an angle ; thus vers A means I— cos .4. The versed sine of an arc BG is the line Bm=AB-^ 174 FUNDAMENTAL TRIGONOMETRICAL RELATIONS. Am. Also covers A means 1— sin A. In reference to the arc BC, it is the line Dn. 217. Complement. — ^Whatever must be added (algebraically added) to an arc or angle to make up 90°, is called the comylement of that arc or angle : thus, Z)(7is the comp. of BC, and DC' (taken negatively) is the comp. of B&. Also 50° is the comp. of 40°, and —50° is the comp. of 140°. The cosine, cotangent, &c., of an arc or angle, are no other than the sine, tangent, &c. of the complement of that arc or angle, regarding the complements of arcs as measured from D. It must be remembered then that if two arcs or angles p, q, together make up 90°, each is the complement of the other, and that therefore sin ^=cos q, cos^=sin q, tan p=cot q, cot p=sin q, &c. 218. Supplement.— Whatever must be added to an arc or angle to make up 180°, is called the supplement of that arc or angle : thus CB' is the supplement of the arc BC, and C'B' the supplement of BC\ Also 140° is the supplement of 40°, 60° is the supplement of 120°, and so on. And it may here be noticed, in connection with the supplement, that mB' is called the suversed sine of the arc BC : — it is the diameter minus the versed sine mB. Of an angle A, the suversed sine, or as it is more frequently called, the suversine, is 2— cos A. It will have been seen, from what has preceded, that the directive signs (+ or — ) prefixed to a cosine, tangent, or cotangent, have been suggested by the position of the geometrical lines in the diagram. To avoid all confusion and am- biguity about direction, the arc whose sine, cosine, tangent, &c., is to be determined, is alw^ays considered to have its origin or commencement at one and the same point B, and to be measured from that origin in one and the same direction BCDC, &c. For instance, if we had to discuss these lines in reference to the arc CC\ we should conceive the point C applied to B, or which is the same thing, should replace CC by an arc, equal to it, measured from B, in the direction BCD. And this is the way supple- mental arcs are considered : the supplement of BC is CB\ as to length, or number of degrees; but to determine its sine, cosine, &c., we conceive C to be brought down to B, and the arc, thus displaced, to terminate at G\ so that CW is the sine of that supplement, and Am' its cosine. As a supplement is what the arc wants of 180°, or of a semicircumference, it is plain that if BC is equal to CB, we must have BC=B'C\ and Cm= C'm\ as also Am=Am' : hence an arc or angle has the same sine as the supplement of that arc or angle : the cosine too is the same in magnitude for both, but the signs of direction (+ or — ) are opposite : thus, if sin 40°=p, then also, sin 140°=j9; but if cos 40°=^', then cos 140°= — q. The signs of direction prefixed to the trigonometrical ratios, thus have their origin in the linear representation of those ratios : — in a diagram, fixing direction as well as length : we could not with propriety speak of a ratio, or abstract number, taking a directive sign, uidess that ratio or number were derived from geometrical considerations. The lines in the diagram at (213), and which have been defined above, may always be regarded as linear representations of the trigonometrical ratios bearing the same names, the radius representing 1 ; so that many properties and relations of these ratios may be established by help of their representative lines, as in the following article. 219. Fundamental Trigonometrical Relations. — Refer- ring to the diagram at art. (213), and remembering that the trigonometrical FUNDAMENTAL TRIGONOMETRICAL RELATIONS. 175 radius is 1, the 47th prop, of Euclid's first book furnishes the following relations, where, for simplicity of notation, (sin Af, (cos Ay^ &c. are written sin^ A, cos- A, &c. l2+tanM=sec2 A ,'. sec ^=*y(l+tan2^), tan 4 = V(sec2 ^-1) [ [1]. l^-j-cot^ -4=cosec^-4 .•. cosec A=i^(l-\-cot^ A), cot A=y/(cosec^ A—1)) Again : since the sides about the equal angles of equiangular triangles are proportional (Euc. Prop. 4, B. VI.), by comparing together the equi- angular triangles TAB, CAm, and the equiangular triangles fAD, C'Am\ where Am'^zAm^ and Cm!=.Cm, we have the relations tan A sin A cot A cos -4 cos ^ 1 sin A 1 1 cos A^ 1 sin A^ 1 sec J.' 1 cosec AJ that IS, tan ^= -, cot A=.— — -, sec ,4= -, cosec Az=,- — ^ cos J. sin J. cos J. sin J. / [2]. The following pairs of quantities therefore, namely, tan A, cot A ; cos A, sec A ; sin A, cosec A, are the reciprocals of each other; and this it will be necessary to recollect. The foregoing fundamental relations show that from the two trigono- metrical quantities sin A, cos A, all the others may be derived, as well as regards sign, as numerical value : we thus see, for instance, that the sign of the secant of an angle is the same as that of the cosine, and that the sign of the cosecant, is the same as that of the sine. 220. All the quantities in the preceding equations are numbers : — the trigonometrical ratios : they apply exclusively to angles. But they may easily be converted into geometrical relations, applicable to the sines, cosines, &c., of arcs belonging to any circle. If we write the terms SINE, COSINE, &c., in capitals, when we refer to arcs instead of angles, we have only to replace sin A, cos A, &c., above, by — 5—, — — — , &c., where R represents the radius of the circle to which the arc A belongs. And to this R we may, of course, afterwards attribute any numerical value we please. We shall now give an example or two in reference to the re- lations marked [I] and [2] above. (1) Required the sine of 45°. Here ^=45o=90^— 46°: hence (217) sin A-=:co3 A .*. [1], sin^ -4-i-cos^ .4=2 sin^ A=l .'. sin ^=^^- = -^2=cos A. (2) Required the tangent, cotangent, &c., of 45°. Dividing the sine by the cosine, we have [2], tan 45°=-V2-r--\/2=l, also cot 45°=1 ; and 2 2 since the secant is the reciprocal of the cosine, and the cosecant of the sine, we have sec 45°=l-f--v'2=v'2, and cosec 45°=>/2. (3) Required the sine, cosine, &c., of 60°. This is the measure of each angle of an equilateral triangle, since the three equal angles amount together to two right angles, or 180°. If from the vertex of such a triangle a perp. be drawn, it will bisect both the base and the vertical angle, dividing the equil. triangle into two equal right-angled triangles. 1 76 TABLES. Let CAO\ in the diagram at p. 171, be an equil. triangle, CC being equal to the radius of the circle passing through G, C" : then BAC=iiO°, therefore to radius AB='[, Am=-=sui 30°, .-. [1], Abo[2],ta.30'=S||i:=i-34v3, cot 80-^8, Since the complement of 30** is 60", we infer from these values (217) that 8in30''=co8 60"=-, cos 30"=sin 60<'=-^/3, tan 80<'=cot 60''=-\/3, li (i O cot 30''=tan 60"= v^ 3, sec 30"=cosec 60"=- ^3, cosec 30"=sec 60"=2. o 221. Tables. — But it would not be possible, in this way, to deter- mine the trigonometrical ratios for all values of the angle A. For this general purpose, certain series have been investigated, and certain alge- braic expedients devised, which will be explained hereafter. With such aid, the sines, cosines, &c., for all values of A, from A=i)°, up to ^=90", have been computed to several places of decimals, and arranged in Tables. For values of A above 90", such computations would be superfluous, for the sine, cosine, &c., of an angle above 90", are the same, in numerical value, as the sine, cosine, &c., of an angle as much below 90". A mere inspection of the diagram at p. 171 is sufficient to show this. There are two kinds of trigonometrical tables. The first contains the several trigonometrical ratios, computed as above hinted. This is called a table of natural sines, cosines, &c. The second is a table of logarithmic sines, cosines, &c. These are not strictly the logs of the natural sines, cosines, &c. : — they are these logs each increased by the number 10, which addition to them is made solely to avoid the inconvenience of negative logs. The trigonometrical sines and cosines, as we have already seen, are all less than 1 ; so that the logs of these are all negative : by increasing each log by 10, we in effect compute the table to the radius jR=10^^, instead of to 7^=1. And we have seen (220) that the trigonometrical ratios remain the same whatever be B. 222. Before concluding these introductory observations, there is one other inference, from the diagram at p. 171, to which the attention of the student must be directed. The chord CO" of an arc CBC" is obviously double the sine Cm of half that arc ; and still regarding the circle to which the arc belongs as the trigonometrical circle — or the circle whose radius is the linear representation of the numerical unit — we infer that the chord of any angle is twice the sine of half that angle, or that did ^=2 sin - A, Now by Euc. 8, VI., the chord CB is a mean pro- portional between BB' , Bm ; that is, CB^^^BB' x Bm, giving to the lines their numerical values, but BB'=2, .'. 2(1— cos A)=:2 vers ^=(2 sin -Ay .'. vers A =2 sin' -A, ii 2 RIGHT-ANGLED TRIANGLES. 177 a property that will be called in request in investigating the third case of oblique-angled triangles. Having thus disposed of the necessary preliminaries, we may now pro- ceed to the business of Trigonometry Proper. Part 1. Solution of Plane Triangles. 223. Right-angled Triangles. — There are two cases to be con- sidered : — 1. When an acute angle and a side are given. — Kule 1. With the vertex of the given angle as centre, and the given side, terminating in that vertex, as radius, imagine an arc to be described, and notice what trigonometrical name, in reference to that arc — whether sine, or cosine, or tangent, &c., the sought side takes. 2. Enter the table of natural sines, &c., with the given angle, and take out the number under that name. Multiply the given side by that number : the product will be the sought side. [Note. — When one of the acute angles of a right-angled triangle is given, the other is virtually given : for if A be one, 90" — A is the other]. 2. When two sides are given to find an angle. — Rule 1. With the vertex of the required angle as centre, and a given side, terminating in that vertex, as radius, imagine an arc to be described, and notice what trigonometrical name the other given side takes. 2. Divide this latter by the former : the quotient will be the tabular number of the same name, over which, in the table, the value of the required angle will be found. For, in the right-angled triangle ABC, let, first, the side AB=^c, and the angle A be given : then \ if with centre A, and radius AB, we imagine an ^ arc to be described, as in the diagram, we see that ly^ EC or a becomes the tangent of that arc, and AC y^ or h, the secant ; and we know, whatever be the ^ y^ length of our radius, that we have the constant ^ ratio -= tan A, and -=sec A. Next, let the hypo- c c ^ tenuse AC=b, and the angle A, be given : then if i-^'^^^^ with centre A, and radius AC, we imagine an arc ^^ to be described, it is plain that BC or a becomes ^^^ the sine of that arc, and AB or c the cosine ; and we know, whatever be the length of our radius, that we have the constant a c ratio -=sin A, and -=cos A, .-. a=c tan A, b=c sec A, a=b sin A, c= b cos A, and 4. A ^ A ^ - 4 <^ A ^ tan A :=-, sec A =-, sin A =-, cos A =-. c ebb results which sufficiently establish the foregoing rules. From the expres- sions for tan A, and sin A, and from (219), it is plain that cot ^=-, and cosec A=i- ; and therefore that c=za cot A, and b=a cosec A. a Note 1. — The following examples, on the solution of right-angled tri- angles, are all worked by the table of natural sines, cosines, &c. In the N 178 RIGHT-ANGLED TRIANGLES. case of right-angled triangles, the operations are, in general, easier by the natural numbers than by logarithms. Prefixed to the tables will be found all the necessary directions for using them. In the common applications of plane trigonometry it is but seldom requisite to compute the angles nearer than to degrees and minutes. 2. In the table of natural sines, &c., the secants and cosecants are not inserted: for since sec A= -, and cosec A=- , we may change cos A sin A multiplication by sec A, or by cosec A, for division by cos A, or by sin A ; and division by sec A, or by cosec A, for multiplication by cos A, or by sin J. In all multiplications and divisions by large numbers, where several decimals are concerned, it is better to use the contracted methods, as explained in the Arithmetic. (See VVeale's Eudimentary Treatise.) When any two sides of a right-angled triangle are given to find the third side, the solution is effected by the 47th of Euclid's first book ; and no aid from trigonometry is required. Putting for the sides opposite to the angles A, B, C, the corresponding small letters a, b, c, if B be the right angle, we have, by the prop, referred to, J2=a2^c2 .-. &=>v/(a2+c2), a=V(&2-c2)=v^{(6+c)(6-c)}, c=^ {{h-\-a){h-a)} . (1) In the right-angled triangle ABC, are given c=174 feet, and A= 18° 19', to find the sides a and b. Here a=c tan A, and b=c sec A=c-^cos A (art. 219) tan 18° 19'= -33104 cos 18° 19'= -9,4,9,3,3) 174 (183-29=6 c, reversed, = 471 94933 33104 79067 23173 75946 1324 3121 a=57-601 2848 273 190 83 Hence the side BC=57-6 feet, and the side ^0=183-29 feet. (2) From the edge of a ditch 18 feet wide, which surrounded a fort, the angle of elevation of the top of the wall was taken, and found to be 62" 40': required the height of the wall, and the length of a ladder necessary to scale it. Here c = 18 feet, and ^4=62" 40'. And tan 62" 40' X 18 = 1-93470 X 18=34-82 feet=«, the height of the wall, 18-4-cos 62" 40'= 18-^-459 17 = 39-2 feet=6, the length of the ladder. (3) The hypotenuse ^C is 643-7 feet, and the base AB, 473*8 feet: AB required the angles A, C, and the perpendicular, BC. Here -— - =cos A, that is, 473-8-T-643-7=-73606=cos 42° 36' .-. ^=42" 36', and C=90"— ^=47" 24'. Also (Euc. 47 of I.), s/(643-7^— 473-8'^)=435-87 feet=^C. (4) The tower DC oi an enemy's fortress cannot be safely approached nearer than B : at this point the angle of elevation CBD of the top is found to be 55" 54'. The observer, upon going back a further distance BA of 100 feet, finds the angle of elevation A to be 33" 20' : required the height of the tower and its distance from B. RIGHT-ANGLED TRIANGLES. 179 DA=CD tan ACD, DBz=CD tan BCD, that is, DA=^CD cot A, DB=CD cot GBB .'. AB=DA— DB= CDicot 33° 20'— cot 55° 54')=CD (1-5Q043 — •67705)=100 .-. CZ)=100-^ •84338=118-57 ft., the height of the tower. And ..CD cot CBD= 118-57 X •67705=80-28 ft., the distance BD. (5) From the top of a mountain BA, m miles high, the angle of depression EAC, below the horizontal line AE, of the re- motest visible point of the surface of the sea, is taken, and found to be : it is re- quired to determine the radius OG of the earth. Since OAE:=:90\ EAC is the comple- ment of OAC, but is also the complement of OAC .-. 0=EAC=Q. Now OA=:OC sec 9=:0B + BA .•.BA = m = OC sec ^-OC=zOC(secQ^l).\OC= m sec 0—1 or, smce sec 0=l-f.cos 0, DC' m cos miles. 1— cos9 Note. — This would be a very expeditious method of determining the earth's diameter, if the refraction of the atmosphere, near the horizon, could be accurately estimated. But owing to the fluctuations of this horizontal refraction, it is of uncertain amount; so that the foregoing method cannot be regarded as giving more than a somewhat rough approximation to the truth. The radius of the earth being known, however, from other investiga- tions, it is easy to find — sufficiently near for the purpose, and without the aid of trigonometry — at what distance at sea the top of a mountain of known height just emerges above the distant horizon. For, calling the height h, the distance d, and the radius of the earth B, we know by Euclid, Prop. 36, B. III., that [':iE^li)h=d^. As K- is comparatively insignificant, we may neglect it: we shall then have d-=^2Rh. Thus, R being about 4000 miles, if /t be 1 mile, 6^=^/8000=89 miles. Also, the distance being known, we may compute the height of the mountain from hz=-— ; thus, if a ship just loses sight of the top of a mountain, ii±\i when at the distance of 30 miles from it, its height will be 900 9 ., ■=— - miles =594 feet. 8000 80 Examples for Exercise. (1) The base of a right-angled triangle is 246 feet, and the angle at the base 33° 45' : required the perpendicular and hypotenuse. (2) The hypotenuse is 430 yards, the perpendicular 214 yards : required the angle between them. (3) Given c=73, ^=52° 34' to find a, h. (4) Given c=288, ^=63° 8' to find a, 6. (5) Given a=85, 6=111-4 to find A, c. (6) Given a=75-18, c=53-42findJ, A, C. (7) Given c=138-24, (7=62° 57' find A. (8) The perp. is double the base : required the angles. N 2 180 OBLIQUE-ANGLED TRIANGLES. (9) A tower 53J feet high, is surrounded by a ditch 40 feet broad : the angle of the elevation of the top of the tower from the opposite bank is 53" 13' : required the length of a ladder sufficient to scale the tower. (10) If the top of a mountain, known to be 6600 feet high, can just be seen at sea 100 miles off, what must be the diameter of the earth ? (1 1) From the top of a ship's mast, 80 feet above the water, the angle of depression of another ship's hull was found to be "20°: required the distance between the two ships. (12) A ladder, 40 feet long, is so placed as to reach a window 33 feet from the ground ; and upon being turned over, the foot remaining in the same place, it reaches a window 21 feet from the ground, on the opposite side of the street : required the breadth of the street. (1 3) From the top of a castle standing on a hill by the sea- side, the angle of depression of a ship at anchor was observed to be 4° 52' ; at the bottom of the castle, or top of the hill, the angle of depression was 4" 2': required the horizontal distance of the ship, as also the height of the hill, that of the castle itself being 60 feet. (14) The height of the mountain called the Peak of Teneriffe is about 2^ miles ; the angle of depression of the remotest visible point of the surface of the sea is found to be 1" 58' : required the diameter of the earth, and the utmost distance at which an object can be seen from the top of the mountain — that is, the length of the line from the eye to the remotest visible point. (15) The angles of elevation of a balloon were taken by two observers at the same time ; both were in the same vertical plane as the balloon, and on the same side of it ; the angles were 35° and 64°, and the observers were 880 yards apart : required the height of the balloon. 224. Oblique-angled Triangles. — Every triangle has six parts as they are called — three sides and three angles. If any three of the six be given, except they be the three angles, the remaining three may be determined by computation. It is plain that the three angles would not suffice for the determination of the triangle, because there may be an infinite variety of equiangular triangles all different in size. There are three cases to be considered. 225. Case I. When two of the three given parts are opposite parts — an angle and a side. Let A BC be any triangle, the sides being denoted by the small letters a, b, c, and from either ver- tex, as (7, let a perpendi- ^ C cular CD be drawn to the opposite side c. If the ly^ angles A, B, be both acute, y^ CD will fall within the tri- ^/__ angle : if one be obtuse, as A *^ DBA ^ B ^D in the second diagram, it will fall without the triangle, but in each case ADC, BDC, will bo right-angled triangles. From the first of these we have CD=b sin A : from the second, CD=a sin B, the sine of an angle being the same as the sine of its supplement. . ^ - . . o, sinA , ...-,. .'. a sm Bz=.h sin A .'. rr=— — -, or a : b : : Bin A : sin B, sm B OBLIQDE-ANGLED TEIANGLES. 181 that is, in any plane triangle the sides are as the sines of the angles opposite to them: hence the following rule, which is applicable alike to oblique-angled and right-angled triangles. Rule. As one side of a triangle is to another, so is the sine of the angle opposite the former, to the sine of the angle opposite the latter. And this proportion it is usual, and in general, for oblique-angled triangles, more easy, to work by logarithms ; that is, to subtract the log of the first term from the sum of the logs of the second and third terms : the remainder is the log of the fourth term. Note. — If a side be required, the first term of the proportion is, of course, the sine of a given angle; but if an angle be required, the first term is a given side. As the fourth terra is then a sine : — the sine of the sought angle, and as to every sine there corresponds two angles, the acute angle given in the tables, and the supplement of that angle (218), a doubt may sometimes exist as to which of these two — the acute angle or the obtuse one — really belongs to the triangle in question. Under the follow- ing circumstances, however, there can never be any ambiguity : — 1. When the given angle is obtuse; for a triangle cannot have two obtuse angles: the sought angle must, therefore, be acute. •2. When the given angle is acute, and it so happens that the side oppo- site to this is greater than the given side opposite to the sought angle ; for, by Euc. Prop. 18, B. I., this latter angle must then be less than the former, and therefore acute also. Moreover, in actual practice, the shape of the triangle will, in general, be suflBciently marked to enable us to pronounce at once whether any particular angle is acute or obtuse : but when mere numerical measure- ments are given, unaccompanied with a geometrical representation, or when the triangle is not actually figured before us, and the given angle is acute, and the side opposite to it happens to be less than the other given side, the example is said to belong to the ambiguous case; for whether we take, as belonging to our computed sine, the acute angle furnished by the tables, or the obtuse angle which is its supplement, either will subsist with the given conditions ; and two distinct triangles, each satisfying those conditions, will be determinable. Example 3, following, is an instance of the ambiguous case of plane triangles. (1) In a plane triangle ABC, are given the base AB or c=408, and the angles ^, jB,=58° 7', and 22" 37', respectively: required the other two sides. The angle C opposite to the side c, being the supplement of ^-1-5=80*' 44', sin C=sin 80" 44' (218). Hence by the rule, As sin C= sin 80° 44' V Logs. 9-994296 As sin (7= sin 80^ : sin^=sin22° : : AB=^m : ^(7=159 44' 37' Logs. 9-994295 : sin ^ = sin 58° : : AB=iQ9> 9-928972 2-610660 9-584968 2-610660 12-539632 12-195628 : BC=ZB1 2-546337 2-201333 Note. — In these operations the rule has been strictly followed: the first log in each proportion has been substracted from the sum of the other two logs. But a subtractive log may always be replaced by an additive one, and the work thereby shortened, thus : Instead of writing .down from the table, the subtractive log, write down what that log wants 182 OBLIQUE-ANGLED TRIANGLES. of 10, in this way: Looking at the leading figure, write what it wants of 9, then what the next figure wants of 9, and so on, till the last significant figure is reached, when there must be put down what it wants of 10. It is plain that in this way we shall get what the entire log wants of 10. This defect from 10 is called the arithmetical complement of the log, or its co-log. Now, if this colog be added, instead of the log itself being subtracted, the result will evidently exceed the true result by 10, which excess is allowed for without any trouble — we have only to dismiss 10 from the erroneous result. With this modification, the work of the above example will be as follows : — As sin (7= sin 80° 44' -005705=00% : sin^ = sin58°7' 9-928972 : : ^£=408 2-610660 £C=Z5l 2-545337 As sin (7= sin 80° 44' 0-005705=colog : sin 5= sin 22° 37' 9-584968 : : ^5=408 2-610660 AC=159 2-201333 With a very little practice, a colog may be written down as quickly as the log itself can be transcribed. 226. We shall now exhibit the work of an example in which seconds are concerned. In the Table of log sines, cosines, &c., there will be found several columns headed " D : " these are columns of differences, each difference being placed between two consecutive numbers in the table. In some tables, these differences are those due to 1 minute, or 60 seconds, and are got by simply subtracting the greater of the two numbers from the less. The difference d, due to any smaller number (a) of seconds is found, from such tables, by the proportion 60 : a : : D : d, so that £2=——. But in the tables intended to accompany the present work, the differences are those due not to 60 but to 100 seconds; so that in in these tables, fi=--—; and thus d is found, somewhat more readily. The most convenient way of correcting for the odd seconds is this : Take from the table the number answering to the degrees and minutes, and against it write the tabular D. Observe whether this D is additive or subtractive, and put against it the + or — accordingly.* When all the logs, with the corresponding tabular differences, have been transcribed, then compute for the seconds thus : multiply each D by the number of seconds in the angle against which that D stands ; add up all the pro- ducts, reject the last two figures of the sum, for the division by 100, and the remaining figures will furnish the proper correction of the result, as in the following example. Note. — Observe that for an angle greater than 90**, the sign of I) must be changed, and that, for a colog, D takes a sign opposite to that for the log. For the method of determining a required angle to seconds, see ex. 3. * As regards acvie angles, D is always additive except for a co-quantity, when it is always subtractive. In the case of an obtuse angle, we enter the Table with the supple- ment of the degrees and minutes, and multiply the corresponding D by the given seconds, changing the sign as directed in the Note. If we replace the obtuse by the acute angle which is its complete supplement, the D will of course preserve its proper sign. OBLIQUE-ANGLED TRIANGLES. 183 (2) Given A=4.r 13' 22^ ^=71° 19' 5'', a=55, to find b. Logs. 2) (Parts for sees.) As sin ^= sin 41° 13' 22" 0-181176=colog -240 -5288 : sin ^= sin 71° 1^5" 9-976489 +71 + 355 : : a = 65 1740363 -49.33 1-898027 — 49=:Correction for the seconds. & = 79-063 1-897978 If the correction for seconds had been additive, we should have placed it under the three tabular logs, and have added all up together. In using tables where the differences are to 60'^ instead of 100^', in place of cutting off two figures from the sum of the "parts for seconds," we should divide by 60. (3) Given a=178-3, 6=145, and JB=4L° 10', to find Ay C, and c. As 6=145 7-838632=colog : a=178-3 2-2.51151 : : sin B= sin 41° 10' 9-818392 sin ^ = sin 54° 2' 22" 9-908175 9-908141=sin54«»2' .♦. A -\-B= 95° 12' 22" 180° 2)=153) 3400(22" .-. C= 84° 47'! As sin B= sin 41° 10' 0-181608= colog (Parti.) : sin C= sin 84° 47' 38" 9-998197 i>=20 7,60 : : J=145 2-161368 8=Cor. for the sees. c=219-37 2-341181 c=219-37 But the present example is in the ambiguous case. The log sine 9*908175 belongs equally to the angle 54° 2' 22'', and to its supplement 125" 57' 88"; and we have nothing here to guide us as to which of these is the angle A. We have proceeded above to calculate c on the supposition that A is acute. If, however, it be obtuse, then (7=180''— (125° 57' 38" + 4P 10') = 12°52' 22", and sin (7=9-347890. Substituting this, there- fore, for sin C above, the sum of the three logs will be 1-690866, which answers to the number c=49076. Hence the triangle in question may be either of the two, £ ABC, A' BO, figured in the margin, where / \v CB=a, and CA = CA'=b, the angle B being / \\ common to both triangles. In the larger tri- \^ / \?\'''' angle, the angle A is 54° 2' 22", and the side \/ \^^ ^B=219-37, the opposite angle C being 84° ^ ^--. ..,-"''^ ^ 47' 38". In the smaller triangle, the angle A' is 125° 57' 38", and the side ^'J5=49-076, the opposite angle C being 12° 52' 22". 184 OBLIQUE-ANGLED TRIANGLES. Examples for Exercise. (1) Given ^=22° 37', J5=134° 46', c=351, to find C, a and b. (2) Given ^=44° 13' 24", 5=79° 46' 38", c=368, to find a and b. (3) Given 5=153, c=137, J5=78° 13', to find C and a, the former to seconds. (4) Given a=117, 6=216, ^=22^37' to find B to seconds. Is the case am- biguous ? (5) Given two sides of a plane triangle 50 and 40 respectively, and the angle op- posite the latter 32° ; to determine the two triangles, to each of which these parts belong. 2Q7. Case II. When the three given parts are two sides and the included angle. Let ABC be any plane triangle, in which the two sides CA, CB, and the included angle C, are given. If the given sides are equal, the solution comes under the case of right-angled triangles, because a perp. from C, bisecting both the angle C and the opposite side, we shall have the hypo- thenuse ACj and the vertical angle [^Cj given to find the base ( -AB J. Each of the base-angles A, B, will be half the supplement of C Let, then, CA be the greater of the two given jp sides, and from it cut off CD=CB: join BD, and draw the perp. CE", which produce to F: lastly, draw EG parallel to AB. The perp. CE bisects the angle C, and also the line DB : hence (Euc. 2 of VI.), DQ=:QA,\ CO=^{BC-\-CD-\-DA)=-{BC+CA)=^Jialf the sum of the sides. Also AQ=-{QA-CB)=.halS tU diference of the sides. Again (A-\-B-]-0={CBD-^CDB-\-0 .'. CBD=half the sum of A andB, and ABD is half the diference, because, being added to the half sum, it gives the greater, B. Now, by right-angled triangles, CE=BE tan CBD, and EF=BE tan ABD .'. CE : EF:: tan CBD : tan ABD. But >jB) Qji-{Ar^B) ' A A OBLIQUE-ANGLED TRIANGLES. 185 (1) aiven a=137, 6=153, C=40° 33' 12", to find the other parts. ^+5=180°-(7=139°26'48".-.i(-4+-5)=69° 43' 24". Asa+&=290 7-537602=colog. : aoo6= 16 1-204120 D (Parts.) : : tan 69° 43' 24" 10-432291 -f 648 165,52 156 Cor. for sees. tan 8° 29' 37" 9-174169 Adding, .*. B= 78° 13' 1" 9-173634=tan 8" 29' Subtracting, A= 61° 13' 47" D=1442) 53500(37" D {Parts.) As sin 5, 78° 13' 1" (colog) 0-009250 -45 -45 : sin Cy 40° 33' 12" 9-812988 246 2952 •: &=163 2-184691 29,07 29 Cor. for sees. c= 101-615 2-006958 Examples for Exercise. (1) Given a=526, J=378, C7=32° 18' 26", to find the other parts. (2) Given a=l7-802, J=21-704, (7= 26° 12' 16", to find the remaining parts. (3) Given J=2065, c=1637, 4=132° 7' 12", to find the remaining parts. (4) Given a=500, 6=400, C=88° 30', to find the remaining parts. (5) Givena=960, c=1686,£=128°4', to find A, C, and h. (6) Given a=1672, J=1014, C=45° 1' 3'', to find A, By and c. Note. — A method of computing the side c, without first finding the angles Ay B, will be given hereafter. (See p. 204.) 228. Case III. When the given parts are the three sides. By Euclid (12 and 13 of II.), we have the following values of AC^y in the triangles at page 180, namely. When B is acute, A (P=BCf^-\-BA^- 2BA . BD, When B is obtuse, AC^=B(P-\-BA^-{-2BA.BD. Now BD=:BC cos CBD, and this angle being the supplement of B, when obtuse, its cosine is minus cos B. Hence, introducing the minus in this case, we have, whether B be acute or obtuse, the single formula b^z=a^-\-a^—'iiac cos B, which is a general expression for one side of a triangle in terms of the other two and the included angle, but it is not so well adapted to computation by logs as the method already given. As b is any side of the triangle, expressions similar to this have place for a% and c'; so that, for the cosines of the angles, in terms of the sides, we have 524.c2_a2 a2+c2-62 a2+&2_c2 COS A= — — , COS B= , cos C= — t-t — •••[IJ. 26c ' 2ac ' 2a6 "■ -^ expressions which are very easily remembered: they may be employed with advantage whenever the three given sides are small numbers. The student is, of course, aware that, to adapt an expression to logarithmic 186 OBLIQUE-ANGLED TRIANGLES. computation, it should consist entirely of factors and divisors : no additive or subtractive operations should be implied in it. In each of the above, the denominator is in a suitable form already; we have, therefore, now to convert the numerator into factors; and this we may do by simply subtracting each side of the equation from unit : , a2-(62-2Jc+c2) a2-(5-c)2 (a-5+c)(a+5-c) Thus, l-cos^= — =— 2j^ = 26^ > that is, (216), versin ^^ (^-^+^)('^+^-^\ But (222) vers A=2 sin^^, 1 l(a-6+.).i(a+6-c) •■•^"'2^= Fc ' or, putting 8=-(a+6+c).'. s— 5=-(a— &+c), and s— c=-(a-f 6— c) 2 2 A V=J - /™^ [21 where 6, c, are the sides including the angle A, and s the half sum of all the tfiree sides. Similar formulae may, of course, be written down, from this, for the sines of the other half angles. Formulae for the cosines of the half angles may also be readily deduced : thus, Since -(1 — cos .4)= sin^ -A .-. 1 —-(1 — cos .4)=-(l-}- cos A) =1 — sin^ -A — cos^ -A. Hence, adding 1 to each side of the first equation [1], we have &2+26c+c2-a2 (5+c)2-a2 (a+6+c)(6+c-a) . .1. 1+ C0Sil= — = — = — =2C082-il 26c 26c 26c 2 ••-4^=v/*-^-W- Ana,p]^[3],ta„^.=,/(ir|)^>...W. Instead of the half angles, we may now find at once the whole angles, thus : (1— cos^)(l-H cos.<^=sin2^=4 sin^-^ cos^-^ .*. sini4=2 sin --4 cos-^ ^2 2 2 .•.[2]x[3],Bin^=A^{s(s-a)(s-6)(s-c)}...[5]. As sin A belongs equally to the angle A in the tables, and to its supple- ment, when a is the greatest of the three sides, there may be some doubt as to which of these angles really belongs to the triangle, for there can be only one triangle having the three given sides (Euc. 8. I.). But half an angle of a triangle is always acute, and, therefore, one or other of the three formulae before given is, in general, to be preferred. These are easily retained in the memory : the numerator of [2] consists of two factors : 8 minus one side, and s minus another. If we were to write s minus the opposite side for one factor, then the other factor would be ambiguous : it would be s minus either of the adjacent sides: this is sufficient to apprise us that the two subtractive sides in the num. are the adjacent sides, and these same sides form the factors of the denom. In the nume- rator of [3], there is only one subtractive side : it cannot be an adjacent side, because the adjacent sides have each an equal claim : it must, there- fore, be the opposite side, which cannot be one of the two sides in the denom., because the other side would then be ambiguous. To preclude OBLIQUE-ANGLED TEIANGLES. 187 mistake, as to which of these two forms belongs to sin -A and which to cos -4, we have only to observe that the expression for cos -A is shorter than that for sin -^, and to connect in the memory the o in cos, with the in shorter. In every practical example, the student should write down from memory the formula he intends to employ, and work from it, instead of from a verbal rule. The formula [4], arising from dividing [2] by [3J, can at any time be recalled from these. 229. It may be noticed here, that it is not always matter of indifference which of the forms [2], [3], be employed. If the sought angle A is fore- seen to be very near to ISO'', then -A will be very near to 90**: and the sines of angles very near to 90° differ so little from one another that, for five or six places of decimals, the sines of several such angles all agree ; so that the tables would give a choice of angles increasing in magnitude, second by second, through a range of some minutes. In the case, therefore, of an ill-conditioned triangle of this kind, [3] is the preferable formula. On the contrary, when Jl is a very small angle, [2], or [4] should be employed. 230. If it be required to determine all the angles, we may find two only, by one or other of the above formulae, and then subtract the sum of these from 180°. But this course is not recommended. It is better to compute all three, in order that the accuracy of the work may be tested by trying whether or not their sum amounts to 180°, as it ought — at least, within a second or two — if our results are correct. In such a case, [4] is the most convenient formula to work from ; for, since 2 V 8{s—a) ' 2 V 8(s— 6) ' 2 V s{s-c) 1 s — a 1 1 s — a 1 .'. tan ^B=. — - tan -A, tan ^C= tan -A. Or in logarithms, Ji s — 2i Z s — c A log tan-B= logtan-.4-i-log(s— a)-log (s— &)=logtan-il— colog(s— a)— log(«— 5) + 10 log tan -. C=: log tan -A -J-log (s — a) —log (s— c) =log tan -A — colog U—a) —log (s— c) -f- 1 0. Since colog (s— a), log {s—h), and log (s— c), have all been previously employed in computing log tan - J, the additional work for finding log tan - B, and log tan - (7, is very trifling. This work, arranged in the most convenient manner, is exhibited in the solution of example 2, next page, where the colog, and the two succeeding logs are marked [p], [^J, [r], respectively, and log tan - ^ is marked [TJ. 188 OBLIQUE-ANGLED TRIANGLES. (1) Given a=l-372, &=6, c=5-523, to find A. ^^^2^=v/ {s- l){s-c) he a= 1-372 b=z 6 c= 5-523 2)12-895 s= 6-4475 8-6= -4476 s-c= -9245 colog 9-221849 colog 9-257825 1-650793 i -965907 2)18-096374 smU, 6° 24' 55" 2 9-048187 Bin 6° 24' = 9 047154 JD=1875)103300(55" .•.-4=12M9'50" (2) Given a=95 '12, 6=162-34, c=i find A, B, C. 2 V s{s-a) a= 95-12 6=162-34 c= 98 2)355-46 s =177-73 colog 7-750239 s—az= 82-61 colog 8-082967|>] s-6= 15-39 1-187239[(?] s-c= 79-73 l-901622[r] 2)18-922067 tan -^,16° 7' 27" 9-461033[T] tan 16° 7' = 9-460823 i)=789)21000(27" l-378066=Lr-i)] tan \b, 67° 12' 4" 10-190827=[2'-i)H 2 tan ^(7, 16° 40' 29" 9-476444=[T-i)]-r 2 Verifi- cation 90° C 0" Note. — In determining -J, in each of these examples, two cologs, or arithmetical complements, have been employed ; it might seem, there- fore, that two tens, or 20, should be deducted from the sum of the four logs. But it must be remembered that, in the logarithmic tables, each log sin, log cos, &c., is increased by 10 (221), so that, in logarithms, the above formulae are log sin-^=-{log (3— 6)-f log (s—c)— log 6- log c} +10 log tan -^=- {log (s-6)+ log (s—c)— log s— log (s—a)} -flO. After the division by 2, in the foregoing operations, the results, in con- sequence of two cologs being incorporated in the sum, each exceed the true log sin -A, and log tan -A, by 10, and it is by this quantity that the log sin and log tan, in the table, are each actually increased, so that no correction of our results is necessary. In deducing tan -By tan -(7, in 2 <« the second example, 10 is added to the remainders [T—p]—q, [T— ;?]— r, agreeably to the logarithmic forms in last page. MISCELLANEOUS PROBLEMS INVOLVING PLANE TRIANGLES. 189 Examples for Exercise. (1) Given a=15-236, 5=12-414, c= 9-018, to find B. (2) Given the three sides 4, 5, 6 re- spectively: determine the angles. (3) Given a=101 -616, 5=153, c=137, to find A, B, a (4) Given a=33, 5=42*6, c=53-6 to find Ay B, C. 231. Miscellaneous Problems involving Plane Triangles. (1) In order to find the distance between two in- accessible objects A, B, a base line CD was mea- sured = 536 yards, and at C, D, the following angles were taken, namely, ^CjB=57°40', BCD= 40° 16', ^DC=42° 22', ADB='m' : required the distance AB. 1. In the triangle y^CD are given CD and all the angles, to find DA. 2. In the triangle BCD are given CD and all the angles, to find DB. 3. In the triangle ADB are given DA, DB, and the included angle, to find AB. These computations being performed, we find ^15=939^ yards. (2) Two eminences C, D, can be easily reached, but from the nature of the intervening ground, the distance between them cannot be conve- niently measured. For the purpose of finding it, two spires, known to be 6594 yards apart, were observed, and the following angles taken, namely : — ACB=S5°W, BCD=z2Z°5&, AI>C=ZV iS', ADB=6S° 2' ; required the distance CD. Assume any point c in AC, and regard cd, parallel to CD, as unit ; then, drawing db parallel to DB, the two quadrilaterals A BCD, abed, will be similar. Compute Ab by last problem, from the given angles equal to those observed ; then since the sides about the equal angles of AB similar figures are proportional, Ab : cd : : AB : CD=--jr; since cd=\ , CD is thus found = 4694 yards. (3) The distances of three objects A, B, C, from each other are J.^=462 yards, ^(7=328 yards, and ^(7=297 yards. A person at D, in the continua- tion of CA, wishing to know his dis- tance from each object, takes the angle ADB, and finds it to be 24" 16' 21"; required the distances DA, DC, DB. (4) The distance AB (fig. to Prob. I.) is one mile, the angle ACD= 76" 30', BCD=U> 10', ^Z>(7=8P 12', ADC=AQ'' 5': required the distance CD. (5) The angle of elevation of the top of a steeple is 40°, when the observer's eye is on a level with the bottom ; and from a window 1 8 feet directly above the first station, the angle of elevation is 37° 30': required the height and distance of the steeple. 190 APPROXIMATE QUADRATURE OF THE CIRCLE. (6) Three objects, A, B,C, whose distances apart are AB — \^ miles, AC=^ miles, B0=7} miles, are visible from a station D, in the line joining A,B: the angle at I) subtended by AC is 107'' 56' 13'' : required the distances of the objects from Z>. (7) From the top of a hill 360 feet high, I observed a column of in- fantry halting in a straight road directly before me : the angle of de- pression of the foremost rank was 57°, that of the most remote 25° 30' : required the length of the column. (8) Wanting to know the height of a castle on the top of an inacces- sible hill, I took the angle of elevation of the top of the hill, 40°, and of the top of the castle 51° ; then retiring in a direct line a distance of 180 feet, I again took the angle of elevation of the top of the castle, and found it to be 33° 45' : required the height of the castle. (9) Given the distances AB=S0O, AC=QOO, BC= 4.00, between the three objects A, B, C, and the angles subtended by the two latter distances at a point P in the same plane as the objects, namely, ^P(7=33° 45', BPC=22'' 30', to deter- mine the respective distances of P from each. (10) Given the angles of elevation of an object B, taken at three stations A, B, C, in a horizontal line, equal to «, a', a", respec- tively, the distances between the stations being AB=a, BC=b : prove that the height h of the object JE is ^=\/; ah(a-\-b) a cot^ ct"-\-b cot- a — {a-\-b) cot^ «' (1 1) In the last problem, if the measured distances AB, BC, are equal, the expression for the height h is h=a-T- \/ ( 5 cot^a-j — cotV— cotV ). It is required to calculate h, when a=36° 50', a'=21° 24', a"=14°, and the two equal distances AB, BC, 84 feet. (12) Suppose the distances are ^5 = 100 yards, 5(7=400 yards, and the angles at A, B, C, 5° 24', 6° 27'-5, 8° 36', respectively : required h. Note. — The foregoing problems are all solvable by aid of the principles established respecting the plane triangle in the preceding a,rticles. But, with the additional assistance derived from a more comprehensive know- ledge of trigonometrical properties, some of them may be solved with less numerical labour, as we shall show hereafter. 232. Approximate Quadrature of the Circle.— By the geometrical quadrature of the circle, is meant the finding of a rectilinear figure, or of a square, which shall be equal to the circle in surface, or area. The attempt to solve this problem has long been abandoned, but it would be wrong to pronounce the solution as impossible, since there is no APPROXIMATE QUADRATURE OF THE CIRCLE. 191 doubt that a rectilinear area equal to the circular one exists. The numerical quadrature of the circle, means the determination of its area in Jinitg numbers, when the numerical value of the radius is given. This problem is impossible, since no finite expression for the area, in numbers, exists. But the approximate quadrature may be obtained, to as many places of decimals as we please, as follows. Imagine a regular polygon of n sides to be inscribed in the circle. Let ah be one of those sides, and let the surface of this inscribed polygon be p^. Then the tangents aA, bA, will be each half a side of a similar circumscribed polygon. Call the surface of this P„. Let M be the middle of the arc aMb ; then will the chords aM, bM, be two sides of an in- scribed poh^gon p.^„, of 2n sides, and the tangent BMC will be a side of a similar circumscribed polygon F^,,. All this is proved in Euclid, Book IV. Now being the centre of the circle, in and about which the regular polygons are described, it is plain that the triangle -. By similar right-angled triangles, ODa, OaA, I OD : Oa : : Oa : OA }► [1] or, OD -.OM.'.OM: OA I and since triangles of the same ^.i altitude are to one another as their J bases, ODa : OaM : : OaM : OaA. Consequently the numerical measure, or area, of the triangle OaM, is a mean proportional between the areas of ODa, OaA. And since the wholes are as their like parts, it follows that — The area of pjn is a mean between the areas of pn and ?„, that is, P2n=N/PnPn-..[2]. Again : since the right-angled triangles ODa, BMA, are similar, the angle BAM being =zOaD, since AOa is the complement of each, we have OD .Oa : : BM : BA, or OD : OM : : aB : BA and since triangles of the same altitude are as their bases, it follows that ODa : OMa : : OaB : OBA .: ODa : ODa-\-OMa : : OaB : OaB+OBA ODa is the 2»th paxt of the polygon pn that Oailf ,, „ ,, p^ that Oa-4 ,, ,, ,, Pn and that Oa^ilf ,, ,, ,. J>„„ •••[1], i3« :i5«+iV • -^^^n P„.,^,„=!^...[3]. Pn-^2hn The expressions [2], [3], enable us to compute the areas of inscribed and circumscribed polygons of Q,n sides, when those of polygons of n sides are given. Let 1 be the numerical value of the radius of the circle ; and, to begin with, let the inscribed and circumscribed polygons be squares, that is, let n=4 : then we have given j?4=2> aiid P^=4:; and therefore from [2], [3], above, ^8=^8=2 -8284271, ^3=^-^^= -J^— =3 '3137085 2^-^/8 .-. ^,e=^(3 -3137085 V8)=3'0614674, P,6= 4-8284271 2x3-137085^/8 ^8+3-0(514674 :3 -1825979. 192 UNIT OF CIRCULAR MEASURE. And in a similar manner may p.^^', P32 ; Pev Pqv &c., be computed in succession, and thus the following table formed : — Surfaces of Inscribed and Circumsci'ibed Regular Polygons. Radius =1. No. of sides. Surf, of ins. Polygons. Surf, of circ. Polygons. 4 2-0000000 4-0000000 8 2-8284271 3-3137085 16 3-0614674 3-1825979 32 3-1214451 3-1517249 64 3-1366485 3-1441184 128 3-1403311 3-1422236 256 3-1412772 3-1417504 512 3-1415138 3-1416321 1024 3-1415729 3-1416025 2048 3-1415S77 3-1415951 4096 3-1415914 3-1415933 8192 3-1415923 3-1415928 16384 3-1415925 3-1415927 32768 3-1415926 3-1415926 &c. &c. &c. 233. It thus appears that the numerical expressions for the surfaces of inscribed and circumscribed regular polygons of 32768 sides each, agree as far as seven places of decimals ; and as the surface of the circle itself is intermediate between these two polygons, it follows that the area of a circle of radius 1, as far as seven places of decimals, must be 31415926. Now the area of any regular polygon of n sides, whether inscribed or circumscribed, is the sum of all the n equal triangles formed by lines from the centre of the circle to the extremities of the sides : the area of each triangle is half the product of a side of the polygon by the perpen- dicular upon it from the centre. In the case of a circumscribed polygon, this perp. is r, the radius of the circle : hence, calling the semi-perimeter of the polygon tt, the area is t-tt, which, when r=l, is simply tt. Now when n is indefinitely great, we have seen that the area tt is 3-1415926... But in this case the perimeter 27r of the polygon, and the circumference of the circle, become confounded ; therefore 7r=3-1415926...is the nume- rical expression for the semicircumference of a circle of radius 1 ; and as we have just seen, this number, as so many square units, instead of linear units, as here, expresses the area of the same circle. It will be re- membered therefore that tt units expresses the length of 180" of the trigonometrical circle, or of the circle whose radius is one unit. Cor. It follows from the above that the area of a sector aMbO is equal to the radius multiplied by half the length of the arc. 234. Unit of Circular Measure.— We have just seen that the numerical measure of J 80° of the trigonometrical circle is the number tt. Hence the length of one degree of the trigonometrical circle is 3*1416-;- 180, and consequently the length of J° is -3-57^- ^, therefore the length of A"" of a circle whose radius is R times 1 , or jB, is Length of A° to rod. R, : -1416 180 AxR-. -5236 30 AxR. UNIT OP CIRCULAR MEASURE. 193 For ex., the length of 15° of a circle whose radius is 10 feet, is x oO ] 50=2*618 feet. If the length of an arc of given radius B be known, we may also readily find the number of degrees in it ; for calling that number J, we have Length of arc 30 Length of arc ^ If the arc be equal in length to R, then there must be 57*295.., degrees in it, or more correctly 57°-2957795. And this arc is called the unit of circular measure : in any circle it is the radius, bent round that circle. The angle subtended by this unit-arc is taken as the unit of angular measure : the number of such units m any angle, subtended by an arc of _._.-- - , Length of arc ^. . _ any radius E, is therefore expressed by ~ . It, then, we drop the angular unit, which is the angle subtended by 57°-2957795, we may ex- press every angle by the abstract number which is the above ratio. When B=l, this ratio is simply ''Length of arc," which for A° is 3-1416 -— , 180 as already noticed above : suppose, for ex., that .4=60, then we should have angle 60°=7r -— - = 1-04719..., which is the length of 60** of the loO trigonometrical circle ; and the number thus expressing the length of any arc, of this circle (the circle whose seraicircumference is tt) is always what is here called the measure of the subtended angle. It would be well to call this the arcual measure of the angle, and its measure in degrees, the gradual measure, as proposed by De Morgan. Perhaps, to preclude mispronunciation, the latter term might be written gradeal. It is plain that although the arcual measure is not the same thing as the gradeal measure, yet that the sine, cosine, &c., of the one is the same 180° 180** as the sine, cosine, &c., of the other; so that sin , cos , =2 Part II. Theory of Angular Magnitude in General. 236. As introductory to the more general theory, which, dismissing the triangle, treats of angles without restriction, we shall return for a moment to the triangle itself, and deal with it in a manner more strictly algebraical. Referring to the diagrams at page 180, we see that c-=a cos B-\-h cos A, and similarly for the other sides a, and h ; that is, a=h cos C-l-c cos j5, 6=c cos ^ + a cos B, c=a cos B-^b cos ^^...[l] EXTENSION OF THE TRIGONOMETRICAL DEFINITIONS. 195 three algebraic equations from which all that concerns the plane triangle may be deduced. This is obvious, for any three of the six quantities entering these equations being given, the other three (except the given quantities be the three angles) can always be found by the rules of algebra. The exception here stated is suggested by the equations themselves ; for writing ma, mb, mc, for a, b, c, amounts to the same as merely multiplying each equation by w, so that cos A, cos B, cos C, would be the same for any equimultiples of a, b, c, as for these quantities themselves ; therefore the angles being given or fixed, does not determine fixed lengths for the sides. In order to find the cosines of the angles in terms of the sides, multiply the equations, in order, by a, b, c; we then have a^=ah COS 0+ac cos B, b^z=:bc cos A -\-ab cos C, c^=.ac cos B-\-hc cos A. And subtracting each of these from the sum of the other two, 62_^c^_^2_2Sc cos A .'. cos 4=:^!±H!z^ 26c a2+c2- J2=2ac cos B .-. cos B=—^ 2ac o2_j_j2_c2_2aS cos .\ cos C7=^^±^^ zao If the three sides are equal, we see that the angles must be equal : the cosine of each being - ; that is, cos QO''^ sin 80°=-. Though the three li lit equations [1] involve the whole theory of the plane triangle, yet we shall not further develope that theory. Like the formulae for the cosines, just deduced, all our results would present themselves in a form ill-adapted to logarithmic computation. The practical facilities offered by logarithms have caused additional labour to devolve on the theorist, inasmuch as his algebraical results must appear in a form fitted, as much as possible, for logarithmic computation. 237. Extension of the Trigonometrical Definitions. — An angle of a triangle can never become so great as 180"; and, in the popular sense, the term amjle is always understood to be an opening, within this limit. The meaning of the word is now to be extended : we are henceforth to regard any multiple whatever of an angle, as also an angle : and this extension is countenanced by Euclid, in Prop. 33 of Book VI., where he speaks of " any multiple of the angle BGG " as also an angle. In like manner, if angles be combined by addition or subtrac- tion, that is in any way united by the signs +, — , the result of the com- bination is an angle. Now for the symbols of algebra to be generally applicable to any species of magnitude, those symbols should always be interpretable in reference to that magnitude, and since the result of such a combination of angles as that just referred to may be negative, it is necessary for us, by appropriate conventions at the outset, to establish the meaning of a negative angle, consistently with the meaning given to a positive angle. What is laid down at page 173, respecting negative cosines, &c., will in some measure prepare the student for the extension of the term negative to angles. 2 196 EXTENSION OF THE TRIGONOMETRTCAL DEFINITIONS. Employing again the diagram at page 171, the conventions necessary to bring the general theory of angular magnitude under the dominion of algebra are as follows : — 1. An angle is regarded as generated by the revolution of a straight line AC, starting from a fixed position AB : it is a positive angle if the line revolve in the direction BCD, &c., and a negative angle if it revolve in the contrary direction BC'C, &c. : for it is agreed, in the former case, to mark the algebraic symbol for the angle, 'plu$, and therefore, consistently with the nature of the algebraic opposites, +, — , the symbol for the angle in the latter case is marked minus. 2. The line AB, the initial .position of the revolving line, is always marked + ; the directly opposite line AB' is therefore marked — . In like manner it being agreed to call AD, +, the opposite line AD\ is marked — . 3. The line DD' is regarded as a fixed boundary from which all lines parallel to AB, or AB' are measured; those parallels which like AB all lie to the right, are marked +, those which like AB' all lie to the left, are marked — . The line BB' is another fixed boundary from which all lines parallel to AD or AD' are measured; those which like AD are above this boundary are marked +, those which like AD' are below it are marked —. 238. These are the only conventions which are necessary. Being adopted by universal consent they render the theory of analytical trigono- metry universally intelligible : the algebraic symbols of quantity serve to denote the magnitudes of the lines and angles employed, and the algebraic signs +, — , mark the directions or positions of these in reference to fixed boundaries, while they at the same time discharge their office of signs of addition and subtraction. The lines to which we have just re- ferred are replaced in trigonometry by their numerical values, conformably to the hypothesis that AB i^ unit, and these numerical values take of course the same signs as their linear representatives. Thus : If the revolving line AC stop at C in the diagram, mC is plus, and Am is also 'plus ; therefore both the sine and cosine of the angle CAB is +. If the revolving line stop at C, m'C is plus, and Am' is minus ; there- fore sin CAB is ■\-, and cos CAB is — . If the revolving line stop at C", m'C" is minus, and Am' is also minus, therefore the sine of the angle generated is — , and so is the cosine. In like manner, if the revolving line continue onward to C", the sine of the angle generated is — , and its cosine -j- . And from sine and cosine all the other trigonometrical ratios are derivable (219). Again, if the revolving line starting from AB proceed in the opposite direction, generating the negative angle C"'AB, the sine of this angle is — , and its cosine -h. Of the negative angle CAB both sine and cosine are — . And it is plain that we have only to notice in which of the four quadrants BC, DB', B'D', D'B, the revolving line stops to determine the proper algebraic sign for the sine or cosine of the angle generated, whether that angle be positive or negative : — the sine and cosine, both as to numerical value and algebraic prefix, being respectively the same, whether the revolving line has terminated a positive angle or a negative angle. It is also obvious, that when the revolving line has performed more than a complete revolution, the sine and cosine of the angle generated is also the sine and cosine of the angle which remains after subtracting EXTENSION OF THE TRIGONOMETRICAL DEFINITIONS. 197 from the former as many times four right angles as there have been revo- lutions : thus sin 583*'=sin (583°— 360'')=sin 43", and cos 583°=cos 43*^; sin 762*'=sin (762''— 7 20*')= sin 42^ and cos 762°=cos 42«. In like manner sin 486''=sin 126", and cos 486"=cos 126°. The sine in this case is +, and the cosine — . A glance at the diagram shows, 1. That if the revolving line stop in the first or second quadrant, the sine of the generated angle is + ; if it stop in the third or fourth the sine is — . 2. If the revolving line stop in the first or fourth quadrant, the cosine of the angle is + : if it stop in the second or third, it is — . Every angle expressed in arcual (or circular) measure must be compre- hended in the general form 2w7r±:0 where measures less than two right angles : if the revolving line terminating the angle lie in the upper semi- circle, the upper sign applies, if in the lower semicircle, the lower sign applies. And since in determining the trigonometrical ratios for an angle the multiple 2w7r may be rejected, as mentioned above, and as the diagram sufiiciently shows, it follows that, however large the angle, the sine, cosine, &c., of it are equally the sine, cosine, &c., of an angle below 180°, either positive or negative: thus sin (720°dbl24°)=sin (±124°), and cos (720°±124°)=cos (±124°)=co9 124°. And generally, measuring less than two right angles, sin (2n/(l— sinM), and sin ^= ^(1— cos"^^)- Here cos A, and sin A, ought each to have two values, numerically equal, but opposite in sign. The double value for cos A is justified by what is shown in Part I. ; for as an angle and its supplement have the same sine, the angle A, under the radical, may be either an acute angle or its supplement, while in the former case, the cosine is -j-, and in the latter — : the double 198 SINE AND COSINE OF {A±B). sign is thus accounted for. But the double sign for sin A is not in- telligible upon any principle delivered in Part I. : now, however, we know that the cosine of A is the same whether A be positive or negative, but that the sine is in the one case +, and in the other — : the double sign for sin A is thus accounted for by a reference to the foregoing con- ventions. 239. What has been said above, in reference to angles and the trigo- nometrical ratios, equally applies to the arcs which subtend the angles, and to the linear sines, cosines, &c., of those arcs. By the definition (214), tangents of arcs all originate at B, and are measured along the touching line in one or other of the two directions BT, BT. The tangent limits the secant, so that wherever the arc originating at B may terminate, the tangent and secant of that arc always form, with the radius AB, a right- angled triangle situated to the right of the centre A. Moreover, the sine and cosine of the arc are always the same, in magnitude, as the sine and cosine of the arc, less than a quadrant, which terminates in the secant : thus the arc of which AT is the secant (p. 171), terminates either in C, or C, and it is plain that the sine and cosine Cm, Am are respectively equal in length to C'm' and Am'. Similarly the arc of which AT is the secant, terminates either in C"\ or C\ and the sine and cosine of one of these arcs are respectively the same in length as the sine and cosine of the other. And from such geometrical considerations, the trigonometrical truths, stated in the last article, might have been deduced, like as the relations at (219) were deduced from the trigonometrical circle, which re- lations we now know to have place, whatever be the magnitude of the angle A. 240. Sine and Cosine of (^±^).— Let AD, AE, in the an- nexed diagrams, be the tangents of any two arcs A, B, and OD, OE, their secants. Wherever the two arcs may terminate, their tangents and secants cannot fall otherwise than are here represented,* and they are equally the tangents and secants of the arcs AB, AC, each less than a quadrant : so that the one pair of arcs may replace the other as regards their tangents and secants. Now (228) and Euclid, Props. 4 and 7, B. II, we have Dir-=0D^-\-0E^-10D . OE cos D0E...[1'] DE^=AD^-{-A£^±2AI) . AE [2] Let the radius OA = l, then OD^—AD'^=lsLnd OE'—AE^=^, Hence subtracting [2] from [1], 0=1 -|- 1+2 tan A tan J5 — 2 sec A sec B cos (A±B), ^^ 1-f- tan A tan B . _ ^^ , .*. cos (A±B)=—^ — : Ti — = cos A cos B^^ sin ^ sin £...[1']. ^ '^ sec -4 sec B * Of course, in the second diagram, there may be below OA, a similar construction to that above it ; but it is unnecessary to encumber the figure with it. EXPRESSIONS INVOT-VING TWO ANGLES. 199 Again: by Euc. Prop. 35, B. I., the surface of a triangle is equal to half the rectangle of its base and altitude ; and any side of a triangle may be considered as the base. Taking DE for the base, OA is the alt., and taking OD for the base, OE sin DOE is the alt. : hence OD . OE sin DOE=DE. OA .«. for 0^=1, sin (A±,B)=. ^^ =:sin A cos B+ cos A sin B..J2'.'] sec A sec B 241. In the preceding investigation, the tangents and secants of the unrestricted arcs A, B, as noticed at the outset, are correctly represented as in the diagrams; but in deducing the expressions for sin {A±B), cos {A-^B), we have regarded these sines and cosines to be the same as those of the arcs [AB:hAC), where each component arc is less than a quadrant. To show that this was justifiable, put 0, 0' for these compo- nent arcs, regarding the circle as the trigonometrical circle. Then the arcs {A:hB), of which the components have the same tangents and secants as those of (9 ±6'), will all be comprehended in one or other of the forms 2n7r+(0±e'). or (2M4-l)7r + (9±0O. Now it has been seen (p. 197) that sin {2n7r-f (0±6'} = sin (9±e'), and that cos {2/i7r + (9±e')}= cos (0±fiO. Also that sin {(2ri + l)7r + (9±90} = — sin {{^±^')} and cos {(2ri + l)7r + (0±9Oi- = — cos (9±G'). This latter form applies to the case where only one of the arcs, as AB, is increased by it or by an odd multiple of it, and it is proper that the results should take the minus sign, for sec A will then be negative in [1'] and [2'], notwithstanding its geometrical position in the diagrams, because sec A= —. And here we may remark that, as concerns negative arcs or angles, it is not true that sec A=^ r'- the 1 in the numerator is the numerical repre- cos A sentation of that linear radius, of which the prolongation is the secant (see page 196) ; and in the case here spoken of it should be —1. 242. Expressions involving two Angles. — The expres- sions [T], [2'], to which the foregoing remarks apply, form the basis of the entire theory of angular magnitude ; we shall here write them sepa- rately for future reference. sin {A-\-B)=. sin A cos B-\- cos A sin J5\ sin {A —B)= sin A coaB— cos -4 sin 5 [ rj n cos {A-\-B)= cos A cos B— sin A ain Bt cos {A—B)= cos A cos B-\- sin A sin B) And from the same general expressions [Tl, [2'j, we further see that , , , „, tan A + tan B , , ^. tan A — tan B tan (A-^B)=:- -^ -, tan (A -B)=:-—- -.— ^...[11.] ^ ^ ^ 1— tan^tanii' ^ ^ l+tan^tanii and dividing num. and den. of each fraction by tan A tan B, and taking the reciprocals, we have ^,,,„^ cot^coti?-l ^,. „ cot^cotj5+l cot M-t-i?)=: — • — ---, cot (A—B)= — —5 7— f-...[IIL] ^ '^ cot B-\- cot ^ ' ^ cot £— cot A 200 EXPEESSIONS INVOLVING TWO ANGLES. If A+JB+a=lSO°, then tan {A-\-B)=- tan C, .'. [II. J> tan A tan B tan C7= tan ^+ tan 5+ tan C. If ^+5+C=90°, then tan {A+B)= cot C=- cot(^+5) .*. equating this with the reciprocal of the fraction for cot (A-\-B) in III., we have cot A cot B cot (7= cot A + cot B+ cot C. The first of these deductions shows that the product of the tangents of the three angles of a triangle (or of any three angles into which ISC' may be divided), is equal to their sum ; and the second, that the product of the cotangents of any three angles into which 90'' may he divided is also equal to their sum. And it is readily seen that the former property applies whenever the three angles amount to any even multiple of 90^ and the latter, when they amount to any odd multiple of 90°. If A, B, (7, be any three angles, unrestricted, we leave the student to prove, from [II.], that , . n ■ /-»v_ taii A 4" tan J54- tan C— tan A tan B tan G an ( + +^)— 1_ tan A tan B- tan A tan C- tan B tan C Other useful deductions from the fundamental formulae [I.] are the following : — 1. By adding and subtracting sin {A-\-B)-^ sin {A —B)= 2 sin A cos B\ sin {A-^B)— sin (^ — B)= 2 cos A sin -B f r-jY.] cos \a +B)-\- cos {A —B)= 2 cos A cos bI cos (A-^B)— cos {A—B)=—2 sin A sin BJ 2. By multiplying, sin {A+B) sin {A—B)= sin2 A cos^ ^-cos2 A sin^ B cos {A-\-B) cos {A—B)= cos^ A cos'^ ^— sin^ J. sin^ B and since sin^ ^-f- cos^ ^=1, and sin^ B-{- cos^ B=:l, these, by substitution, become sin {A +B) sin {A -B)= sin2 ^ -sin^ ^=cos2^- cos^ A\ ^^ cos {A+B) cos (^ — 5)=cosM-sin2 ^=cos25- sin^ ^j ■••'■-' so that sin {A-\-B) sin {A—B)z=.{sm A-{- sin J?) (sin J.— sin B). Again : since any angles may be put for A and B, substitute -{A-\-B) for the former, 2 and -{A — B) for the latter ; then [IV.] becomes sin A-\- sin i?=2 sin -(^4-^) cos -{A -B) ' ^ 2 sin A — sin ^=2 cos -{A + 5) sin -{A —B) cos A 4- cos 5=2 cos -{A +B) cos -(^ —5) A 2 cos 5— cos -4=2 sin -(,4 + 5) sin -{A —B) A A , J. . . sin 4+ sin 5 ^/ a t t,\ sin 4— sin 5 ^ 1,. j,. whence, by division, r— ;! = tan-(4.+5), ; -= tan -(A—B). cos 4+ cos -S 2' ' co3A-\- cosB 2^ tan^{4+5) and dividing the first of these by the second, -: — ;; : — =r= — . tan- (A—B) .[VI.], EXPRESSIONS INVOLVING SINGLE ANGLES AND THEIR HALVES. 201 . , . - a sin ^ ,„„, a+b sin ^4- sin 5 Now in the tnangle, -=- — - .*. (38) r=-: — -. : — jr, 5 sin jS ^ a—b sm-4— sm^ that IS, -= — -* tani(^-5) which establishes the rule, otherwise arrived at (227), for solving the second case of oblique triangles. T Ti V J- • • cos 5— cos -4 ^ 1/^ , CVA ^/A D^ In like manner, by division, -, -=tan-(44-^) tan -(A—£)t cos J. + cos ^ 2 2 cos B— cos A 1, , ^, . .^ . n = tan ^{ATB), sin-4±sin-fi 2^ sin ^ 4- sin -B cos B— cos A , sin ^ — sin B cos B— cos A so that = , and = ■: : — r. cos -4 + cos ^ sin A— sin B cos ^ + cos J5 sin -4 + sm B But these may be got from principles still simpler, thus since sin^ A + cos^ A= sin^ B-\- cos^ B .'. sin^ A — sin^ 5= cos^ B— cos^ A sin J[ + sin 5 cos B— cos ^ . , . , sin ^ — sin B cos B— cos A • I , ^— from Wiiicii "-^ — ^ . ' * cos B-\- cos A sin ^ — sin ii' cos B-\- cos A sin J.+ sin B' The above are the most useful deductions, involving two angles, from the fundamental formulae [I.]. But as any of the operations of algebra may be applied to these formulae, it is obvious that a variety of other results may be deduced. Instead of further multiplying these we shall proceed to derive some important expressions involving only single angles, their halves, and their multiples. 243. Expressions involving Single Angles and their Halves. — Putting A for B in [I.], and remembering that cos^J. = l — sin^^ and sin^J=l— cos'^^, we have sin 2^=2 sin A cos A, cos 2A= cos2 ^-sin2 ^=1-2 sin^ A=2 cos2 ^-l...[VII.]. Putting ^ for 5 in [XL], tan 2.4= ^_^°^7^ ...[VIII.] These expressions for sine, cosine, and tangent of 2A may be changed for those of -A u by putting this last for A^ we thus get sin ^=2 sin-^ cos -^, cos ^=cos2-^- sin2 ^/l=l-2sin2-^=2cos2-^-l...[IX.] 2tan-^ And tan -4= — ...[X.] l-tan2-^ If the first of [IX.] be added to and subtracted from 1= sin'* -A + cos2 -A^ A A (1 IV /I 1 \" sin -^-f- cos -^) , 1— sin^=f sin--4— cos-^ ) ...[XL] And preceding in like manner with the second of [IX.], we have 1+ cos A= 2 cos2 \a, 1- cos ^=2 sin^ i^...[XII.] A A These last expressions, for angles in general, were found useful at page 186, where A is an angle of a triangle. From [IX.] and [XII.], by division, we have .[XIV.] .[XV.] S02 TRIGONOMETKICAL AMBIGUITIES. sin ^ , 1 . 1— cos -^ /I— cos A ^^^-r^ ^ — -= taii-^= — : — -—=./— -...rxiii.] 1+coB^ 2 sm ^ V l-i-cos-4 " ■" It is worthy of notice that since sin 45°= cos 45°=*/-, therefore * A sin (^±4 5°)= (sin A± cos ■^)\/-^ .'• sin ^± cos A= sin (^+45'') . ^^2 In like manner, since cos (^±45°)=(cos A+ sin ^)\/-^ •'• cos -^+ sin A= cos (^±45°)v^2. Also, since tan 45"=1, .-. [II.], <.n (45«+^=l±^, tan (45»-^=;-^ .-. tan (45°+^- tan (45°-^)= ^^*^^f^ =2 tan 2^ 244. Expressions involving Multiple Angles.— Refer- ring to the formulae [IV.], put A for B, and [}i—\) A for A, and we then liave sin nA-\- sin (w— 2)^=2 sin (w— 1)-4 cos .4) [yVI 1 cos nA-\- cos (ti— 2)-4=2 cos {^—VjA cos A)" And writing 2, 3, &c. for n, we have for the sines and cosines of multiple angles, sin 2-4=2 sin A cos A, as found above. Also, since cos^ A=l— sin^ A, sin 3-4=2 sin 2 A cos -4— sin .4= 4 sin A cos^ -4— sin A=d sin -4— 4 sin-"* -4... [XVII.] &c. &c. cos 2A=.2 cos^ -4—1, as found above, cos 3-4=2 cos 2 -4 cos .4— cos A=4: cos^ -4—3 cos -4... [XVIII.] &c. &c. Making the same substitutions in [II.], we have tan (?i-l)^+tan^ rvrv i tan nA=- -~- — -. ——.... [XIX.] 1— tan -4 tan {n—l)A ^ -" 2 tan A which for %=2, 3, &c. gives tan 2-4=—— — — -, as before found, tan 2-4+ tan A 3 tan A— tan^ A „ ^^„ ^ tan dA=- — - — ^^j- — — = -, Q , o . > &c....[XX.] 1— tan A tan 2-4 1—3 tan^ A ^ ■* 245. Trigonometrical Ambiguities-— In the trigonometrical expressions established in the foregoing articles it has been seen, con- formably to what was stated at page 171, that the trigonometrical ratios, and not the angles to which they belong, alone enter the various formulae ; these angles A, B, &c., occur in the notation, but it is sin A, cos A, tan A, &c., which furnish the numerical values concerned. Now as these values equally belong to a variety of different angles, ambiguities may present themselves unless a linowledge of the angle itself, or at least of the limits within which it lies, precludes such ambiguities. Thus taking the expressions [XII.], 1 , /I— cos -4 1 ^ /14- cos -4 2^=\/— 2— '^^^2^=V— 2 • sm Here sin --4, cos --4, are each ambiguous, as to sign, in the absence of all knowledge respecting the magnitude of the angle A, to which the pro- TRIGONOMETKICAL AMBIGUITIES. 203 posed cosine belongs : for it may belong to either of several angles. If A be known to lie between the limits 360^ 2.360^ or 3.360^ 4.360° ; or 5.360^ 6.360", &c., then sin -A must be negative : for other limits it is positive. Again, if A lie between the limits 180°, 3.180"; or 5.180", 7.180"; or 9.180", 11.180", &c., then cos -A will be negative, and for other limits positive : the radical sign very properly provides for these different cases. And in all such instances of ambiguity the proper sign for the trigono- metrical ratio will always be readily found by noticing in what quadrant the angle to which it belongs terminates. In the following formulae [XXI.] derived from [XL], putting 2 A for A, there is a double ambiguity, sin A-\- cos A=^{l-{- sin 2A}, sin A— cos -<^=,^(1— sin 2A) sin ^—A n/(1+ sin 2^ + ^(1 -sin 2A) cos ^=^1 n/{1+ sin 2^) — ^(1— sin 2 J) .[XXL] By giving the double sign to each radical four values would be implied, and we might assign, as above, between what limits the angle must lie for any one of these four to be the true value ; but as already observed, by knowing the limits beforehand, we can always readily afiBx the proper sign ; we may just remark, however, that restricting the signs to those actually exhibited above, the forms will be correct for all angles A between the limits 180", 3.180"; and if the signs of the second radical in each be Interchanged, they will be correct for all angles between 0, 180" ; or 3.180", 4.180". The student will remember that the ambiguities here commented upon all arise from this source, namely, that sin ^= sin (27ur+^)= sin {2n-]-l)^—^}, cos ^= cos {2nir+fi). 246. We shall now give a few applications of the preceding formulae. (1) Required the sine and cosine of 15". By [IV.], we have cos (30°+ 15°)+ cos (30°-15°)=2 cos 30° cos 16° .'. cos 45°=(2 cos 30°-l) cos 15°. But (220), cos45°=^s/2, and cos 30°=iv'3 .-. cos 15°=--^^— -=]>,/ 2{>^S-^1). Again, by [IX.], sin 15° cos 15°=- sin 30°=- (page 195), .*. sin 15°= ^ = ^ =i*./2(*y3— 1), 4 cos 15° v^2(V3+l) 4^^^ ^' (2) Required the sine and cosine of 18". By [IX.J, sin 36"=2 sin 18" cos 18", but sin 36"=co8 54"=cos 3. 18"=4cos"^ 18"— 3 cos 18"by [XVIIl.]: hence, dividing by cos 18", 2 sin 18" = 4 cos^ 18"— 3 = 4 (l-sin^ 18")— 3=1—4 sin^ 18". Transposing, 4 sin^ 18" + 2 sin 18"=1, and the roots of this quadratic are -(—1± V 5), of which the positive root only is admissible. 204 APPLICATIONS OF FORMULAE. .-. sin 18"= (— 1 + V5). And since, as just shown, 2 sin 18°=4 cos^ 18°— 3, /. l(-l+v'5)+3=4 cosn8° /. cos 18°=ly5±^=^^(io-{-2V5). (3) Given cosec ^=sin J.4-tan -^, to find sin A. Transpose, then, by [XIII.]. 1 1— cos ^COS^ ^ . o M 1 o-# 9^. A ^ Sm A=— : -— =-: ; .'. COS -4=sin'^-^=l— cos''^ .*. COS'^^+COS ^=1 sin A sin A sin A .-. COS A=zh-1±^5) .'. sin ^=^/cos A=l^{-2±2>^5). 2 ^ (4) An object 6 feet high placed on the top of a tower subtends an angle, of which the tangent is '015, at a place in the same horizontal plane as the bottom of the tower, and 100 feet from it: required the height of the tower. CD Put fi for the angle DAC, and / for BAC, then tan 6= -— , and tan ^=f^ .'. tan ^-tan ^/=-?-=-06. Now [11. ], tan (6-^) = tan ^— tan / "06 ._ _ ^ :^ 'Olo, 1 +tan ^ tan ^ 1 +tan ^ (tan ^— '06) ' 4 tan2^--06tan^+l =1 .-. tan2 ^— -06 tan ^=3. Solvingthis quadratic, tan ^=1-7623... .-. tan ^=tan^— •06=1-7023... andlOOtan^= 170-23... feet, the height of the tower. (5) In a plane triangle ABC are given two sides a, b, and the in- cluded angle C to find the third side c, without first finding the angles A, B. By art (236), and formula [XII.], c2=a-+62-2a& cos C=a^-\-b'^-2al(l-2 Bm^hj\={a-Vf-\-iah mi^-0=z (a-6)2 + (2^/(irsiiilcy=(a-6)^|l + (|^sinic7y} Put now the second term within the brackets, (the square), = tan'' (p, ^ being an angle determinable from that square of its tan being known ; then G'^={a—hf sec'^9 .*. c={a—b) sec can never be greater than a-\-b, or than (^ar^^bf + ^^ab, otherwise (s/ar>j,^bf would be negative, so that the square within the brackets can never be greater than 1 ; we may therefore put it = sin^ =a, BCI)=a^, CnB=|i^, ACD=a^ ADC=(i^. From the triangles CBD, CD A, I)B=-A^-^, DA=^Ar''.'"l . • sin(«,+ /3,) sin(a2+/SJ Therefore two sides DA, DB, and the included angle of the triangle ADB, are given to find AB : and this we may do by logs, as explained in example 5. (8) Give a general formula suited to computation by logs of problem 4, page 178. Put CBD— a, A=ai, and BA=a, then DB=CD cot a, BA=CD cot a, .-. DA—DB=a=CD (cot aj— cot a) .'. CDz= cot a, — cot a sin a cos asj— cos a sin a, .*. DB=.CD cot azzzCD- — =a- -' by [1.1 sin a sin (a— a^) (9) Convert the expression cos a cos ^ + sin a sin ^ cos y into a form suited to logs. Since — =tan .-. sin=cos . tan ; hence the expression is cos a cos /3 (1+tan a tan /3 cos y). Put the second term in the brackets = tan 0, and the expression is n /i I X A - cos ^+sin ^ ^sin (^+45°) ,^ cos a. cos /3 (l+tan ^=cos « cos (i =cos a cos /3 -; — s/2, cos 6 cos by [XIV.], which is the form required. 206 APPLICATIONS OF FORMULiE, (10) Eliminate 6 from the equations sec 9— cos 0=m, cosec S—sin Q=n; that is, find the relation between m and n. _ 1 ^ 1— cos^^ sin* ^ 1 cos^^ Here m= -—cos e:= -— =: -, n=- — -—sin 6=— — - cos e cos fi cos 6 sin ^ sin ^ .'. m . n^=cos^ 6, and n . ifnP=siii^ 6 .'. (m . n^p-\-(n . m^)^=cos^ ^-f-sin^ ^=1. (!]) Prove that 2 cos {A +45«) cos (^— 45«)=cos ^A. By [XIV.] cos (^+45°)=— 7-(cos ^— sin^), cos (^—45°)=— -(cos -4+sin A) .-. 2 cos (^+45°) cos (^-45°)=cos2^-sin2^=cos 2J, by [VII.] Examples for Exercise. Prove the truth of the following expressions : namely, (1) cos*-4— sm*-4=sin 2 A. (2) tan -^+ cot r-4=2 cosec A. 2 2 « sin (A-{-B) (3) tan ^+tan B= \ ^ . cos A cos B /i\ . ^ ■„ sin (A—B) (4) tan ^— tan B= \ i. cos A cos B tan A +tan B sin {A-\-B) sin ^±sin B ^, .. x>\ (6) -=tan -AA+B). ^ ' cos ^+cos B 2> ' (7) cosec i4=l+tan ^ tan-^. A (8) cos 2^4- cos 25=2 cos {A-\-B) cos {A-B). (9) cosec 2-4 ^-sec vl cosec A. ^ ' tan ^— tan B sin {A—B)' (10) sin (^-f5+C)=sin ^ (cos 5 cos C— sin B sin C) H-cos A (sin 5 cos (7+cos B sin C^. (11) If ^+5+ (7=180°, then cos2^+cos25+cos2(74-2 cos -4 cos B cos C=l. sin -^+sin 3-<^-|-sin 5A (12) tan 34 = cos -4+cos 3-4+cos 6A' (13) Given sec A=zi sin ^, to find A. (14) Given cosec -^=sin -^+tan-4, find A. 2 (15) Given sin 2^=sin2 3^, find A. (16) Given 4 sin x sin 3a;=l, find x. (17) tan (a:+a)4-cos (^+a)=sin (a;— a)+cos {x—a), find a?. (18) Given tan -44-sin -^=m) ^ j , , i x- i. j. i ^ ' ^ . . ^ ?■ fiiid the relation between m and ». tan .^— sm A=.n ) (19) a sin^ A-\-h cos^ A=m\ h sin'^^+a cos^ B^n i show that — | — = 1 — a tan ^=6 tan 6 j a b m n (20) Show that by means of a subsidiary angle b, may be put in the form following, adapted to logs, namely, 2sin^-^+45°yv/a. INYERSE TRIGONOMETRICAL FUNCTIONS. 207 247. Inverse Trigonometrical Funclions-~If we had sin A='p, and wished to express the angle A itself, we might do so thus : " ^ = the angle whose sine is p." But it has been agreed that the nota- tion sin-'2? shall stand for the same thing, namely, " the angle whose sine is py In like manner cos~'g' means " the angle, w-hose cosine is 5." Similarly, log~'n means the number whose log is n, and so on. These are called inverse functions.'^ 3 4 (1) Prove that sin"^ - + sin"* -=90°. Since ^-l-5=sin-i (sin^ cos ^-|-cos A sini?), 6 5 by putting - for sin ^, - for sin B, and consequently - for cos A, and - for cos B, we 5 5 5 o have sin-. I + sin-.i = sin- (- . -+- . -)= sin-. 1=90". (2) Prove that tan - » - -f tan-^ r.=^^°- Since A + B= tan-» -^ —^ — -, by Z o 1 — tan A tan x> putting - for tan A^ and - for tan B, we have 2 o W- 1 1 2 3 tan-» - -f tan-i - = tan-i — — =tan-» 1=45°. 2 o ^ 1 i 2*3 From the above general expression for A-\-B,d>xi interesting theorem, due to Euler, may be deduced. Putting in that expression t^ for tan A, and ^2 for tan By it becomes tan-i <,— tan-i <^ = tan-^ -^— — ^ ...[1]. Now the following is obviously an identical equation, namely, ^,=(^,-^^+(^3-^3)+(^3-^4)+...+(^»-.-^«)+^«...[2]; that is, it holds, whatever be put for Ai, A^ &c. Put, therefore, .4,=tan-i<,, ^2=tan-' <2, ^3=tan-i t^, &c., then by [1], tan-i f.=tan-iii5^+tan-i -^+tan-i :^+ ...+tan-^ i-t-ijtj i-j-tgta ^~r^^^4 [The identity [2], above, has enabled us to arrive at this formula of Euler in a manner much more simple than is usually done]. Let 71=2, and take x—q=:0. 3. aP-{-px-\-q=0. 4. x^-j-px—q=0. 1. The roots of the first are a;=-'| 1+y//^ 1 l)f- l^^t q be less than -p% then we may assume ~^=sm^

Jq . tan ^ 0. 2 tan ^ ^ tan ^ cm ^ ^ ^ 1 ' ^ ^ 2 tan-^ Therefore, for the computation of the two values a„ a^ of x, we have log «! =2 log ?-log tan - ^+10 ) ^ J 11' ^^®^® ^°* *^° ^=log 2+-log y — log p+10. log (-aa)=2log y+logtan-^-10 J CONSTRUCTION OF TRIGONOMETRICAL TABLES. 209 The last two forms of the equation become the same as the first two by changing the sign of p : hence the roots of the third and fourth equations are the same as the roots of the first and second, respectively, with changed signs (141). Note. — Having determined one root of either equation as above, the Other root may be got by subtracting the found root from the coef. of x with a changed sign (68). We see that the determination of a root, by this method, requires five references to the tables : it would be found more readily by Horner's method, however large the coefficients p, q, may be. 249. Solution of a Cubic by Trigonometrical Tables.— Let the equation, deprived of its second term, be a;^—qx—r=^0, and put V 8 1 a!=^,theny^-qn^i/—rn^z=0. But[XVIIL],cos=^, aa=2(|) cos(^y+-)), a3=2 (|) cos(y-^). This method of solution applies exclusively to the case in which all the roots are real — Cardan's irreducible case : for since cos 3^ is less than 1 , Example. Let the equation be a?^—147ar— 285-5=0. Here cos3«=1142(^-i^Y=^=-4162=cos65°24', .-. ^=21° 48', .'. cos ^= -92849, \19o/ 2744 cos (120°4-21° 48')=cos 141° 48'=-cos 38° 12'=- 78586, cos (120°-21° 48')=cos 98° 12'=-cos 81° 48'=- -14263. And multiplying each of these by 2(49)5=14, we have a,=12-9988..., a2=-ll-0020..., a3=-l-9968. 250. Construction of Trigonometrical Tables.-— Although the existing tables of the trigonometrical ratios supply every practical want, and render the labour of constructing them anew superfluous, yet it is well that the student should have some idea of the facilities which analysis affords for accomplishing such a work. We shall, therefore, give here a brief sketch of the mode of proceeding, first establishing one or two necessary preliminaries. 1 An arc of a circle, less than a quadrant, is greater than the sine of that arc, and less than its tangent. 210 TO COMPUTE SIN 10" AND COS 10''. Let AB=AG, be an arc less than a quadrant; then Bm=:^Cm, is the sine of that arc : and since a straight line is the shortest distance between two points, BmC arc BA: hence the arc is less than its tangent. We may regard the arc here referred to as belonging to the trigonometrical circle, and to be the arcual measure of the angle 9, and may therefore conclude that, of the three values sin 0, 0, tan 9, the middle value always lies between the other two values ; therefore, of the 1 three values 1, -; — -, -, the middle value always lies between the other sm 9 cos two values, however small be taken. But the more is diminished, the nearer does approach in value to 1, which it ultimately becomes when is reduced to zero : hence -: — -, by diminishing 0, may be made to ap- proach as near to 1 as we please, and it actually becomes 1 when is reduced to zero. 1 0"^ 2. Although when is of any value, sin < 9, yet if < -^, sin 9 > 0— — . For [IX.] sin ^=2 sin - ^ cos 7; ^•••[1], and as shown above, tan - ^> - fi, or sin - ^> - ^ cos 7: ^ .'• [1], sin 6>6 cos^ -6, or > ^ ( 1 — sin^ - ^). 1 /I \2 ^3 ^\A%m^-6^--. 251. To compute sin lO'' and cos 10".— The arcual measure of 10" is 180.60.6 •x (art. 234) = 64800 •000048481368110. ..>sin 10". Also -?--i(^-?—Y= -000048481368078.. ....[3]. (cos ^=cos X COS y— sin x sm y) Also[l], [2], ^me={x-yy)-^A,{x-^yf-{-A,{x^-yf^-A^{x-[-y)T-\-...[4:l cos0= 1 +^2(^+2/)2+^4(^+y)*+^6(^+#+."[5]. Substitute the developments of sin x, cos a?, as given by [1] and [2], in [3], then the developments of sin 0, cos become ^ {y+A,y''-\-A,f+...){l+A,a?+A,x'+...)...i^ cose={\^A^u?-\-A,x'+A^a^+...){l-\-A^y''+A,y*-^A^f +...)- {x-\-A^a?-{-A,x^^...){y+A,y^+A,y'+...)...[1l Now, by the Binomial Theorem, the coef. of y in [4] is the series l^ZA^x^-\-5A^x^-\-lA^x^-^...[^] ; and from the last two factors in [6], we see that the coef. of y is 1+ A,x^+ A,x^+ A^x^-ir...[n The series [8], [9], being each the cojef. of ?/, in the two identical series [4], [6], are equal, whatever be the value of x : hence the coefficients of the like powers of x in [8], [9j, are equal. In a similar manner, by equating the coefficients of y in the two identical series [5], [7J, we have the two equal series = - X- A.^3i^- ^5:^5-. ..[11]. Equating now the coefficients of the like powers of x, in [10], [11], and in [8], [9], we have THEOREMS OP EULER AND DE MOIVRE. 213 2^2=— 1, 3^3=^2, 4^4=— ^3, 5^5=^4, QAq=—A^, &c. . __1 1_ . __}_ . _ 1 1 '— 2' ^~ 2.3' '""2.3.4' ^""2.3.4.5* ^~ 2.3.4.5.6' 63 05 g7 2.3^2.3.4.5 2.3.4.5.6.7^ 02 04 66 cos0=l 1- — h... 2 ^ 2.3.4 2.3.4.6.6 ^ ...[A]. 253. This latter series, in addition to its application in higher in- quiries, is of use in solving Case II. of plane triangles, whenever the given angle is so large as to render the Tables inefficient in giving its cosine with accuracy. Thus, C being the given angle, and tt— 6 its arcual measure, 9 will be very small ; so that in the series for cos 0, the third term, and the terms following, may be neglected. We may, therefore, write -l)(->-2)(..-3) ^„_, ^ ^^, ^_ sin n 6=71 cos™-* 6 sin 6 — cos"-^ 6 sin^ 6-\- Ji.O ^^_l)(^_2)(,i-3)(7i-4) „.,.,, -i ' „ ' ^ cos^-s ^ sin^ ^— . . . 2.0.4.5 cos 2^=:cos^ ^— sin^ 6 cos 3^=cos-'^ 6— 3 cos 6 sin^ 6 cos 4^=cos* 6— 6 cos^ 6 sin^ ^+sin'' 6 cos 5^=cos5 ^—10 cos^ 6 sin^ ^+5 cos 6 sin*^ &c. &c. .'. sin 2^:=2 cos 6 sin 6 sin 3^=3 cos2 6 sin ^— sin^ 6 sin 4^=4 cos^ 6 sin 6— 4 cos 6 sin^ 6 sin 5^-=5 cos* 6 sin ^—10 cos* 6 sin^ 6-\- sin^^ &c. &c. 262. Development of in powers of tan 0.— By Euler's forms, we have e*^=cos 6-\-i sin 6, e~*^=cos 6—i sin 6. Dividing the first by the second, g.fl cos^+tsin<> 1+ttan^ m i • xi. i i- i. -j g''»«'= — - — -r—, — -=:; — — Taking the nap. log of each side, cos ^— I sin ^ 1— ttan^ o x- & id=- log -i4^^=i tan ^+-(» tan fff-^-Ai tan fff^^i tan 6)1 -\- ... A 1 — % tan B o o i by (112). Dividing by t, and remembering that i^, i\ i^, &c., are alternately — and +, * In these expressions, as also in the analogous expression in the preceding article, the exponentials might have been replaced by the forms [5]. Thus, this last might have been written (u— %-^)"=(w"— w"")— %(i4"~2_i4-cn-2)j^,^. EULER*S AND MACHINES SERIES FOR Jw. 217 111 ^=tan fi-- tan' ^+- tan^ 6 -- tan' 6-\-... Ill fw. or, tan-i <=«_-<3_^_<5__<7+... This is called Gregory's series for the arcual measure of an angle in a series of ascending powers of its tangent, it having been first discovered by James Gregory (though certainly not in the above manner), in 1771. [Gregory held the chair of Mathematics in St. Andrew's College, Aber- deen.] The series is convergent only so long as tan does not exceed unit : beyond this limit it is useless for the purposes of calculation. When tan 6=1, that is when G= -, we have 4 a series which, though convergent, is too slowly so for the numerical deter- mination of ir. The following are more suitable. 263. Euler's Series for ^^r. From the general series [3] page 207, putting 1 for «p and limiting the number of terms to two, we have tan~*l=tan~*T— — ^+tan~*«2» where «2 may be any value we please. Put *2 = Q» ^^^^ W6 hOi.'VQ o ^=tan-U=tan-il+tan-ii= by [1], (^-^-13+^.-...)+ which are Euler's series for ^tt. 264. Machin's Series for J-^r. — Referring to the same general series at page 207, and taking five of its terms, we have t^-.l=tan-.g+.^-.^+tan-.^^+tan-.ii=|+tan-.,„ where any values we please may be put for the fractions. Let each of them be=- ; then we easily get 2 7 9 1 11 0R ^^r. page 207, and the series [1], above, developments of 6=tau~'i^ for any value of not exceeding - tt, may be multiplied to any extent. In the first series all the tangents t^, t.^, &c., except one (which may be amj one) may be chosen at pleasure, or all the quantities, or angles except one, connected with the symbol tan"', may be chosen at pleasure. 265. Development of i 6 in Sines of its IVIultiples.— In the logarithmic theorem [3], at page 81, Nap. log m=={m—m~^)—-{m^—m~^)-\--{m^—m-^—{m*—m~*')-\-... iS o 4 put c^« for m ; then (254), i0=2t sin 0— -(2i sin 20)+ (2i sin 30)— (2t sin 40) + ... 2 3 ' 4 therefore, dividing by 2i, -0=sm 0— -sin 20+-sin 30— -sin 40+.., 2 2 o 4 which, for 0=^^, gives the same series as that before found (p. 217). 206. Development of sin 0, cos 0, in a series of Fac- tors. — By Algebra (p. 109), if a^, o^, &c., be the roots of ^X + ---=0, Now by [A], page 213, if sin 0=0, that is, if 0=0, or ^nm=z\-\-2niei- 2 2.3 ' 2.3.4 ■ 2.3.4.5 2.3.4.5.6 i (27i*)» (27i,r)* (2^«f_, \, /- (2n^)3 , (27i^)'> 1 =1^ — r-+TO--2:3X6i+-"j+i2^'--2T-+2-:3X5--"-r The first of these two series is the development of cos ^nnr, and the second the development of sinSnTr (page 213) ; and since cos 2w7r=l (n being or any positive or negative whole number), and sin 2ri7r=0 .*. e~«'^^= l...f 1]. Hence taking the nap. logs, 2w7r2=log I. If w=0, this is = log i, which being the only real value of log 1, is that employed in arith- metical computation. But when logs are applied to expressions where imaginary quantities are recognized, the imaginary values of log 1, which are comprehended in 2w9ri, must not be neglected if our conclusions are expected to be general. Nor, in such a case, may the logs even of negative numbers be neglected ; for that such logs exist under an imaginary * For an interesting deduction from Wallis' s approximation to the value of «•, which finds its application in the Theory of Probabilities, see De Morgan's Trigonometry and Double Algebra, 1849, p. 61. S20 KUMBERS OF BERNOULLI. form, will appear from putting Sn + l for 2n, in the above development, and remembering that cos (3?i + l)7r= — 1, and sin (27i + l)7r=0, so that e(2»+l)7rj=:_l .'. (2?i+l)a-i= nap. log— 1, .'. No real log exists. Multiplying by N, • iV^e(2»+l)7u=_iV.*. nap. log iV+(27i+l)5ri=nap. log— iV. The logs here are Napierian, but by introducing the modulus M, corre- sponding to any base a, and considering 2n and Qrz + 1 to be either positive or negative integers, as we are at liberty to do, we may generalize the foregoing results thus : — log N= real log N±2Mn^^-l, log-N= real log N±{2n+l)^^-l...l2]. 269. The conclusions arrived at in the preceding article enable us to give to the development of 0, marked [1] at page 217, a generality which at present it does not possess. It is plain that 6 or tan~'i, has any one of the infinite number of values comprehended in diW9r-|- 0, for the same value of tf whereas the development [I ] gives but one value, and that the true value only when 6 does not exceed -tt, even admitting divergent series. The discrepancy arises from suppressing imaginary logs, when applying logs to an imaginary quantity : — it may be rectified thus : — iV=-log — r- — -=±ntri+i tan ^4--(i tan ^^+-z{i tan ^)*4- &c. Ji X. — ii tan o o .'. ^=±7i?r-f-tan ^— -tan^^-J-- tan^^— &c. o 6 Rejecting angles whose tangents exceed 1, as leading to diverging series, and putting dots for the " &c.," the second member of this equation repre- sents the first in all its generality. The development in (265) is, in like manner, defective : — it should be 1 11 -^=±»jr-f sin ^— -sin 2^+-Bin 3^— ... 2 2 o ' And here we would invite the student's attention to an apparent analy- tical paradox for which he is perhaps but little prepared : he might think it a fair inference from the above that 0=±2ww+2(sin 9—- sin 29h-...) li has all the generality of the equation from which it is deduced, but such is not the case: for writing 20 for 0, in that equation, we find that 6=±W9r+sin ^0—^ sin 40-|-..., which two expressions for cannot be equal, inasmuch as wtt has twice as many values as 2?i7r : — odd multiples of TT being excluded from the latter. 270. Numbers of Bernoulli.— There are certain numbers thus called, from the name of the analyst who first invited attention to them, which are of frequent use in certain trigonometrical and exponential developments : we may arrive at them by investigating the development of -i--. Since (110), ^ NUMBERS OP BERNOULLI. 221 unity divided by it must give a series of the form 1 -e' A e- _^ =——= {-Aq—A^x+A^x^—A^x-^-A^x*— ...,'. Addiugy ^3+-'+->7r^+jr^-r^=0, &c., l-^=:-l=2Ao-h2A,x'-[-2A,x*+... .-. (94), .lo=-i A^z=0,A,=0, &c. Now multiplying [1] and [2] together we have which being an identical equation, the product furnished by the second member of it must be such that the coefficients of the powers of ^ must each be zero, A being =1. We may arrive at the coef. of x by reversing A A the first two terms — \-Aq of the multiplier, writing them thus Ao-\ — , and then multiplying each by the term above it. In like manner for the coef. of x\ reverse the first three terms of the multiplier, and then multiply each by the terjn above it : and so on. We thus get A.A.Aq 1 2 "^2.3 2. 3. 4"^ 2. 3. 4. 5 from which conditions we have to determine only A^ -4,, ^3, &c., for before seen, A^, A^, &c., are each 0. • • ^0—2' ^""6 • 2' ^~ 30 • 2.3.4' ^~42 * 2.3.4.6.6* *°* 6' 30' 42' [3] thus, putting for uniformity Bq for Ao, and remembering that B^ B^, &c., are each 0, from which, by multiplying as before, we get the following relations, namely, 2^0+1=0, 35,+35o+l=0, 452+6J9,+45o+l=0, 5B^-\-l0B^+10B^-\-5Bo+l=0,6B^-\-15B.^-]-2QB^-{-15Bi-{-6Bo-\-l=0, and so on, the numerical coefficients being those of the developed binomial, the leading coef. being absent. In these equations of condition we already know that B^, B^, B^, &c., being all connected with the even powers of x, are each 0, and deter- mining the others, in order, we readily, find that And these are the numbers of Bernoulli. The development of -^37 is therefore 222 SUMMATION OF TRIGONOMETRICAL SERIES. 271. From the results of the foregoing investigation, it appears that for any BernouUian number we have the general expression ^»4.9N7?_ 14.1/«4-9N (^4-2)( r.+l) » {n-\-2) {n^l){n){n -l) 2:3 ^«-^- [Note, — In some works, Bernoulli's numbers, when all taken with positive signs, are denoted by B^, B^, &c. It may be remarked here that, since B.j=B^, the latter need never be computed.] Eequired the development of tan ^ in terms of ^ by Bernoulli's numbers. , sin ^ 1 eiO—e-iO 1 e2J»— 1 1 /, 2 \ tan ff=z =— . — =:— . ^— . I 1 I. cos^ i ei^-\-e-^0 i eSifl^i i \ eM-^l/ 1 e^—1 e'+l e'+l e'—l e^'—l J—=l-?LZlB,x-?—^B,a^- ^'~-^ B,3^- &c. e'+l 2 2 ' 2.3.4 ' 2.3.4.5.6 * Substituting 2i4 for x, and multiplying by 2, this becomes 2 22(22—1) . 2^(2*— 1) o-g ■■ ■=1 ^ Bi{^-\ — B.^ifi~... Subtracting from 1 and dividing by t, 272. Summation of Trigonometrical Series.— l. Let it be required to find S, the sum to n terms, of the sines of a series of angles in arith. prog., namely, iS=sin ^-hsin (^+«)+sin (^+2a) + ...+sin {^+(w-l)a}. By [IV.], page 200, 2sin-«sin^ =cos ( ^— - a ) — cos ( ^-fn* ) 2 sin -a sin {^-\-a) =C0S ( ^+^a ) —cos ( ^4-^* ) 2 sin -a sin (^+2a)=cos ( ^+xa ) —cos ( ^+^« ) 2 sin -« sin {4-\-{n—l)ct}=cos | ^+-{2n—Z)a [ -cos | ^+-(2w— l)«l By adding up these, we have 2S sin -«=cos ( ^— -« J —cos | ^+-(2n— 1)« l .^na cos( ^— "« ) —cos] ^-}--(2m— 1)« [ sin ] P+~{n—l)a [sin -7 •. fe ^ /^ -= ' ' ^ ' ...[I]- 2 sin -a sin « * The author is not aware that this formula has ever been given before. SUMMATION OF TRIGONOMETRICAL SERIES. 223 2. Let it be required to find the sum S of n terms of the cosines of a series of angles in arith. prog., namely, S=coB ^+cos(^+«)+cos (tf+2«)+...-|-cos {^+(7i-l)«}. By [IV.], page 200, 2sin-«cos^ =sm{^+-aj—8m(i—-aj 2 sin -a cos (^+a) =sin ( ^+-» ) — sin ( ^+n* ) 2 sin -a cos (^+2a)=sin ( 6-\-- . wa;— 1 u~^x—\ Bringing these fractions to a com. dem., 2%S=. — ^ ; — a;^— («*+%- »)a;+l w-ia;(w-'»+i ««+ ' — M- "a;"— Ma;+l) _ (it"— 'M-")a; «+2— {u'^-'r i —%-(«+ »)a;''+ > + (w— ■jt-^)ar a:*— (t4-f%-')x+l x^— (it+%-")a;+l — — — a;"+-sin w^— a;""*''sin {n-\-V)6-\-x sin ^ ~ x^— 2x cos ^+1 Examples for Exercise. (1) Required the cuhe root of cos 3^-|-sin %6^J—\. (2) Required the fifth power of cos 2^— sin 26^—1. (3) Prove that sin n6=Q0B" sfn tan ^- ^ ^~ ^ ~ ^ tan3^-t-...\ cosw^=cos"^( 1 i— — ^tan^^H — ^^ ii- — i^^ ^tan*^— ... ) \ 2 2.3.4. / (4) Required the sum of n tei-ms of the series sin ^— sin (^-}-a)+sin (^-|-2«) — ... /ex Ti- J XI. 1 » sin^+sin 3^+sin 5^-l-...+sin(27i— 1)^ (6) Find the value of — ■— — ; )■- -f . ^ ' cos^-|-cos 3^+cos 5^-i-...+cos(2w— 1)^ (6) Find the value of 5'=:cosec ^-fcosec 2^-f-cosec 4^-|-cosec 8^-|--- -+00380 2"-'^. (7) Find /S'=tan ^+2 tan ^-|-22tan 22^-H23tan3^4-...+2«->tan 2«-'^. (8) Find S=:x cos ^-|-^^cos 2^-|-a:^cos Z6+...-\-x^ cos nd. * The student will notice that the second of these fractions is got from the first by simply changing u into w"*, and in like manner is the second of the two fractions following got from the first of them. IMAGINARY ROOTS OF ±1. 225 273. Imaginary Roots of ±1. — Let x stand for either of the nth roots of positive unity indifferently, that is let ^"=1. By De Moivre's Theorem, . . ^ X- 2m . . 2m ., - m a;=(cos 2m^+z sm 2mir)"=cos — ^r+i sin — it=:1" ...LlJ» % n which general expression for 1» is real only for such values of n as will render sin — tt zero, that is, for such values only as will render — tt either 0, or a multiple of w. Now if n be odd, we shall keep clear of these real values by putting, for the arbitrary integer m, the successive values 1, 2, 3, &c., stopping when 2»i=7i— 1, that is when m=-(n—l); 1 L ■we shall thus get - {n—l} pairs of imaginary values for 1", that is w— I imaginary roots, and no more ; for if after getting the imaginary value due to 2?/i=7i— 1, we were to continue the substitutions, the former values _i would recur, as is obvious. Hence, when n is odd, !« has n—l imaginary roots, all included in the form [1], and a single real root, namely, cos = 1. But if n be even, we shall avoid the real values in [I] by putting for m, the successive values 1, 2, 3, &c., stopping when 2?n=n— 2, that is when m=- (n— 2); we shall thus get x (n--2) pairs of imaginary values for 1 , that is, n— 2 imaginary roots, and no more; for if after getting the real value due to 2m =n, we were to continue the substitutions, the former imaginary values would recur. Hence, when n is even, 1" has n— 2 ima- ginary roots, and two real roots, namely, cos 0=1 and cos 9r= — 1. If we combine the factors of j;"— 1, as given by [1], in conjugate pairs (139), recollecting that cos^ + sin^=l, we shall have the following decom- position of a;"— 1 in quadratic factors, united, when n is odd, to the simple factor (x— 1), namely: — For n oddf x«— l=(a;— 1) | (^x^—2 cos —x-\-l\fx^—2 cos — x+1 Y.. For n axn, x'-l=(x'-l)\{x'-2(!oa-^+l)(x'-2ix>s —x+l\.. the quadratic factors within the braces being, in the first case, -(n— -1) in number, and in the second case, -n in number. From each of these, equated to 0, a pair of imaginary roots will be obtained. For the n values of (—1)", that is, for the n roots of «"+l, we have by De Moivre's Theorem, Q 226 CONSTRUCTION OF LOG SINES AND COSINES. a;={cos (2m+l)«'±isin (2m+l)«-}'*=cos ^±i sin — — *=(—!)".. .[2], which general expression for ( — !)"• is real only for such values of n as will make the sine zero. If n be odd, then for m-= -(n—i), we have x= — 1, the only real value, the -{n — 1) pairs of imaginary values being got by putting successively 0, 1,2, &c., up to ~{n—l)—l, for m. But if n be even, there can be no real value included in [2] ; for then 9r cannot become either 0, or a multiple of v : all the n values are therefore imaginary, and the -n pairs are found by putting in succession 0, 1, 2, &c., up to -(n— 2)for vi. Hence, as above, /i For n odd, x»+l=(x-^l) I (^xi-2 cos-x-\-l\^ cc^- 2 cos— ^+l\.. For neven, x''-\-l=\ fx^—2eoa-x-\-l jfaP— 2 cos— x-\-l)... Ex. Required the cube roots of 1, and —1. From a;:=cos — — ±* sin , n n ^ 2jj, 2vt by putting for m, and 1 successively, we have x=l, and a;=cos — ±i sin — = 3 o cos 120° ±1 sin 120°= -cos 60° ±i sin 60°=-l±'-^=^^^^^—=A 2 2 2 * • r- (2m+l)cr , . . (2m+l)* ^ .,. ^ , Again : From a; = cos +i sm —, by putting for m, n n we have x=- + i sin -=cos 60°±* sin 60°=-+——=: ~ , o o 2 2 2 -1 ±n/-3 .. ^.. , l±v/-3 , and (-1)J=_1, or . Examples for Exercise. (1) FindiV^i, oTl^l/N. (2) Find(-iV)i, or(-l)*V^. (3) Find the values of IK (4) Resolve ic^"— 2cos ^ . «"+! into its n quadratic factors, and thence show that all the roots of the equation x^^'—px-]- 1=0 can be found whenever p is not greater than 2. 274. Construction of Log Sines and Cosines.— At a 12, we promised to advert to the method of computing log sines, and SPHERICAL TRIGONOMETRY. 227 log cosines, without first finding the natural sines and cosines. The method is this : — From sin ^=^r 1 2)0~"9^^~^ )( "'■""^T^ /■"' ^* P^S^ 218, we have by putting — . r f or ^, and taking tlie logs of both sides, ,0, sin = . ^=.0. =+H I+.0. (l-^)+lo. (l-^,)+.o. (1-^,)+... . , . . , w^ . , «' . , (2n-{-m)(2n—m) Now the first three terms of this series are log — |-log o+^^g ' Wl = log r+log m+log (2n,+m)+log {2n—m)S (log %+log 2). The developments of the terms which follow these three, when con- veniently arranged, are (p. 8J) -fG-^e.sV-)G)'-- And by giving suitable values to m and n, the log sine of any angle may thus be approximated to, to any degree of exactness. Log cos — . - may be found, in a similar manner, from the factors of the cosine at page 218, and thence the log tangents, and log secants. [On this subject the reader may consult the Appendix to " Woodhouse's Trigonometry."] END OF THE PLANE TRIQONOMETRZ. III. SPHERICAL TRIGONOMETRY. 275. Preliminary Theorems. — Before entering upon the theory of the Spherical Triangle, it will be necessary to establish a few elementary propositions concerning the circles of the sphere, the arcs of which form the sides of every such triangle. The following principles (1, 2, 3,) sufficiently prepare the way for these. 1 . Three points anyhow situated in space all lie in the same plane ; for having joined any two of the points by a straight line, we may con- ceive a plane to pass through the line, which plane by being turned round the line, as upon an axis, must at length arrive at the third point : so that in that position all the three points must lie in the plane. Any three points, therefore, not in a straight line, taken in a plane, completely determine the position of that plane. [We thus see, what has hitherto been tacitly assumed, that the figure inclosed by three straight lines is always a plane triangle,] 2. The common intersection of two planes is a straight line ; for if among the points in the intersection there be three which are not in the same straight line, since these three are in both planes, the planes must coincide and form but one plane. 3. Definition. — A straight line is said to be perpendicular to a plane, when it is perpendicular to every straight line in that plane, drawn through its /oof, that is, through the point where it meets the plane. Q 2 2*28 PRELIMINARY THEOREMS. 276. Theorem T. If a sphere be anyhow cut by a plane, the section is a circle. Let C be the centre of the sphere, and ADB any plane section. Draw Cc perpendicular to the plane, and from c, the foot of this perpendicular, draw any line cD in the section, terminating in some point 1) in the surface of the sphere : then the angle CcD must be a right angle (by 3, above). Draw CD ; then wherever on the surface the point D may be, CD is always of the same constant length, namely, the radius R of the sphere, .*. cD= ^{R'—Cc') : hence the length of cD, that is, of any straight line drawn from c to the boundary ADB, &c., is constant, .'. c is the centre, and ADB, &c., the circumference of a circle, cD being the radius r. The less Cc is, the greater does rz=^{R'—Cc") become : it is, there- fore, the greatest possible when Cc=0, that is, when the section passes through the centre C of the sphere. On this account a circle, the plane of which passes through the centre of the sphere, is called a great circle of the sphere, while every circle of which the plane does not pass through the centre C is called a small circle of the sphere. Through any two points on the surface a great circle may be drawn, because these two, with the centre, make three points, which (by 1 above) are all in one plane. . A circle of the sphere of one kind or other may always be drawn through three points on the surface, because a plane may be made to pass through them ; and every plane, as just shown, produces a circular section of the sphere. If the line Cc, perpendicular to the circular section, be prolonged to pierce the surface in the points P, F', these points are called the poles of the circle ; if the centre c of this circle do not coincide with C, the centre of the sphere, the poles of it are unequally distant from its plane ; but if c coincides with C, that is, if the section be a great circle, its poles are equidistant from the plane of that circle. Since PP' is a diameter of the sphere, and since planes may pass in all directions through PP\ cutting the surface in great circles, it follows that an indefinite number of great circles may be drawn through the poles of any other circle. The distance of any circle from either one of its poles — measured upon any one of the great circles through both poles — is constantly the same, that is, the arcs PB, PD, &c., are equal, because Pc is the common versed sine of all those arcs : hence the other arcs P^B, P'D, &c., must be equal. Every point, therefore, in the cir- cumference of a great circle is 90° distant from each pole of that circle : so that if a circular radius equal to the quadrant of a great circle be applied to the surface of the sphere from any fixed point, and the other ex- tremity be made to describe a line, that line will be the great circle of which the fixed extremity of the circular radius is one of the poles. 277. II. Two great circles always intersect in two points at the distance of a semicircle, that is, the circumferences bisect each other. For since the plane of each passes through the centre of the sphere, the intersection of these planes must be a diameter of the sphere com- mon to both circles, and it is at the extremities of this diameter that the circumferences cross each other. Hence, if from any point on a sphere two quadrantal arcs can bo drawn PRELIMINARY THEOREMS. 229 to two points in any great circle — \Yliich points are less than a semicircle, or 180° apart — then the first point must be a pole of this great circle. For it is necessarily a pole of some great circle passing through the pro- posed points, and as only one great circle can pass through two points less than 180° apart, the pole must belong to the great circle mentioned. 278. Measure of a Spherical Angle. — The sides of the spherical triangles considered in Trigonometry are all arcs of great circles only, and the angle included between two such arcs, that is, a spherical angle, is measured in a manner similar to that in which a plane angle is measured. For the measure of a plane angle we take the intercepted arc of the circle whose centre is at the vertex of the angle : the number of degrees in this arc is the gradeal measure of the angle : the number of units in it (radius being 1 unit) is the arcual measure. Similarly, for the measure of a spherical angle we take the intercepted arc of that great circle whose pole is at the vertex of the angle, the linear radius of this great circle being that of the sphere : thus, the spherical angle JBPD is measured by the arc QF, of which P is ihe pole, and CQ, the radius of the sphere, the linear radius : expressed in degrees, this arc is the gradeal measure : expressed in units of the radius, it is the arcual measure. But the plane angle QCF is also measured by this same arc : hence the plane angle at C, formed by perpendiculars to the line of intersection PP\ of the two planes through C, in which the sides of the spherical angle are situated — a perpendicular being in each plane : this plane angle C has the same measure as the spherical angle P. • It is obvious that the plane angle at C, here referred to, is the angle which would be presented by a section of the two planes in which the arcs PB, PD, are situated, by a third plane through C, and perpen- dicular to the common edge PP^ of the two former. Every plane parallel to this third plane, that is, perpendicular to the edge PP', would give for section an equal plane angle, however the two planes meeting in PP' be prolonged. But the plane of the angle TPt is equally perpen- dicular to PP' : hence the angle TPt is also equal to the spherical angle P, to the sides of which the sides of the plane angle are tangents : so that the measure of the plane angle formed by two tangents to the sides of a spherical angle, at its vertex, is equally the measure of the spherical angle itself. 279. III. Any one side of a spherical triangle is less than the sura of the other two sides. Let be the centre of the sphere, and ABC a spherical triangle on its surface. Draw the radii OA, OB, OC; then the three angles at will be in three distinct planes : we may consider the angle AOB to be in the plane of the paper, and the other two to be in inclined planes above the paper. These angles will be mea- sured by the arcs AB, AC, BC, respectively, the radius, OA, or OB, OC, being unit. Let ^^ be the greatest of these three arcs : then it will be only necessary to prove that AB<{AC-\-CB), or that the angle AbB<(AOC+BOC). In the plane of AOB draw any line A'B\ and make an angle BOD in that plane equal to BOC in the inclined plane : make also OC^=OD, and draw A'C\ B'C. Then by construction, the two sides OB', OD^ and the in- cluded angle, are respectively equal to the two sides OB', 0C\ and the 230 PBELIMINAR-X THEOREMS. included angle, .-. B''D^=B'C\ But in the plane triangle A'CB\ A'B'<{A'C'-{-B'C'): take away the equals B'D, B'C\ .-. A'n of the former is less than the third side A!G' of the latter, the angle A'OD180''— C, .'. 180°>^ + B—G.'. 90°>aS'— C, and so of the others. Neither is there any am- biguity about the radical signs : they are always positive, because ^ * ^^ always < 90^ (280). S34 FORMULAE FOR SIDES IN TERMS OF ANGLES. The product of the first and second of [9] gives for the sine of a side sin a= . ^ . ^ n/{-cos S cos (S-A) cos (S~B) cos (,5- (7)} ...[101. sm ^ sin C/ 288. The preceding formulae for sides in terms of angles, have been derived from the like formulae for angles in terms of sides, by a reference to the supplemental triangle : and by availing ourselves of this latter, we may always pass from any expression involving whole angles and sides only to one similarly involving sides and angles. All we have to do is to write for sin, cos, in the one formula, sin, —cos, of the opposite quantity, whether side or angle, in the other ; thus the fundamental relations [A] become in this way changed into the following, which are equally true : — cos -4 =— cos B cos C+sin B sin C? cos a \ cos 5=— cos -4 cos (7+ sin ^ sin (7 cos J V ...[B], cos C?=— cos-4 cos 54- sin A sin j5 cos c j from which we infer, like as at p. 232, the following geometrical truths, namely, 1. If two angles are equal, the sides opposite to them are also equal. 2. If two triangles have two angles and an interjacent side of one, equal respectively to those of the other, the remaining angle in the one will be equal to the remaining angle in the other. 3. If the three angles in one be equal, respectively, to the three in the other, the sides of the one will be respectively equal to those of the other : the equal sides being opposite to the equal angles. 4. From the first of these inferences, combined with its converse at p. 232, it may be easily proved that in every triangle the greater side is opposite to the greater angle, and conversely. Let the angle A, in the triangle DAB' nX, page 239, be greater than the angle B' ; and conceive ^^ to be drawn, making the angle B'AB=B'; then BA=BB\ Add DB to each, then DB+BA=:DB\ but (279), DB-\-BJ>DJ, .-. J)B'> DA. Conversely: let DB'>DA, in the triangle DAB'; then A>B'. For if these angles were equal, then (by ]) the opp. sides would be equal, which they are not ; and if B' were greater than A, then, by what is proved above, DA would be greater than DB\ which it is not, hence A>B\ 289. As in what follows we shall have frequent occasion to pass from a triangle to its supplemental one, it will save trouble to the student if we here place before him, for the purpose of ready reference, the changes necessary to be made in thus passing from one triangle to the other, in the cases that will be required. We have noticed above — what is suf- ficiently obvious — that when only whole sides and angles enter, uncon- nected with their multiples, or submultiples, or with their sums and differences, the only change necessary is to write —cos for cos, and con- sequently, —sec for sec, — tan for tan, —cot for cot. But as seen at (287), these are not always the changes when other functions of the angles enter : we here give the directions necessary for what follows : — 1 . For sin a write sin A, and for cos a write — cos ^, and vice versa. 2. For sin -a write cos -A, and for cos -a, sin -A, and vice versa; A A A A and consequently, for tan - a write cot- A, and vice versa. Napier's analogies. 235 3. For sin {a±:h) write —sin (A-±B)j and for cos (a±5) write cos (A±B), and vice versa. 4. For sin or cos of -^{d+h) write sin or -- cos of - (A + B), and for sin or cos of r (a— 6) write —sin or cos of - (A—B), li li and .-. —tan - [A-±.B) for tan - (a±5). 290. Napier's Analogies: Two parts and the part between them given. — In the expression [A] for cos a, substitute for cos c its value as given by the third equation : we shall then have cos a=cos a cos^ J-fsin a sin 5 cos h cos C+sin 6 sin c cos A .*, cos a (1— cos^ 6)=cos a sin^ 6=sin a sin h cos 6 cos C+sin h sin c cos A .*. cos a sin 6:=sin a cos 6 cos C+sin c cos ^ .*. sin c cos ^ =cos a sin 6— sin a cos 6 cos C. ) ..^ _ Similarly, sin c cos 5=cos 6 sin a— sin 6 cos a cos C. ) **■'• Adding, sin c (cos ^ +cos jlS)=:sin {a-\-h) (1— cos C). Now [3], sin A sin c=sin a sin C, sin J? sin c=sin & sin C .-. (sin A ±sin J5) sin c=(sin a±sin&) sin C [12]. Dividing [12] by the preceding equation, we have sinyl+sinJ5 sin a ± sin & sin C cos A +COS B" sin (a+6) * 1 —cos C* . A . ' -^ H . . . , COS -(a— 6) ■r. , / ««,^^ sin^±sinB . 1,, ^, , r^^ -, sma+sino 2 Bat (p. 200), -= -=tan-(^±5); and [VI.], . ,, = — :; . ^ -"cos^+cos-B 2^ '' '• ■" sin(a+6) 1, * cos-(a+6) . _ sin-(a— 6) . „ 2sin-Ccos-C sina— sinJ 2^ ' ., sinC 2 2 1_ _-_ — — — = — . Also -= =cot-C. sin(a+6) .1 1-cosC ^ . -1_, 2 sin-(a+6) 2sin2-C 1 cos-(a-&) ^ ^ sin -(«-&) ^ Hence, tan -{A-\-B) = — cot - C, tan -{A—S)= — cot - C. cos-(a+6) sin-(a+6) And from the supplemental triangle (see art. 289) we have the cor- responding forms J cos-(^-^) J ^ sin- (4 -5) ^ tan - (a+h)= — tan c, tan - {a -h) = — tan - c. cos- (A+B) "" sin- (^+5) These four equations furnish Napier's four analogies, namely, cos- (a+J) : cos - (a—b) : : cot- C : tan - (A+B) ) I I I I -P^i an- (a+5) : sin - (a—b) : : cot- C : tan - (A—B) cos-(i4+5): cos-(4— -B): :tan-c : tan - (a+6) A Z A A mi-{A-\-B)\ sin -{A—B): :tan-c : tan - (a- J) .[14], 236 SOLUTION OF ETGHT-ANGLED TEIANGLES. which supply solutions to the cases in which the three given parts are two sides and the included angle, or two angles and the interjacent side. From the first of these analogies it is plain that cos -{a + h), tan -[A -\-B\ must have the same sign: hence a + 6, A-\-B, are either both greater or both less than 180°, an inference which will be useful hereafter. 291. Solution of Right-angled Triangles.— Let ABC be a spherical triangle, right-angled at C: then, since sin C=l, and cos (7=0, we have from the formuloB [3], [A], and [BJ, the five following equations, namely, sin a=sin c sin ^, sin &=sin c sin J5, cos c=:cos a cos 6, ^^ cos A =cos a sin B, cos B=cos 6 sin -4 . Let these be written in order one under another, and supply a third member to each, thus : Substitute in the second member of each equation the values for its two factors as given by the other equations: for instance, for sin c, sin A, in the first equation . . ^.^ ^ sinb cosB . , , sm a=sm c sm A, substitute — — 7-, ;-, as eiven by the second and sm B cosb fifth equations, and so on : we shall thus have 1. sin a =sin c sin A =tan b cot B. 2. sin b =sin c sin jB=tan a cot ^. 3. cos c =cos a cos 6 =cot A cot B. 4. cos yl=cos a sin J5=:tan h cot c. 5. cos ^=cos b sin A =tan a cot c. It is plain that these five formulse supply solutions to all the varieties that can occur as to the three given parts {two besides the right angle) in a right-angled sph. triangle : but the whole of them may be compre- hended in two short and easily-remembered precepts which we shall give in the next article : we shall here deduce one or two useful inferences from them. 1. The shortest great-circle arc that can be drawn from any point on a sphere to a given great circle, is the perpendicular from that point. It is evident that this perpendicular (a) cannot exceed 90° : suppose it less than 90°, and c any other arc from the point making the angle A with the great circle ; then, since from (1) above sin «^0% then D^>90% and conversely. Hence the angles adjacent to the quadrantal side are of the same affection as the sides opposite to them. 2. The triangle will be ambiguous whenever the given parts in the right-angled triangle ABC are one of the perpendicular sides, AC, BC, and the angle opposite to it (p. 238). (1) In the triangle DAB, where 2>^=90% are given ^=54° 43', and D=42" 12', to find ^i5 and J5. Here since DAB (Parts for seconds.) — 23 — 184 ^ -1768 -56576 [=-=«"" + 322 +12236) + 223 +12934 J =+25170 20-505775 -316 2)20-505459 10-252730 681 /)=494) 4900(10" -315.90 sin («-a), colog '927694 sin («-6), 9-738434 10-666128 Correction for sees. —443 tan-^, Subtract 10-665685 10-252730 6815 Z)=626) 23000(37" sin (s-a), colog 0-927694 sin («— c), 9-836745 10-764439 Correction for sees. —436 tan-^, Subtraa 10-764003 10-252730 tan- (7, 17° 7' 31 "4 9-488727 492 i)=748)23600(3r'i Hence the angles are A=12r 36' 20", 5=42° 15' 14", (7=34° 15' 3". Note. — The correction for seconds is got from the " Parts for seconds " above. Examples for Exercise. (1) Given a=47° 45' 51", 6=73° 54' 13", c=44° 30' 22" to find A. (2) Given a=48° 27' 8", J=53° 55' 21", c=48° 28' 57" to find A. (3) Given a=60° 18' 48", 6=32° 37' 20", c=47° 47' 14" to find A, (4) Given a=63° 60', 6=80° 19', c=120° 47' to find A, J?, and 0. (5) Given a=40° 0' 10", 6=50° 10' 30", c=76° 35' 36" to find A, B, and C. B 242 THE THREE ANGLES GIVEN. 296. II. The Three Angles given.— The formuloe for «, any one of the three sides are (295) 1 y-cos S coa{S-A) 1 _ /cos (>S^--B) cos (.9-C) sin --«=*/ : — ^— : — -pz , " 2 V smB sm O cos -a: ■V- sin B sin 1 _ . —cos S cos (S—A)_^ an 2 '^-Vcos (S-B) '^i^-C)'' when all the sides are to be determined, apply the third formula to find one, a, and find b, c, by the equations ^ 1^ COB (S-B) 1 1 cos(-S'-0, 1 tan-6= — — — — tan -a, tan - c= — — --, tan -a, 2 cos(-S— ^) 2 ' 2 coB{S—A) 2 like as in the former case, the logarithmic formuloe being as follows (See the example). logtan-6=log tan~a— {logcos (o&) : : cot-C : tan-(^+J5), 2 A A jL asxd sin j:{a-\-l) : sin -{aojh) : : cot-C : tan -{A(^B). A A A A And A, B, being thus found, the side c is found by either of the other two analogies. 1 1 11 qob-{A(\jB) : cos -{A-{-B) : : tan -(a+5) : tan -c. A A A A 11 11 or sin -{A(:>jB) : sin -(^+5) : : tan -{aoJb) : tan -c. 2 A 2 2 Note. — Tn the iirst of the above analogies, if the leading term is minus, the final term will of course be also minus, and conversely ; that is, -(A + B) will be acute or obtuse according as ■x{ci-\-h) is acute or ob- tuse : there can, therefore, never be any ambiguity in this case. (1) Given a=80° 19', 6=120° 47', and C=51'30', to find the other parts. 1. 27ie angles -4, B, computed. cos i (a+ J), 100' 33' colog 0-737327 sini(a+J), colog 0-007404 A A COS- (a<^&), 20 14 9-972338 sin-(a(>oJ), 9*538880 A A cotic7, 25 45 10-316644 coti(7, 10-316644 A 2 tani (A+B), 95" 22' 37" 11-026309 tan| (AcsiB), 36° 6' 17" 9*862928 5791 854 i)=2257) 61800(23" i)=442) 7400(17" By addition and subtraction of the half-sum and half- difference we have ^=131° 28' 54" and J5=59° 16' 20". Observe that- {A-\-B) is obtuse because- (a +5) is, A A The angles A, B, being now known, we may proceed to find c by one or other of the second pair of analogies. But when the third side only is required to be computed, it is not the shortest way to arrive at it by first computing the two angles adjacent to it, as above ; it is better to employ for the purpose a subsidiary angle, as explained in the next article. 298. Side computed by a Subsidiary Angle. —In the general formulsB [^4], we have cos c=cos a cos h+sina sin 6 cos C=cos a (cos 6-f tan a sin 5 cos Cf), R 2 244 SIDE COMPUTED BY A SUBSIDIARY ANGLE. Assume tan a COS C=:cot «=-; — , sm M sin a cos 6+sin I cos a cos a sin (6+to) .•. cos C=:COS a : = : . sin ea sm a Hence, knowing a, b, and c, we have the following formulae for com- puting c, namely, /> o cos a sin (5+ «) __ 1. cot «=tan a cos C, 2. cos c= : — ^^ ^...[11. sin a> We shall compute c in the last example, first by Napier's Analogy, assuming A, B, to have been found as above, and then, by the formula just deduced, independently of A and B. 2. The side c computed hy Napier's Analogy. {Partsfor 1 «^'^-') ^?> The table is, of course, cosj(^oo^), 36° 6' 17" colog '092594 - 153 - 2601 entered with the sup- 2 ^ 1 plement of - (^4-^), cos -(^+5), 95 22 37 8-972289 -2238 -51474 2 and the difference tani (a+6), 100 33 0* 10729923 ""^^^^ ^ . i)=-2238, 2 is multiplied by 9-794806 60-37=23. _, . - , , „^ It would have come to Correction for seconds -^89 the same thing if we had entered the table tanL, 31° 54' 46" 9-794317 ^'^f }^T Z.^^^" ^V-' 2 and had then multi- 2 101 plied the same Z), taken with opposite sign, by /. c= 63 49 32 D=469)21600(46" 37. See (226). * This tangent need not be taken from the tables : for, logtan=:logsin—logcos+lO=(10—logcos) — (10— logsin)+lO=cologcos— colog sin+10, and these cologs are already exhibited above, in the work for A and B. Or the tangent may be taken out at the same time with the sine and cosine, in anticipation of the work which is to follow : whenever seconds occur in the arc, it is better to do so. We take this opportunity of remarking that, in the solution of triangles, whether plane or spherical, the work will be expedited if a hlanh form or skeleton of the whole operation be prepared before the tables are touched : This done, the blanks left for the several tabular quantities should then be filled up, so that all the use possible may be made of the tables when once in hand. If two or more extracts are to be made from the same page, although these may have to be inserted in remotely-different parts of the work, yet they should aU be taken out with the first extract, and each put in its proper place in the blank form, in anticipation of the future demand for it. Room should always be left towards the right of the blank form for the corrections for seconds, in imitation of the plan adopted in this work. Such an orderly and systematic mode of arranging this part of the operation will preclude aU confusion, as well as facilitate a revision of the entire process. It may be mentioned here, that when, as in the case of - (^A +-8) above, we have to 2 take out the difference D for an obtuse angle, it will be prudent, directly the angle is written down, to put some mark (as an asterisk) against the seconds, as a reminder that since we enter the table with the supplement, the D taken out is to be multiplied by what the seconds want of 60". We consider this to be the preferable mode of proceed- ing in all sucn cases, as there will be then no risk of taking the contiguous I> for that which is really the correct one. TWO ANGLES AND INTERJACENT SIDE GIVEN. 215 TJie same computed hyformvlcB [1]. tana, 80° 49' 10-767935 cos a, 9-225833 cos C, 61 30 9-794150 sin a, colog 0-578143 -768 -33024 sin (6+«), 9-840854 219 + 3723 cot «, 15° 19' 43" 10-662085 6=120 47 437 9-644830 -293,01 Correction for sees —293 (6+«)=136 6 43i>=826)35200(43" cose, 63° 49' 33" 9-644537 680 i)=428)14300(33" The small difference of 1", in the two results, arises chiefly from u being 15" 19' 42''A, instead of 15'' 19' 43'', as we have here taken it. Where fractions of a second are disregarded in the several items of the work, small discrepancies in results obtained by different methods are to be expected. Examples for Exercise. (1) Given two sides 60° 10' 27", and 40° 0' 14", also the included angle=121° 36' 24", to find the remaining angles. (2) Given a=38° 30' J=70°, 0=31° 34' 26", to find the remaining parts. (3) Given a=84° 14' 29", J=44° 13' 45", (7=36° 45' 28", to find the other parts. 299. TV. Two Angles and Interjacent Side Given. — This case is also solved by Napier's analogies ('290), which need not be here repeated. The second pair determines the sides, after which the re- maining angle may be found by either of the other two analogies, or inde- pendently, by means of a subsidiary angle : thus, A and B being the given angles, and c the given side : — Since cos C=cos -4(tan A sin B cos c— cos B), put tan A cos c=cot «=-; — , _. R\n B cos a»— sin a cos B cos A sin (B—u) .'. cos C=cos A : = : — ^ •• sin e* sm at Hence knowing A, B, and c, we have the following formulse for (7, namely, 1 . ^ . „ „ cos ^ sin (B—u) rm 1. cot fl»=tan il cos c ; 2. cos C= ^ ^...[21. sin u (1) Given ^=39° 23', J5=:33« 45' 3", c=68° 46' 2", to find a, 6, C. 1. 2%€ ddes a, b, computed hy Napier's analogies. cos h^A ^B\ 36° 34' 1" -6 colog 0-096198 sin \{A +5), colog -224926 2 2 co8i(l(N;5), 2 48 68-6 tan^c, 34 23 1 9-999475 9-835243 ^xu\{AcsoB\ 8-691374 9-835243 tan^(a4-&), 40° 23' 49" 9-929916 708 tani(a<^^)» ^° 1^' ^S" 8-751543 49740 2)=427)20800(49" i)=3749) 180300(48" Adding and subtracting, a=43° 37' 37", 6=37" 10' 1". 246 TWO ANGLES AND INTERJACENT SIDE GIVEN. •rt* - 00 00 c» >o J— ' o" oo I— 1 c> tH CO o 00 i>. ^ = ? n-{A-{-B) : cot -(7, COB -{AnoB) {A+B) : tan-(a+&) tan-c. Since two angles answer to the same sine, the result of the proportion [1] is ambiguous. It is necessary, therefore, to ascertain from other considera- tions what the character of the angle B must be when only one triangle can exist with the proposed data, and under what circumstances two triangles may involve them. The formulae [A] at page 232 give cos 5=(cos 6— cos a cos c) —-sin a sin c, from which we may derive the following conclusions, namely, 1. If cos b be numerically greater than cos a, the second member of the equation must of necessity take the same sign as cos b. In other words, B and b must be of the same affection if sin fe<8in a. Hence: An angle is of the same affection as its opp. side, if the sine of this side is less than the sine of the other given side. 2. If cos b be numerically less than cos a, then the sign of the second member of the equation will depend upon the value of cos c : this may be such as to render the product sin a cos c still less than cos 6, or such as to make this product greater than cos b : thus cos B will be + or — according to the value of c, or cos B may be taken either + or — con- sistently with some admissible value of c. Hence : An angle is ambiguous if the sine of its opposite side is greater than the sine of the other given side.* In addition to these the following principles also will be found useful. 3. If a+6>180°then^+5>180^andif a + 6<180nhen ^ + ^< 180° (p. 236). 4. The greater side is always opposite to the greater angle, and con- versely (p. 234). (1) Given a=80° 6' 4'', 5=70° 10' 30^ ^=83° 15' 7", to find the other parts. 1. Computation of the angle B. D. -37 76 sin a, 80° 5' 4" colog 0-006538 Bin 6, 70 10 30 9-973444 sin^, 33 16 7 9-739013 , 44 sin5, 3r34'38" 9-719039 8909 D =343) 13000(38" 321 {Parts for sees.) -148 *, 2280 2247 43.79 From either of the principles (1), (4), above, it follows that the angle B, here de- termined, is acuie. * The author conceives that these tests will be fonnd more simple and convenient than those, from Legendre, given by most modem writers on Spherical Trigonometry. 248 SUBSIDIARY ANGLE FOR COMPUTING C, C. 2. Computatimi of the anffle G. cos -{a(Kjh), 4° 57' 17" colog 0-001623 cos-(a+6), 75 7 47 tan ^(^+5), 32 24 52 1^ cot-C, 80 42 38| D. {Parts for sees.) 0-001623 18 306 9-40.9682 -793 -37271 9-802513 465 24180 -28 -27,85 9-213690 4198 •. 0= 161° 25' 17" D=1Z21) 50800(38"^ 3. Computation of the side c. 2). {Tarts for sees.) cosi(^(v^B), 0°50'15" colog 0-000046 03 045 cosi(^+-B), 32 24 52 9-926511 -134 -6968 teiil(a+6), 75 7 47 10-575497 848 39856 329 .. tan|c, 72 32 30 J 10-502383 159 829,33 .•.c= 145" 5' 1" D=736)22400(30"4 Note. Cos - (A-\-B), and tan -(a +6), employed in this last operation, are both to be taken from the table when the extracts for tan -(^+B), and cos - (a-j-b)y used in the previous operation are made. (See the foot- note, p. 24:4.) Examples for Exercise. (1) Given a=63° 50', 5=80° 19', ^=51° 30', to find the other parts. (2) Given a=40° 36' 37", 6=91° 3' 25", ^=35° 57' 15" : show that £ is ambiguous, admitting of two values, and find the angle C for the case of B obtuse. 301. Subsidiary Angle for computing C, c— If either of these alone be required, we may effect the solution, without first finding £, by aid of a subsidiary angle, thus : ^y [11] ps^e 235, sin c cos-4=cos a sin 6— sin a cos h cos C. Divide each side by sin a, and then in the first put -: — r for -: — , and we shall have sm-4 sin a TWO ANGLES AND AN OPPOSITE SIDE GIVEN. 249 sin c cot A =cot a sin J —cos & cos C. Multiply by tan C and transpose, sin C+tan A cos b cos C7=cot a sin 6 tan A. Assume tan a>:=tan A cos 6, tten sin./^,\ ixz- sin CH COS C=cot a — r , or sm Ccosw+smw cos(7=sm(C+«)=cota tano sin». Hence, for computing (C+«), we have 1. tan a»=tan -4 cos & ; 2. sin(C+«)=cota tanft sin*; and then subtracting «, the angle C is found. Again : By [A], page 232, sin b sin c cos il +cos 6 cos c=cos a. Divide by cos &, then tan 6 sin c cos .4 +cos c=: r. Assume tan «=tan 6 cos A. and we have cos 6 sin* cos a sin c tan a»-|-cos c=sm c f-cos c= = cos ej cos .: sin c sin «»+cos c cos*=cos (C7— «)= rcos u. Hence, for computing (c— «) we have cos COS a 1. tan «=tan b coaA: 2. cos (c— «)= r cos « ; cos 6 and adding tty the side c is found. 302. VI. Two Angles and an Opposite Side Given-— The side opposite to the other given angle is found by the proportion [1] of last case, and the remaining two parts by Napier's analogies. Since the proportion [1] gives a sine, there may be a doubt as to ■whether the side to which it belongs is acute or obtuse, or whether it may be either indifferently. To decide this it is suflScient to observe that in the formula cos 6= (cos ^+cos A cos C)-r sin A sin C, if cos B be numerically greater than cos A, B and b must be of the same affection : but if cos B is numerically less than cos A, then values may exist for C that will render cos h either positive or negative. Hence 1. A side will he of the same affection as its opp. angle, if the sine of this angle be less than the sine of the other given angle. 2. A side is ambiguous if the sine of its opposite angle is greater than the sine of the other given angle. (1) Given ^=33« 26' 7'', 5=130° 5' 22", a=U° 13' 45" to find the rest. 1. 27ie side h computed. D. {Parts for sees.) Bin A, 32° 26^ 7" sin B, 130 5 22 sin a, 44 13 45 colog 0-270578 9-883617 9-843466 141 -331 177 216 9-997802 797 -2317 6726 9720 141,29 Here h is ambiguous, because sin ^> sin A, its values are either &=84° 14' 25", or J=95° 45' 35". The former value is taken in next page. i)=20) 500(26" 250 SUBSIDIARY ANGLE FOR COMPUTING C, C. 2. T/ie side c computed. cos -(AcK^B), D. (Parts for sees.) 48° 49' 37"4 colog 0-181464 241 9038 coB^iA+B), 81 15 Uk 9-182196 -1369 -60921 tani(a-f6), 64 14 5 10-316321 -492 538 2690 -491,93 tan^c, 25° 33' 5"^ 9-679499 71 c= 61° 6'10"| D=541)2800(5"^ 3. The angle C computed. 1 cos-(a+J), 64 14 5 tan i (4+5), 81 15 44^ cot - C, 18° 22' 42"! D. (Parts for sees.) g 0-027014 77 1540 9-638197 -486 -2180 10-812720 1402 62389 617 617,49 10-478548 849 D=704)30100(42' '1 C= 36° 45' 25"^ Note. Cos - (a+b), and tan -{A + B) are to be taken from the table at the same time that tan ^ («+^) and cos -{A-{-B) are taken in the pre- ceding operation. (See foot-note, page 244.) Examples for Exercise. (1) Given 4=70° 39', B=48° 36', a=89° 16' 53"i, to find the rest. (2) Given 4=103° 59' 67" '5, 5=46° 18' 7" '25, a=42° 8' 48", to find C. (3) Given 5=59° 16' 20", C7=131° 28' 54", c=120° 47', to find a. 303. Subsidiary Angle for computing C, c— As in the last case, these two parts may each be computed, independently of the part b, thus : In article (301) it was seen that sin C+tan A cos h cos C=:cot a sinS tan A. Apply this to the supplemental triangle, remembering (289) that cosines, tangents, and co- tangents change sign, and there results sin c— tan a cos B cos c=cot A sin B tan a. Assume tan «=.tan a cos 5, then cot A sin B sin et sma* Bin c cos c= ^ cos« cos 5 cos . ^ ^ cos^ . ^ . ^ . /^ V cos^ . -: sm C— cos C= .♦. cos u sm C— sm <» cos C;=sin (C— *)= ;: sin tt. sm « cos 5 cos B Hence, for computing (C—a), we have 1. cot «=tan £ cos a : 2. sin(C— «)= ?;sin*». cos 5 Note. — In applying the several formulsB involving subsidiary angles, care must be taken that the influence of algebraic signs be not overlooked : suppose, for instance, that of the three given parts A, B, a, the only one that is obtuse is B ; then cot u in the formulae just deduced would be nega- tive, and if we were therefore to take u obtuse, we should evidently have a negative value for sin (C—u), to avoid which (though the avoidance is not absolutely necessary), we must regard u as negative, so that C—u will in this case be C+w. As an illustration, let (7 in the example at p. 250 be computed by these formulae : tanJ5, 10-074648 cos a, 9-855342 70 D. 427 -205 (Farts.) 16226 -9225 70,01 2). cos J?, colog 0-191031 250 cosil, 9-926351 -134 sin •», 9-881584 179 162-69 (Parts,) 9500 -938 6707 152.69 sin(C-«), 9-999118-69 8 2)=13) •69(5" .•.C-*»= 86° 21' 5" •=-49° 35' 37" J /. C= 45' 27"i, as before, [nearly. cot«, 9-930060 220 i)=427)16000(37"*47 .». .=-49° 35' 87"i Here *» is negative because tan B is nega- tive, and because sin », in next opera- tion, in order to give a positive value for sin {C—u), must be negative, seeing that cos B is negative. Note. — The angle C—a is ambiguous, being given by a sine, and sin ^>sin ^ : — the acute value of it is taken. The difference of 2'', be- tween the former determination and that here arrived at, is easily accounted for: the value of D, namely 13, is so small, that an error even in the figures usually rejected from the *' Parts," is sufficient to affect the accuracy of the correction for seconds. It would have been over-refine- ment to have included these figures — as they are nearly always inaccurate — every D being only an approximation, and consequently, when mul- tiplied by the seconds, giving a product the last two figures of which may be widely erroneous : — it would have been over-refinement to have pre- served these, but for the purpose of actually giving practical proof to the student of the desirableness of avoiding sines of very large acute angles, and cosines of very small ones, in fact, all such trigonometrical ratios, of which the differences (D) are very small (see art. 229). We may here notice, in conclusion, that the student will find, upon 252 AREA OF triangle: spherical excess. examination, that all the formulae involving the subsidiary angle a, may be obtained by dividing the triangle into two right-angled triangles, and then applying Napier's rules. 304. Area of Triangle : Spherical Excess.— Two great circles of the sphere always intersect each other in two equal angles, at the distance of a semicircumference (277), and the portion of the spheric surface inclosed by the two semicircumferences is called a Lune. It is plain that whatever part the angle of the lune is of four right angles, the same part is the lune itself of the whole spheric surface ; that is, if A° be the angle, 360° : A*" : : surf, of sphere : surf, of lune=r^ xsurf. of sphere. It will be proved hereafter that if R be the radius of the sphere, the ex- pression for its surface is Surface =4:7rB\ so that the surf, of a lune, of A angle ^^ is Lune =— — 27ri^ [1]. Let now ABC loO be any sph. triangle, and let the sides AC, BO he pro- duced till they again meet. In the diagram, the circle ABA'B' is the base of a hemisphere above the plane of the paper : the prolonged sides will therefore meet in a point C in the remaining hemisphere, beneath the plane of the paper: the concealed portions of the meeting semicircles are represented by the dotted lines. Now the triangle A'B'C, in the lower hemisphere, will \ / evidently be equal to the triangle ABC in the upper; c' for AA\ BB\ CC\ are all equal, each being a semi- circle; therefore taking away the part CA\ we have AC=:A'C'. In like manner, BC=B^C\ and the angles at C, C", are equal : the two triangles are therefore equal. The triangle ABC is a portion of the lune BB\ it is equally a portion of the lune AA' ; and the equal triangle A'B'C is a portion of the third lune CC» Hence the whole surface of the hemi- sphere is equal to the sum of these three lunes diminished by the two angles ABC, A'B'C, that is, by twice the triangle ABC. Putting there- fore S for the surface of the triangle, we have [1], ^^.=^^,.n^.,s ... .=d±^-J?5!,^= PI which is the numerical expression for the surface of any sph. triangle, in terms of its three angles and the rad. of the sphere. It follows from it that two triangles are equal in surface, provided only that they are on equal spheres, and that the sum of the angles of one is equal to the sum of those of the other. The numerator of the above expression is called the Spherical Excess : it is the amount by which the three angles of a triangle exceed 180°, and is an element of considerable importance in trigonometrical surveying on an extensive scale. This spherical excess is, we see, an angular magnitude ; and we know that magnitudes of this kind admit of two distinct modes of measure- ment — the gradeal and the arcual : it begets confusion to represent the thing measured by the same symbol when diJBferent units, of measure are employed, and when it is at the same time of importance that we know EXAMPLE FROM TREGONOMETRICAL SURVEY. S53 which unit is exclusively taken in the case before us : the expression for the Spherical Excess, . r, -sr 180= zn (jradeal measure^ is -^= ^3 • in arcual measure, it is e= — ...[3]. If, as in all the operations of trigonometry, we regard the radius to be unit, then we shall have s=S, that is, the number representing the spherical excess, in reference to the angles of a triangle on the surface of a sphere of radius 1, is the number representing the surface of the portion of that sphere which the triangle occupies ; the latter number being of course so many square units. This explanation is necessary, in order that the student may know what writers on this subject mean, when they say that '• the area of a triangle on the surface of a sphere, of radius unity, is equal to the excess of its three angles above two right angles." Given the surface S oi & spherical triangle on the surface of the earth, in square feet, to determine the number of seconds in the spherical excess. Calling here the excess B, we have [3] B=.— . : and taking (as found from mea- surement) 365154*6 feet for the length of a degree of the earth, we have 7> o«r:ifr>.«lSO, ^ _ ^^ ^ 3-14159-Sfx602 i2=365154-6-feet .-. ^=--^^^^^_ degrees==3g^^3^— seconds, .-. log JI=log 5+log 62-83185.. .-2 log 365154 •6=log 5^-9-3267737. Hence the following Rule. If from the log of the area of the triangle in feet, the constant log, 9-3267737 be subtracted, the remainder will be the log of the spherical excess in seconds. [This rule, usually called '* General Roy's rule," is due to the late Pro- fessor Dalby, who communicated it to General Roy while engaged by him on the great Trigonometrical Survey of England and Wales]. 305. It will be easy to compute the spherical excess by the above formula when the surface of the spherical triangle in feet is known. In geodesical operations, the triangle on the surface of the earth — or rather the triangle a little above it, whose surface is parallel to the still surface of the ocean — is necessarily so small a portion of the whole surface of the sphere, that its area, computed as that of a plane triangle, from the actual measurements, cannot be affected with any practical error of conse- quence. Now it is obvious that horizontal angles, taken at three different stations, lie in different planes, for the horizons are different, and more- over that each is the angle of a spherical triangle : — the spherical triangle formed by great circle arcs of the earth joining the points of observation (like the angle A, for instance, in the diagram at p. 231). The three observed angles therefore — if taken with perfect accuracy — ought, to- gether, to exceed two right angles by the spherical excess E, computed as above : and the observer is thus furnished with a ready method of testing the correctness of his observations. The following is an example, taken from the *' Trigonometrical Survey " before referred to. 254 EXAMPLE FROM TBIGONOMETRICAL SURVEY. Names of Stations. A. BuxterHiU B. Dean Hill C. Dimnose Observed angles. 76° 12' 22" 48 4 32-25 55 43 7 Distance of Dunnosefrom Baxter HiU 140580 "4 feet=^ Dean Hill 183496-2 feet=fiC i^^, 70290-2 4-8468947 BC, 183496-2 sin 55° 43' 7", .-. log 8= Subtract 6-2636271 9-9171279 10-0276497 9-3267737 log^= 0-7008760 ^=5" -022, or 5" /. By observation, E=\ '25" 180 1-25 Now considering A JBG as a plane triangle, we have given the two sides AG, BO, and the included angle C, to find the area S. By Mensuration (see art. 308), 8=- AC. BO Bin C, which is computed by logs as in the margin, and from the result we infer that the total amount of error in the three observed angles is 5''— 1''-25=3''-75. Now, if all the angles have been observed under equally favourable circumstances, so that there appears no reason why one should be more erroneous than another, the cor- rection thus found is to be equally distri- buted among them, that is, one-third of the total correction is to be applied to each ; but if there be a suspicion that one angle is less to be depended upon than the others, then to this is to be applied the greater correction. The corrections of the angles being thus made, the required sides of the triangle may then be computed by Spherical Trigonometry, or we may accomplish the object by Plane Trigonometry, by aid of a theorem for the purpose, first given by Legendre.* If each of the true spherical angles be diminished by one-third of the spherical excess, their sum, thus diminished, will amount to 180°, and will therefore belong to a plane triangle ; and Legendre has shown that we may employ this plane triangle, instead of the spherical triangle : the sides of the former being regarded as equal to those of the latter, since the difference of length will be practically inappreciable, in reference to triangles connecting stations on the surface of the earth. And this leads us to remark that in geodesical operations we need never be concerned about ambiguous cases, in reference to our spherical triangles : since, in regard to these matters, we may treat them as plane triangles. Another suggestion also of some interest ofiers itself. The spherical excess, in a geodesical triangle, is very small, simply because the area of the triangle itself is very small — comparatively with that of the whole spheric surface. In plane triangles the area has nothing to do with the magnitude of the angles: these always amount to 180". The sum of the angles of a spherical triangle may have any range between 180^* and 540"* (283), which range, however, becomes gradually more and more contracted as the area of the triangle diminishes, and by continuing this diminution, may be made as small beyond 180" as we please. [In what is done above, the spherical excess has been found from the three angles of the triangle : it may also be found (and of course the area of the triangle) from two sides and the included angle, or from the three sides ; but it would be quite out of place to enter at any length here into * See the Appendix to Brewster's translation of Legendre's Geometry. AREAS OF PARALLELOGRAM AND TRIANGLE. 255 these matters : the subject belongs to Geodesy, and the student is referred for ample information on this important branch of practical science, to Captain Yolland's able treatise in the third volume of the Woolwich Course, to the Geodesie of Francoeur, and to the valuable and profound articles, Trigonometry, and Figure of the Earth, by the Astronomer Royal, in the Encyclopedia Metropolitana.] END OF THE SPHERICAL TRIGONOMETRy. IV. MENSURATION : Part I. Surfaces. 806. Mensuration is the practical science which enables us to compute the surfaces and solid contents of bodies when certain of their linear dimensions are known and expressed in figures : it has therefore two principal divisions : — the mensuration of surfaces, and the mensura- tion of solids. Each of these, too, may be further subdivided under two heads, according as the surfaces and solids are bounded by straight lines and planes, or by curve lines and curve surfaces ; but these, and other distinct portions of the subject, will be sufficiently indicated by the headings of the following articles. 307. Plane Rectilinear Surfaces. — The amount of surface inclosed by the boundaries of any figure, is called the area of that figure, and is estimated in square measure, that is the area is so many square inches, square feet, square yards, &c. The simplest of all rectilinear figures is the rectangle : its area is found by multiplying the number of units of length in one side, by the number of units of length in the other side, the product being the number of square units in the surface. Thus, if the side AB, of the annexed rectangle, mea- sure 6 inches, or feet, &c., and the side AD, 4 of these linear units, it is plain that the whole surface might be cut up, by the cross lines in the figure, into just 6x4=24 squares, the side of each square being the unit of length : hence the area is 24 square inches, or feet, &c. To prevent confusion, linear feet will be denoted by /, and square feet by F; so that if the sides of the rectangle in the margin were a feet, and b feet, respectively, we should have Are£L=ab Feet. In algebraic investigations, however, of the various formulae of mensuration, the denomination Feet, or Inches, &c., though always under- stood, is usually suppressed ; so that we should say that the area of the rectangle whose sides are a, h, is ah. From the expression for the area of a rectangle, we may easily deduce that for any parallelogram, and thence that for a plane triangle, thus : — 308. Areas of Parallelogram and Triangle.— The deter- minations of these merely require that we know the base and altitude of each. For 1, let ^C be any parallelogram, AB being its base, and the perp. BE or CF its alti- tude, then (Euc. 35, I.) the parallelogram AC= the rectangle DF. The area of this latter is EFx ED=DOxED=ABxFD, that is, the area of the proposed parallelogram is the product of its J) — _ "~~ """" c A _ _ _ _ R 256 AEEAS OF PARALLELOGRAM AND TRIANGLE. base and altitude. 2. Let either DB, or AC be joined : then each of the triangles, having the common base AB, will be half the parallelogram (Euc. 41,1.): hence the area of a triangle ilBC will be - AB x CD. Though the altitude be not directly given, yet it is as good as given if the base, an adjacent side, and the angle between them be given. For in each of these figures, the per- pendicular, or altitude p^ is ^=& sin A : hence 1. Area of parallelogram AC=:bc sin A. 2. Area of triangle ABC— - 6c sin ^...[]] which expressions become simply be, and - be, when A is a right angle. Note. — When the measures of the lines are given in more denomina- tions than one, the multiplication of the two measures may be performed either by decimals or duodecimals (see Weale's Arithmetic, p. 173). But when the given sides include an oblique angle, decimals are to be preferred, since sines are given in decimals. (2) Given two sides of a parallelogram 23 f. 9 in. and 11 f. 3 in. ; and the angle between them 58° : required the area. log 23-75 =1-375664 log 11-25=1-051152 log sin 58°= 9-928420 Ex. (1) Given the base and altitude of a parallelogram 23 f. 9 in., and 11 f. 3 in. : required the area. 1. By Decimals. 23 f. 9in.= 23-75 f. 11 f. 3in.= 11-25 Product 267-1875 F.= area 12 2-2600 12 27-00 •. Area=267 F. 27 In. 2. By Duodecimals. 23 f. 11 9 in. 3 261 5 11 3 267 2 3 Ai-ea=267 F. 2 P. 3 In. =267 F. 27 In. log 227-634 =2-355236 172 D=191) 6400(34 .-. Area=227-634 F. (3) Two sides of a triangle are 36 f., and 25 1 f., and the angle between them 58° : required the area. nat. sin 58°= -84805 51X9 = 459=- prod, of sides Area =389-255 F. 12 3-060 12 36-72 .'. Area=389 F. 36-7 In. In using natural sines, contracted multi- plication may be employed. area of triangle : three sides given. 257 Examples for Exercise. (1) How many square feet of flooring are there in a room 30 ft. by 25 ft. 4 in. ? (2) How many square yards of painting are required for a parallelogram whose length is 37 ft. and the perpendicular breadth 5 ft. 3 in. ? (3) What is the area of a triangle whose base is 18 ft, 4 in. and altitude 11 ft. 10 in. ? (4) How many square yards are there in a triangle, two sides of which are 25 ft. and 21^ ft. respectively, and the angle between them 45° ? (5) How much length of carpeting, | of a yard wide, will suffice for the room in ex. 1 ? [Note. — In purchasing a carpet for a room the outlay will in general be greater than what calculation would give from the dimensions of the room : for it must be remembered that a roll of carpeting is never cut in the direction of its length. If so many breadths and a fraction of a breadth would suffice, an additional breadth must be paid for, and where the length of the carpeting necessary is considerable, the expense may thus be a good deal increased. In general one dimension of the floor is more favourable for the demand of an integer number of breadths than the other, and where economy is studied, and the way in which the pattern runs is not considered of consequence, this should be attended to : suppose, for example, a room is 9 yards by 5. If the pattern is to run along the room, then the breadths are to make up 5 yards ; the number 3 2 of them will therefore be 6-^-=6-, so that 7 breadths will be required, 4 o .*. 7x9=63 will be the number of yards of carpeting. But if the g pattern run across the room, the number of breadths will be 9-r j=12, and 12x5=60 will be the number of yards necessary, so that 3 yards will be saved. Of course in either case some little waste may be expected in adjusting the pattern.] 309. Area of Triangle: Three Sides Given.— Let the be a, b, c, and A the angle between 5, c, then (p. 256), area = - he sin A ; therefore, substituting for sin A its value in terms of the sides as given by [5] at p. 186, we have Area=v'{s(s— a)(«— &)(s— c)} [2], where s=-[a-\-b-\-c\ which formula, except when the sides are small, should be computed by logs, to which it is well adapted. Expressed in words, the formula is this, namely : — Rule. From the half sum of the three sides subtract each side separately. Multiply the half sum and the three remainders together, and the square root of the product will be the area. Ex. Required the area of a triangle whose sides a, 6, c, are 47-8, 31 2, 39 6; 258 AREAS OF QUADRILATERALS. 1. By Logs. 2. WUTioutLogs. a, 47-8 log s, 59-3 1-773055 59-3xll-5x28-lxl9-7 6, 31-2 „ s-a, 11-5 1-060698 =377507-0615, and c, 39-6 „ S-&, 28-1 1-448706 V'37507-0615=614-416=Area. 2)118-6 „ s-c, 19-7 1-294466 s, 59-3 2)5-576925 s-a, 11-5 Area 614-41G 2-7884625 s-h, 28-1 51 s-c, 19-7 D=n) 1150(16 Examples for Exercise. (1) How many square yards are there in a triangular pavement, the sides of the triangle being 30, 40, and 50 feet respectively ? (2) Find the area of a triangle whose sides are 4|, 5^, and 6^ feet. (3) How many acres are there in a triangular field whose sides are 2569, 4900, and 5025 links ? (4) How many acres are there in a triangle whose sides are 20, 30, and 40 chains ? 310. Areas of Quadrilaterals.— The simplest four-sided figure, after the parallelogram, is the trapezoid: this has two opposite sides parallel, but not the other two. Let the diagonal DB be drawn, dividing the figure into two triangles : then if AB, DC, be the two parallel sides, these two DC triangles will have a common altitude DE — the perp. distance between those sides : hence the area is \ [AB+DC) D£:...[l]. But if no two sides be parallel, the figure is a trapezium, in which case it will be necessary to draw two perpendiculars to the diagonal — one from each of the vertices opposite to it, and the area will obviously be \ (BE+DF) AC, [2]. We may, however, dispense with the mea- surement of the diagonal, which it is some- times desirable to do, by either adding to, or taking from, the figure, two right-angled triangles, such that the result is a trapezoid. Thus, in the trapezium AC, if perpen- diculars be drawn to AB, or AB prolonged, from 0, D, twice the area of the trapezoid BF will be equal to (FC-\-EI)) EF, and twice the area of the two triangles, will be equal to BF . FC+AE . ED; so that twice the area of the trapezium will be (EF^:BF}FC-^{EF^AE)ED; the upper sign having place when the triangles both lie without the figure, as in the diagram, and the lower when they lie within it. But a single rule, comprehending both ABEAS OF QUADRILATERALS. 259 cases, as also that in which one triangle is within and the other without the figure, may be easily deduced ; it is this : Rule. — Having measured the two perpendiculars upon the side — or side produced — chosen for base, proceed thus : Measure along the base from one extremity up to the farther perpendicular, and multiply that measure by the other perp. Measure, in like manner, from the other extremity up to the farther perpendicular, and multijoly that measure by the other perp. Half the sum of these products will be the area. Of course to get these measures the base need be measured but once, and then the bases of the two triangles. 311. There is still another way of computing the area of a trapezium, which is a variation of the first method. A diagonal is measured as before, but only one perp. dropped upon it : the other being drawn in the remaining triangle as in the annexed diagram. DB, AB, and the two perpendi- culars, being measured, the areas of the two triangles are found and added together. In land surveying, complicated figures are always divided into triangles and quadrilaterals, which are computed separately, and the sum of the whole taken for the area of the figure. In what has been explained above, lengths only are measured, and not angles ; but it is plain that if the diagonal AC (see fig. at p. 258), as also the two sides AB, AD, are measured and the two angles at A, the pair of triangles may be com- puted by [1] at p. 256. Or, instead of this, we may measure both diagonals AC, BD, and the angle G, where they intersect ; any one of the angles at G will do as well as any other, because the sine of one is the same as the sine of any other one. Twice the areas of the four triangles, having G for their common vertex, is DO . GO sin G-^DO . OA sin Q-\-BG . GO sin 0-\- BO . OA sin G ; or, adding together the first and third, and the second and fourth, BB . GO sin Q-\-DB . GA sin G=BD . AC ms. G...\Z'\ ; 80 that the rule is to multiply the product of the two diagonals by the sine of the angle at which they intersect. (1) In a four-sided figure two of the opposite sides are parallel, and their lengths are 32 feet, and 47 feet, respectively : also the perp. dist. between them is 28 feet : required the area. Here by [1], the area is i (32+47)28=79x14=1100 Feet. (2) The diagonal of a trapezium is 132 feet, and the perpendiculars upon it from the opposite corners, 17 feet and 26 feet, respectively : required the area. Here by [2], the area is i (17+26)132=43x66=2838 Feet. (3) In the quadrilateral field AC (last fig. at p. 258), perpendiculars are drawn from A, B, to the side DC: the distance of C from the farther perp. is 745 links : the distance of D from the other perp. is 1 000 links : s 2 260 EQUIDISTANT ORDINATES. moreover the perp. from A is 595, and that from B, 352. What is the area of the field in acres? By the rule p. 259, the area is -(745.352+1000.595)=428620 links=4 acres 1 rood 5*792 perches. Examples for Exercise. (1) How many square feet are there in a plank 124 ^^^^ loJ^o* tl^e breadth at one end being 15 inches and that of the other 11 inches? (2) In the trapezium ABCD (p. 258), the measures were as follows : -4C=161, Z)F=30-1, BE=:24:'5: required the area. (3) In the quadrangular field ABCD (p. 258), only the following measures could be taken, namely, BC=265 yards, AD=220 yards, ^(7=378 yards, 4^=100 yards, and CF=^70 yards : required the area in acres. (4) Required the area of a quadrangular paddock of which the two diagonals are 30 and 40 yards, and the angle at which they intersect 60°. 312. Equidistant Ordinates.—Every plane figure, with all its boundaries straight lines, may be cut up into triangles and quadrilaterals ; and by suitable measurement of the parts of tliese, all the component areas may be obtained, as explained above. But when the boundary at any part is a curve, other methods must of course be adopted. Should the curve be of a kind not subject to any geometrical law, the area of the portion of surface of which it is a boundary can be found approximately only: the approximation may, however, be brought so close as to involve ijo error of practical consequence, by the method of Equidistant Ordinates. Let DC be any curve boundary, and let it be required to find the area of the portion of surface AC, where DA, CB are perpendiculars upon AB. Let AB be divided into any even number of equal parts by the perpen- diculars lb, kc, id, &c. : these perpendiculars are called ordinates : they are here equidistant ordinates. Draw through D, I, the parallels Dn, mp, to AB, and complete the little rect- angles Dl, Ik. Then the area of the two rectangles An, hp, will be {AD-\-U)Ah [1], and the area of the two Al, bk, will be {U-^ci:)Ab [2]. Now it is plain that the curved area ADkc is greater than the first of these and less than the second, that is, it is intermediate between the two ; taking, therefore, half the sum of the two, we have -(AD-^^2bl-\-ck)Ab for the approximate area of the curved portion of surface referred to, and which approximation is the nearer to the truth the closer the ordinates are together. Applying a similar method of proceeding to the portion ch, and then to eC, &c., till the last ordinate is reached, we have Approx. aiesk=f-AD-\-bl-\-ch-\-di-^...-^^Bc)Ab...[d]. Again: Referring to the expressions [1], [2], above, we see that the rectangle mc is greater than the first and less than the second ; the dif- REGULAR POLYGONS. 261 ference being in the one case the rectangle mn in excess, and in the other, the rectangle Ik in defect. Consequently the rectangle w/c, as well as the curve surface ADkc, is intermediate between [1] and [2], and is evidently a closer approximation to the curve surface than either of those areas. Hence, an approximation closer than [3] will, in general, be had by taking, instead of one-half the sum of the former two, one-third of the sum of all three. Now, the rectangle mc, like the rectangles before considered, is made up of two rectangles Jl, bp ; so that, adding these to the former pairs, and taking one-third, we have -{(AD-^2bl-\-cJi:)+(J)l-\-hl)}J!>=]{AI)+iU-\-c]i;)Ah=3iTea.ADkc. o o Similarly, -(ck-^i di-\-ek)Ah=a,Tesi cJkhej ~(eh-\-ifg-\-BC}Ab=a,Tea.ehCBf &c. o o .'.h{AI)-{-BCf) + i{hl-\-di-\-fg-{:..)-^2{cl:-^eh-\-...)}Ab=s^Te&AC, o which, expressed in words, gives the following rule, namely, Rule. — 1. Measure an odd number of equidistant ordinates between the extremities of the curve boundary — the more numerous they are the better. 2. To the sum of the extreme ordinates or boundaries add 4 times the sum of the even ordinates, that is, of the second, fourth, &c., and twice the sum of all the others. Multiply the result by the common distance between the ordinates, and one-third of the product will be the area bounded by the curve, the extreme ordinates, and the line connecting these extremes, and to which the ordinates are perpendicular. (1) Given nine equidistant ordinates, 14, 15, 16, 17, 18, 20, 22, 23, and 25 feet, and the common distance between them 2 feet: required the area of the surface. 14+25=39, 4(15+17+20+23)=300, 2(16+18+22)=112, and I (39+300+112)=30o| F. Examples for Exercise. (1) Given seven equidist. ordinates, 2, 3, 5, 6, 9, 10, and 10^ ft., and the distance between them 6 inches : required the area of the surface. (2) Given eleven equidist. ordinates, 6, 9, 12, 14, 20, 20, 20, 18, 16, 11, Sin,, and the common distance 6 in. : required the area. (3) The ordinates of a curve are 0, 2, 3^, 5|, 6, 7, 8|, 9, 71, 6f, 5, 3, 2^, 1^, 0, feet : required the area, the common interval being 1 ft. 3 in. 313. Regular Polygons. — Let ab, in the diagram at p. 191, be the side of a regular polygon of n sides, and let be the common centre of the circumscribed and inscribed circles : the radius of the former will be R=OM, that of the latter r=OD. The angles subtended at 0, by all the n sides, will be equal, the measure of each being — . Call each of the equal sides, a : then JR sin — =-a, and r tan-=-a, n 2 n 2 1 fr 1 ir .•. li=-a cosec -, and r=- a cot - . . .[11. 2 n 2 n ^ •' 262 THE CIRCLE AND ITS SECTORS. 1 1 Qtt Now the area of the triangle Oab being -ar, or (p. 256)- B^ sin -, if we put S for the areas of all the n triangles, that is, for the area of the whole polygon, we shall have S= nar, or S=-va'^ cot -, or S=ni^tShn -, or S=^-^I^ sin — ;...[2] 2 4 % n 2 n ■" where r is the perp. from the centre of the polygon to one of the sides a, R the line from the centre to one of the vertices, and — the whole, and n n the half of the angle subtended at the centre by any side. The first of the expressions [3], namely, S=-nar, is the semiperimeter of the polygon multiplied by the radius of the circle it circumscribes. (1) Kequired the area of a pentagon, each side of which is 25 feet. Here S=\na^cot -=^5x252 cot 36°=^ X 1-37638=1075 -297 Feet. 4 w 4 4 (2) Required the area of a regular octagon circumscribed about a circle of 20 feet radius. Here -S'=wr2tan-=8x202xtan 22° 30'=3200x •41421=1325-472 Feet. n (3) Required the area of a regular octagon inscribed in a circle whose radius is 20 feet. Here S=lnR'2 sm—=ix20^Xsm 45°=1G00xJn/2=800V2. 2 n 2 ^ Examples for Exercise. (1) Required the area of a regular hexagon, each side of which is 20^ feet. (2) Required the area of an equilateral triangle, a side of which is a. (3) Required the area of a regular polygon of 20 sides, the radius of its inscribed circle being 20 feet. (4) Required the area of a polygon of 50 sides, the radius of the circumscribiiig circle being 25 feet. 314. The Circle and its Sectors,— When the sides of a regular polygon, circumscribing a circle of radius r, are indefinitely great in number, the perimeter of the polygon becomes confounded with the circumference of the circle, the semiperimeter and the semicircumference being alike represented by tt (233), and the area of each being - peri- ls meter x r. If r = 1, the - perimeter or semicircumference is «• = 3-1415926... and so many square units is also the area (233). And since the circumferences of circles are as their diameters, and their areas as the squares of the diameters, by putting d for the diameter (2r) of any circle, we have THE CIRCLE AND ITS SECTORS. 263 1 : -d : : T : - c?a'=the semicircumf erence, 2 2 (1\2 I -) ::«•:- 9rd^=^}^=the area •[!]. Hence, to find the circumference of a circle we have only to multiply the diameter by 3-14159..., or rather by 3141 6. And to find the area, to multiply the square of the diameter by - 7r=-7854, or the square of the radius by 3-1416. For the length of an arc of a circle subtending A"" at the centre, or containing itself J.**, the radius being -d=r, the expression is obviously --—A='0l76rA ; and for the area of the sector, bounded by 180 this arc, the expression is -(•1075ri4)xr=-0175rM-r2,orr xhalf the length of the arc : hence calling the length I, whether it be that of a whole circumference or not, we have where A is the number of degrees in the circular arc. (1) Required the circumference and area of a circle whose diameter is 50 feet. Here [1], circumference=fi?7r=3-141 6 x 50=157-08 feet, and area=-7854rf-'=-7854x 2500=1963-5 feet. (2) Required the area of a circular sector, the chord of which is 24-24, the radius of the circle being 15 feet. Referring to the diagram at page 191 we have lo-io 4-04. Bin aOi) =-=—== -r-= '808 .-. aOI)=5Z° 54' : hence [2], 15 5 _,^ Ir^* ^ 152x3-1416 ^„54 5x3-1416 /^, 1\ -^. oo«^- „ , AreaaOJf=-— ^=_3-^3_x63-=— ^— x(54--)=105-83265 Feet, and the double of this, 211*6653 Feet, is the area of the sector. Note. — Since the circumference of a circle is c=<7rdf .-. d=-, •. Area=^ti2=^( - ) =f-= -0796802... [3], 4 V\^/ itr an expression which is useful in finding the area when the circumference only is given. Examples for Exercise. (1) How many square yards are there in a circle whose diameter is 3^ feet ? (2) What length of arc is there in 12° 10' of a circle of 10 feet radius ? (3) Required the area of a circle whose circumference is 12 feet. (4) Required the area of a sector of a circle of 25 feet radius, the arc being 147° 29'. (5) If the diameter of the earth at the equator be 7924*9 mUes, what is the length of one degree of the equator ? (6) How many square yards are there in the top of a circular table of which the cir- cumference is 178 inches ? 264 SEGMENT OF A CIRCLE. Note. — The number 3141 6, which so frequeutly occurs in practical inquiries about the circle, may be replaced by the fraction — = 3-142857, when much accuracy is not necessary. This fraction for the ratio of the circumference of a circle to its diameter, was first given by Archimedes. 355 But the fraction -— , proposed by Metius, is a much closer approximation, 355 closer even than the ratio 3*1416 to 1, since— — =3*1415929, which llo agrees with 3'] 415926 as far as six decimals. 315. Segment of a Circle. — To find the area of a segment aMb (p. 191), we have first to find the area of the sector, and then the area of the triangle Oah, and to take the sum or difference of the two, according as the segment is greater or less than a semicircle. In the diagram at page 191, «6 is the common chord of the two seg- ments into which that line divides the circle ; the smaller of these, that exhibited in the diagram, is the sector mirnis the triangle ; the remaining part of the circle includes this triangle. (1) Required the area of the smaller segment of a circle cut off by a chord 12 feet in length, the radius of the circle being 10 feet. Here the semicliord-T-r=6-r-10= -6=8111-^^=8111 36° 52'^ .*. 4°=73°-74, A 1 r^cr 5y3-1416 .-. Area of Sector=-— ^=-^a 73 -74=5x1 •0472x12-29=64-35044 Feet. Area of Triangle =^r2sin A°=50 sin 73° 44' |=50x -96000=48 Feet. /. Area of Segment=64 -35044-48=16*35044 Feet. (2) Required the area of a segment of a circle whose height is 46| feet, and the radius 25 feet. Here the segment is obviously greater than a semicircle, and .*. its height minus the radius is the height of the triangle=:21f . Also 21|-7-25=*87=cos I A" .'. ^°=69° 5'=59°^ 2 and 360— 59-5^=300i^=number of degrees in the arc of the segment. .-. Area of sector=i ^ 300 i^=i?^^il^ (301-^)t=1641*2 Feet. Jt loU o Area of triangle=- r^ sin ^=312 *5 x '85792=268 '1 Feet. 2 .*. Area of segment=1641*2+268*l=1909-3 Feet. 316. There is another and more speedy way of finding the area of a segment, when the height is given, by aid of a table of segments to dia- meter I, the area being then found by the following rule, namely: — Rule. Divide the height of the segment by the diameter ; the quotient will be the height of the similar segment in the table, against which the area of that similar segment will be found. Multiply this tabular area by the square of the given diameter, and the product will be the area sought. CIRCULAR RING. S65 Note 1. — When the proposed segment exceeds a semicircle, the similar segment will not be found in the table, which is here annexed. In this case the quo- tient mentioned above must be subtracted from 1 (the tabular diameter), and the segment corresponding to the remainder taken out. This segment must then be subtracted from the whole tabular circle, namely, from •785398, the remainder will be the tabular area cor- responding to that sought. This, therefore, must be multiplied by the square of the given diameter, as the rule directs. 2. If the quotient, or the remainder here spoken of, be not found exactly in the table, we take out the nearest less segment, and correct it as we do a loga- rithm. The following application of the rule to the example just worked will sufficiently explain this : — 46-75 50 =1- -935= -065. Hence, tab. seg. = -02170 To be subtracted from "78540 Against -06 in the table we find -01924 : the difference between this and the next number is -00493, which multiplied by the neglected 5, (and divided by 10) gives •00246 for correction; therefore the correct tabular segment is *02170 : The theory of the an- nexed operation is obvious; the heights of similar seg- ments are as the diameters of the circles, and their areas, as the squares of the diameters. Note. — Instead of multi- plying by 2500, it is better to divide 7637*0 by 4. Area of sim. segment -76370 Square of diameter . 2500 38185 15274 Area of segment . 1909-25 Feet. Examples for Exercise. (1) Required the area of a segment of a circle, of whicli the height is 6 feet, and the diameter of the circle 32 feet. (2) Required the area of the segment of a circle, the arc being a quadrant, and the radius 24. (3) What is the area of a segment, the arc of which is 280°, the diameter of the circle being 60 feet ? (4) Required the area of a segment, whose height is 18 feet, and the diameter of the circle 50 feet. (5) Required the area of a segment, less than a semicircle, whose chord is 16 feet, the diameter being 20 feet. (6) From a circle, whose diameter is 50 feet, a segment, con- taining 54-1475 Feet, is to be cut : required the length of the chord. Areas of Segments. Height. •01 -02 -03 •04 -05 •06 -07 •08 •09 •10 -11 -12 •13 •14 -15 •16 •17 •18 -19 •20 •21 •22 •23 •24 -25 -26 •27 -28 •29 -30 •31 •32 •33 •34 -35 •36 -37 •38 •39 •40 •41 -42 •43 -44 -45 •46 -47 -48 •49 •50 Area, d~l. •00133 •00375 -00687 •01054 •01468 •01924 •02417 •02944 •03502 •04088 •04701 •05339 •06000 •06683 •07387 •08111 •08853 •09613 •10390 •11182 •11990 •12811 •13646 •14494 •15354 •10226 •17109 •18002 •18905 •19817 •20738 •21667 •22603 •23547 •24498 •25455 •26418 •27386 •28359 -29337 -30319 -31304 •32293 •33284 •34278 •35274 •36272 •37270 •38270 •39270 317. Circular Ring.— If a smaller circle be placed wholly within a larger, the space between the two is called a circular ring. If D be the 266 INSCRIBED AND CIRCUMSCRIBED TRIANGLE. diameter of the outer circle, and d that of the inner, the difference of their areas, that is the area of the ring, will obviously be Area=-7854 (D^-^='7854: {D-\-d){D-d\ \7hether the two circles be concentric or not, so that we have only to mul- tiply together the sum and difference of the diameters, and the product by -7854. (1) The diameters of two circles, one within the other, are 10 and 6 : required the area of the space between their circumferences. Here -7854 (10+6) (10-6)='7854x 64=50-2656, the area required. Examples for Exercise. (1) Required the area of the ring between two circumferences, the diameter of which are 10 and 20. (2) In a circular slab of marble, of which the diameter is 73*25 inches, a circular hole, 3 '5 inches in diameter, is cut : how many square inches are there in the upper surface of the remaining ring ? (3) The inner diameter of a circular building is 73 ft. 3 in., and the thickness of the wall is 1 ft. 9 in. : how much ground does the wall stand upon ? 318. Inscribed and Circumscribed Triangle.— Let ABC be a triangle inscribed in a circle. If any two of its sides be bisected by perpendiculars, they will meet in 0, the centre of the circle (Euc. 5. VI.). Put the radius OA=R, then AD=:R smAOD=:R sin (Euc. 20. III.) : hence - c=:R sin C, j:h=R sin B, ^a=zR sin A ; 2 Ji ^ [5] page 186, R= dbc i»s/ {s{s—a){s—b)(s—c)} 4 area of triangle .*. Area of triangle=-^. 4x1 ...[li Again, let the triangle circumscribe a circle: then the centre is the point where the lines AO, BO, bisecting two of the angles, meet (Euc. 4. IV.), and Om, drawn to the point of contact m, is the radius r, and is perpendicular to AB : hence Triangle OAB=-cr, triangle OAC=- iVf triangle OBC=- ar. The sum of these is the triangle ABC=- he sin A =- (a+ J+c)r=«r, 2 2 he . j{s—a){s—h){s—c) .'.r=-sinA, or, r=^^ '-^-^ ^...[2], It is plain that the expression Area=sr, where s is the semiperimeter of the figure, applies to any polygon circumscribing the circle, whether it be r 4(s— a)(s— 6)(s— c) ...[3]. SURFACE OF PRISM AND CYLINDER. 287 regular or irregular. If the triangle be regular, that is, equilateral, — =2 .*. B=2r. But of the equilateral triangle, the area is 11 la' - a* sin 60°=-a^-v/3, and from the expression above, area=-— , .-. ^3=^, .-. i2=-^=-aV3, .-. r=-i2=-aV3, also area =- a V^- MENSURATION : Part II. Solids. 319. To find the solid content or volume of a parallelopiped, prism, or cylinder, the rule is this : Multiply the area of the base by the height, and the product will be the number of the cubic units in the body. Note. — This rule applies, whatever be the boundary of the plane base of the solid, provided the sides rising from this base are all upright, or perpendicular to it. The truth of the rule is obvious : Let the base contain m square feet : then an upright solid on this base, 1 foot high, must contain m cubic or solid feet, and therefore a solid h feet high, on that base, must contain mh cubic feet. (1) The diameter of a rolling stone is 18*7 inches, and its length 4*75 feet: how many cubic feet of stone are there in it? 18 "7^ X •7854=area of base in sq. incJies, .'. 2 log 187+log •7854+log 4-75-log 144=log volume. The calculation is annexed, from which 2 log 18-7 P '271842 it appears that the number of cubic (1 '271842 feet is 9 0595. log -7854 1-895090 (See p. 279.) Note. — The 18*7 being inches, we log 4*75 0-676694 must divide by 12 to bring them to colog 144 7-841638 feet: hence the square of 18-7 must be divided by 144. log ^'0595 957106 •^ 080 2)=48) 260(5 Examples for Exercise. (1) How many cubic feet are there in a block of marble of which the length is 3 ft. 2 in., the breadth 2 ft. 8 in., and the thickness 2 ft. 6 in. ? (2) The diameter of a well is 3 ft. 9 in., and its depth 45 ft. ; what was the cost of sinking at 7s. 3d. per cubic foot ? (3) How many cubic feet are there in a triangular prism, whose length is 10 feet, and the three sides of the triangular base 3, 4, and 5 feet ? (4) The length of a hollow iron roller is 4 ft., the diameter of the outer circumference 2ft., and the thickness of the metal | in. Required the solid content. (5) How many cubic feet are there in a cylindrical column whose height is 20 ft., and circumference 5^ ft. ? (6) Required the volume of an oblique circular cylinder, which, when standing on a horizontal plane, inclines at an angle of 60°, the length being 25 feet, and diameter of the base 30 inches. 320. Surface of Prism and Cylinder.— The surface of any solid bounded by plane rectilinear faces, is found by computing each face separately by the rules given in the preceding Part, and then taking the 268 VOLUME AND SURFACE OF PYRAMID AND CONE. sum of the results. If the solid be an upright prism or parallelepiped, then omitting the two ends, it will be sufficient to multiply the perimeter of the base by the height : if the base be a regular polygon of n sides, then the lateral surface will be found by multiplying n times a side of the base by the height. The curve surface of an upright cylinder is found also by multiplying the circumference of the base by the height, for the base may be regarded as a regular polygon of an infinite number of sides (233). If, therefore, r be the radius of the base, d its diameter, c its circumference, and h the height of the cylinder, the curve surface is S=c7i=7rdh. Thus the curve surface of a cylinder whose height is 20 feet, and the diameter of its base 2 feet, is 5=3-1416 x 2x 20 = 125-664 square feet. If the whole surface be required, we must add to this the areas of the two ends, namely -7854x8 = 6-2832; so that the whole surface is 131-9472 sq. feet. Examples for Exercise. (1) Required the surface of a cube, the length of each side being 15 feet. (2) Required the whole surface of a triangular prism, whose length is 20 feet, and each side of the base 18 inches. (3) The gallery of a church is supported by 16 cylindrical columns of wood, each of which is 4 ft. 8 in. in perimeter, and 12 ft. high : required the cost of painting them at 9d. per sq. yard. 321. Pyramid and Cone. — Since by Euclid (7 and 10, Book XII.) a pyramid is a third part of a prism of the same base and altitude, and a cone is a third part of a cylinder of the same base and altitude, we have this Rule. Multiply the area of the base by the altitude, and take - the product. o (1) The diameter of the base of an upright circular cone is 26^ inches, and the altitude 16 J feet : required the solid content, or volume. Area of base=26-52x -7864=551 "547, and since -^=5 "5, 3 551 •547x5-5 ^, ^^„ ,. ^ /. Volume= —-; =21-066 cubic feet. 144 Examples for Exercise. (1) Required the volume of a triangular pyramid, the height being I43 ft., and the sides of the base 5, 6, and 7 ft. (2) Required the volume of a cone of which the height is 10| ft., and the circumference of the circular base 9 ft. (3) How many cubic feet are there in a pyramid whose base is a pentagon, of which each side is 2 ft., the height of the pyramid being 12 ft. ? (4) The diameter of the base of a circular cone is 42 inches, and its height 94 inches : it is required to cut off two solid feet of material from the top by a plane parallel to the base : what must be the altitude of the cone out off? 322. Surface of Pyramid and Cone.-— The sides of a pyramid are all plane triangles ; its lateral surface, the sum of all these, is there- fore found by the rules for computing the areas of plane triangles before given. If the base be a regular polygon of n sides, the lateral surface of VOLUME OF FRUSTUM OF PYRAMID OR CONE. 269 an upright pyramid is found by multiplying n times a side by the slant height of the pyramid, and taking half the product. As there is no limit 10 the number n, it may be indefinitely great, the sides themselves then being indefinitely small, the polygonal base becoming confounded with a circle (233) : hence the convex surface of an upright cone also is found by multiplying the circumference of the base by the slant height, and taking half the product. Thus, the convex surface of an upright cone, of which the slant height is 50 feet and the diameter of whose base is 8-5 feet, is /S'=3-1416 x 8-5 x 25=^ 3-1416 x 850=667*59 sq. feet. 4 Examples for Exercise. (1) Required the convex surface of a triangular pyramid, each side of its base being 3 feet, and its slant height 20 feet. (2) Required the convex surface of an hexagonal pyramid, each side of its base being 2 3 feet, and its perpendicular height 10 feet. (3) The diameter of the base of an upright circular cone is 26^ inches, and its altitude 164 f^et : required the number of sq. feet in its entire surface. 323. Frustum of Pyramid or Cone.— A frustum is the por- tion of the solid which remains after a solid similar to the whole is cut off by a plane parallel to the base. Let H be the height of the complete pyramid or cone, and h the height of the similar solid cut off : also let A be the area of the base of the larger body, and a the area of the base of the smaller. Then, since the bases of similar pyramids and cones are as the squares of their altitudes, we have A:a: : H^ : h^ .'. ^A : >^a : : E : h .-. (79) ^A-^a : ^A : : ff-h : H, and tJA — tJO' • n/* • * E—K : A. From these two proportions we have Now the difference in solid content between the original body and the part cut off is, (321), -(-ijGT— aA)=Volume of the frustum ; o .-. Volume of frustum=^ ^^f "'"/'' (g-A)=^(^+a+VJa)(g-A).. .[11, u \/ A. — isj Oj o where H—h is the height of the fnistum. Hence the following rule : — Rule I. To the areas of the two ends add the square root of the pro- duct of those areas : multiply the sum by the height, and one-third of the result will be the volume or solid content. II. If the solid be a conical frustum, the shortest way of arriving at the final product will be to add together the squares of the diameters of the two ends and the product of those diameters, and to multiply the sum by the height "and by -7854 : one-third of the result will be the volume. (See p. 263.) III. If, however, the diameters are not given but the circumferences, add together the squares of these and their product, multiply the sum by the height and by -07958, and take one-third of the result. (See p. 263.) E.Kpressed as formulae, the last two rules are as follow, where Z>, d, are 270 SURFACE OF A FRUSTUM. the diameters of the ends, and C, c, the circumferences, observing that •7854=^, and •07958=-^ : 4 49r Yolnme=^^h{D'^-^(P-\-I)d), or Voliime=:i- . -(C2+c2+Cc)...[2]. IZ LA K* (1) The shaft of Pompey's Pillar is a single stone of granite 90 feet in height : the diameter of its circular base is 9 feet and the diameter of the circle at topis 7| feet: how many cubic feet of stone does it contain. Here I -7854^ (D- + d-+I>fi)=30 x '7854 (81 -{- 5625 +67-5)=23-562x o 204-75 =4824-32 cubic feet. Examples for Exercise. (1) How many solid feet are there in a stick of timber whose ends are squares, each side of the greater end being 15 inches, and each side of the less end 6 inches, the length of the stick being 24 feet ? (2) How many cubic feet are there in a conic frustum 18 feet high, the end diameters being 8 feet and 4 feet ? (3) The ends of the frustum of a pyramid are regular hexagons, a side of the greater of which is 1 3, and a side of the less 8 : the length of the frustxim is 24 : required its volume. (4) The largest of the Egyptian Pyramids is only a square frustum ; its height is 480 feet ; a side of its square base is 745 feet, and a side of its square top is 32 feet : requii-ed its weight, allowing 150 lb. to the cubic foot. 324. Surface of a Frustum. — If the frustum be that of a pyramid, each face will be a trapezoid, the area of which is found by multiplying the sum of the two parallel sides by the perpendicular dis- tance between them, and taking half the product : for the frustum of a regular pyramid of n equal faces, we have only to take one of these trapezoids n times, or, which is the same thing, to multiply n times the sum of the opposite ends of a face by the distance between them, and to take half the product. When the frustum becomes that of an upright cone, by the number n of the sides of the polygonal ends becoming in- definitely great, the sum of the two circumferences multiplied by the slant height of the frustum, and half the product taken gives the curve surface ; or, which amounts to the same : — Multiply the circumference of the frustum, taken half-way between the two ends, by the slant height, and the product will be the surface. Let it be required to find the convex surface of a frustum of an upright cone, of which the slant height is 12 J feet, and the circumferences of the two ends 6 ft. and 8-4 ft. Here half the sum of the circumferences, that is, the circum- ference in the middle, is 72 ft. .-. /S=7-2x 12J=90 F. ExAMPiiES FOR Exercise. (1) The slant height of a frustum of an upright cone is 45 ft., and the circumferences of the ends 113 ft. 9 in, and 31 ft. 3 in. : required the convex surface. (•^) The slant height of a frustum of an upright cone is 9 ft., and the diameters of the ends 43 in. and 23 in. : required the whole surface. SPHERE AND SEGMENT OF A SPHERE. 271 (3) The perp. height of the frustum of an upright pentagonal pjrramid is 11 ft., each side of the base is 34 in., and each side of the top 18 in. : how many sq. feet are there in the convex surface ? 325. The Sphere. — Since a sphere is two-thirds of its circum- scribing cylinder [see article 327], .-. Sphere=| ^R\ 2i2=^a-i23=l*i)3_ .5236 2)3=1 ^= '01689 C^ ; 3 3 6 Kt 9r that is : multiply the cube of the diameter by "5236 ; or multiply the cube of the circumference by -01689. [See this value computed at p. 279.] Examples for Exercise. (1) Required the volume of a sphere whose diameter is 12 feet. (2) The circumference of a cannon ball is 18^ inches: how many cubic inches does it contain ? (3) The exterior diameter of an iron shell is 10 inches, and the interior diameter 8 inches : how lauch metal, in cubic feet, is there in the shell ? 326. Surface of a Sphere and Segment of a Sphere. — Let a radius be drawn to any point on the surface, and conceive a small tangent plane at this point : the radius will be perp. to this plane, which plane therefore will form the base of a pyramid of which the altitude is the radius R of the sphere. By thus covering the surface with small tangent planes, we shall have a series of pyramids having these planes for bases, and having B, for their common altitude. The volume of the whole assemblage of pyramids will therefore be found by multiplying the sum of their bases by - B. But by making these bases smaller and smaller — o thus increasing their number indefinitely, the polygonal surface will be- come confounded with the spheric surface : hence the volume of the sphere "will be F:=- RS, where S is the surface of it. o And it is obvious from the above reasoning that the expression F=- US is equally that for a spherical sector (consisting of a segment o and an attached cone), the base of it being S, any portion of the spheric surface, and its vertex being at the centre of the sphere, that is, its height being R. We have seen that for the sphere V=t^R^ .-. \ RS=Ur^ .-. 5f=4«-/22=*2)2...[n, the surface is therefore equal to four times the area of a great circle of the sphere, or it is equal to the convex surface of a cylinder circumscribing the sphere (320). These deductions are admissible, however, only on the assumption in (325), that the volume of the sphere is two-thirds that of the circum- ^■m. D \e y 272 THEOREM OP ARCHIMEDES. scribing cylinder — tlie celebrated theorem of Archimedes. Or if the de- duction [1] be proved, independently, the theorem itself follows : we pro- ceed therefore to prove [IJ without reference to (325). 327. Theorem of Archimedes. — Let ABC be a quadrant of a circle, and draw the perpendiculars BD, CD, completing the square CB. If this square and quadrant be made to revolve about the fixed line ^C as an axis, the former will evidently describe a cylinder, and the latter a hemisphere inscribed in the cylinder. Now two perpendiculars, km. In, to the axis may be drawn so close together that the intercepted arc mn may be taken for a straight line, that is, for its chord, y in which case mn, in revolving, will generate part of a conic surface, /»g generating the corresponding portion of the cylindric surface : the radius of the middle section of the conic surface is represented by the dotted line ae. Now the area of the surface, thus described by mn, is tt x 2ae x mn (324), and ^ the area of the corresponding surface described by pq, is ^r x 2a& x pq. Conceive a perpendicular to Iq to be drawn from m: this will be equal to pq, 80 that pq, mn, will be the perpendicular and hypotenuse of a little right-angled triangle similar to the triangle AEe, the angle included be- tween the little hypotenuse (inn) and the perpendicular (=pq) being equal to the angle A, since the sides of one angle are perpendiculars to those of the other. Hence by similar triangles, mn : pq : : Ae : AE ; but Ae=^ah, and AE^ae, .'. aeXvin^idbXpq, Consequently the surface generated by mn is equal to the surface gene- rated by pq. The same reasoning applies to every other arc m'n' of BC, taken so small as to be confounded with its chord : hence, taking the sum of all the indefinitely small arcs into which BC may be conceived to be divided, we conclude that the convex surface of the whole hemisphere is equal to the convex surface of the cylinder on the same base and of the same altitude : and, moreover, that the convex surface of any portion of the sphere, contained between two parallel planes, is equal to the convex surface of that portion of the circumscribing cylinder which is contained between the same planes. Putting therefore S for the whole surface of the sphere, we have S=i7iE'='7rD' ; and since it was proved above that 1 4 2 V=- BS, .'. V=-7rB^=-'7rB'.2R, that is, the volume of the sphere o o o is equal to two-thirds that of its circumscribing cylinder. For the portion of surface between two parallel planes at the distance h apart, we have S='7rDh. The formulae established above supply the following rules : — Rule I. To find the surface of a sphere. — Multiply the square of the diameter by 3- 1416, or the circumference of the sphere by the diameter. II. To find the surface of a segment. — Multiply the circumference of the sphere by the height of the segment. (1) The surface of a sphere whose diameter is 24 inches is 24' x 3-1416 = 1809 56. (2) How many square inches are there in the convex surface of a seg- ment of a sphere whose circumference is 3 ft. 10 in.: the height of the segment being 6 inches ? Here /S=46 x 6=276 inches. volume of segment and zone. 273 Examples for Exercise. (1) How many square miles are there in the surface of the earth, the circumference of which is 25000 miles ? (2) Required the convex surface of a segment of a sphere of 5 feet diameter, the height of the segment being 1 ft. 9 in. (3) From a sphere 4 ft. 11 in. in diameter two segments, 12 and 14 inches high re- spectively, are cut : required the convex surface of the portion left. Note. — The portion of surface left is evidently the same whether the planes cutting off the segments are parallel or not ; and the same may be said as to the volume left. 328. Volume of Segment and Zone.— As before, D being the diameter of the sphere, and h the height of the segment, which we shall first regard as less than a hemisphere, we have for its surface S='7rDh ; also for the volume of the spheric sector, we have (326) l^=p ^^—a "^^'^ = •5236/>-^. This includes both the segment considered, and the attached cone. Putting r for the radius of the base of this cone, that is, for the radius of the base of the segment, its volume is (321) ^xl{R-h) ; but (Euc. 35. III.), t^={D-h)h, o .-.vol. of cone = x^—r^ x—T 1 r\x^rf . {x-^rf . .'. \olume=-ir =miii. .*. =miii.=M, 3 x—r x—r .'. x"-\-2rx-\-r'^=ux—%tr .'. a;*— (w— 2r)a:+j*'+«r=0 .*. (p. 14S), x= -u—lr Also V ^=n/ { («-2»*)'- 4(rH^'*) } =n/ { w(w- 8r) } =0, .♦. M=8r, the smallest value of u, .*. a:=3r: hence the height of the smallest cone is twice the diameter of the sphere ; and the radius [y] of its base is r v/2. Prob. 6. To determine the greatest cylinder that can be inscribed in a given cone. Let h be the height of the cone, and r the radius of its base. Put x for the radius of the cylinder : then it is obvious, by similar triangles, that r \Ti: : r—x : — ^ =the height of the cylinder, r hir—x) the volume of which is V=.'jrx- =max., r .'. rjr^— a^=max.=M, .'. /(a;)=rx-— ic^— w=0. That this may have equal roots, we must have /,(a;)=2ra?-3x2=0 .-. 2r-3a;=0 .'. xz=?^r. This value of x, put in/2(^)=2r— 6^, renders it negative; the value is therefore such as to render the function 7 a maximum. Hence, the height of the greatest cylinder is -— ^ ^=-7i=one-third the height of the cone. r 3 Prob. 7. To determine the dimensions of a cylindrical measure that shall hold a given quantity under the smallest extent of surface. Let a be the given quantity, or the volume of the cylinder, and put x APPLICATION OF MAXIMA AND MINIMA. 283 for the radius of its base, and y for its height. Then the volume of the cylinder is a=7irxhj, .'. y— — ;; and the entire surface is convex siirf.+base=2flrjry+«'x2= \-Tx^=m{n.=u, .'.f{x)=trj(^—ux+2a=0. X That this may have two equal roots, the condition f^{cc)=^7rar—u—0 must be satisfied. Subtract the former equation from x times this, then ^ cr vex? sr ^ tT ^ ic hence the height of the cyHnder must be equal to the radius of the base, 3 r d and this is the given quantity a / — • That this value of x answers to a minimum, follows from the fact that />(ir)=67ra; is positive for that value (p. 150). The other values of x are imaginary, so that u admits of but one minimum, and no maximum. Prob. 8. The corner of a leaf is turned back, so as just to reach the other edge ; to find when the length of the crease is the smallest possible. Let the bottom corner of the page now before the reader, represented by AC, be turned down, FQ being the crease; then PB=PB\ and QB=QB' ; also the angles B, B\ being right angles, a circle may be described about the figure BB'. And since (Euc. B. VI. Prop. II.) the rectangle of the diago- nals is equal to the rectangles of the opposite sides, B% PQ.BB'=.^BQ ,PB,..[\]. Let PB=x, and AB=a; then (Euc. 12. XL), ^. £fB^=B'P^-\-BP^-\-2AP.PB=2x^+2(a-x)x=2ax, ^ .'. B'B=^2ax. Now BQ'^=PQ''-PB\ .% (1) PQ^.B'B^=iBQ^. PB^=i{PQ^-PB-)PB^=iPQ^ . PB^-iPB\ ,\iPB^={iPB^-B'&)PQ\ that is, ix^=z{ix^-2ax)P(^, 2ar' 2x /(^*'— ^")j the process there followed would have remained EQUATIONS OF A POINT. 285 precisely the same, and without attending to the caution now to be pressed upon his attention, the student would naturally conclude that the maximum value of x—^{a^—x-) is av^2, the value of x, giving this maximum, being a?=:-a^/2, which is an absurd conclusion, for x=^-a\/^ gives, not Z til a\/^, but 0. The cause of the absurdity is this, namely, that in reducing x—u~^{d'—x-)^=0 to a rational form, we have virtually replaced this equation by {x—u—J[a'^—x'^)}{x—u-\-s/{ci^—^')}'=^') which is equally satisfied whether iYie first factor be =0, or the second, and consequently have infringed the express restriction that \hQ first factor must of necessity be ; and it turns out that our solution applies exclusively to the second factor — the factor extraneously introduced to effect a particular purpose, and not to the factor to which we have presumed it to apply. It is thus seen how important it is, when we have to satisfy the conditions of an equation involving a quadratic radical, and instead of dealing with it in the state in which it is presented we first rationalize it, and then deduce our conclusions : — it is seen how important it is to examine whether those conclusions really apply to the given equation, or only to the new equa- tion unavoidably introduced in changing the irrational form to a rational one. And such examination is equally necessary, whatever be the nature of the inquiry. In the particular example here commented upon, the correct inference from the absurdity which such examination leads to would be, that x-^ s/{a^—x-) has neither a maximum value nor a mini- mum value. (See the author's " Theory of Equations," pp. 26, 40.) END OF THE MENSURATTON. V. ANALYTICAL GEOMETKY: The Conic Sections. 338. The main object of Analytical Geometry, or, as it is sometimes called, Coordinate Geometry, is, by representing geometrical lines and surfaces by algebraic equations, to deduce their various properties by the processes of algebra, and thus to express them in a purely symbolical form. Of lines situated in a plane, the most important are the straight line, the circle, and the higher curves known as the conic -sections : these will be discussed in the order here named. Part I. The Straight Line and Circle. 339. Equations of a Point.— Let P be the position of any point on the paper ; the object is to denote this position by the symbols of algebra, so that if the geometrical position P were obliterated, we might recover it from the algebraic representation. In order to do this, two straight AX, AY, intersect- ing at any angle A, are assumed in the plane of the point : these are called axes of reference; then if the parallels to these axes, PB, PC, be of known lengths, and 286 EQUATIONS OF A POINT. these lengths be represented by appropriate symbols — such as the figures of arithmetic, and it be also indicated by such representation in which of the four angles about A it is situated, it is plain that the position of P will be sufficiently recorded ; for if the given lengths AB, AC, be mea- sured along the axes in the indicated directions and parallels to those axes be drawn from B, C, their intersection P will mark the position of the point. 340. The length AB is called the abscissa of the point P, and AC, or BP, its ordinate; when spoken of together, AB, BP, or AB, AC, are called the co-ordinates of the point P, and the axes of reference are hence more generally called the axes of co-ordinates : to distinguish one from the other, AX is referred to as the axis of abscissas, and AY as the axis of ordinates, since it is along these that the abscissas and ordinates are respectively measured in constructing the point. It has already been seen, in the Trigonometey, how lines drawn from one fixed axis parallel to another may be algebraically indicated both in length and direction. There the axes of reference were always two diameters of a circle, at right angles to each other ; in reference to these the geometrical sines, cosines, &c., were measured. The conventions found to be necessary and sufficient for the complete representation of lines thus referred to rec- tangular axes, are equally necessary and sufficient to denote lines referred to oblique axes : thus, let x stand generally for the abscissa of a point, and y for its ordinate, then, if for any particular point The equations of a point. P, a!=a, and y=b, and if each of the four points i>, x= a, y= b P, P\ P", P"', have a, b, for the lengths, or nume- P*, x=—a, y= h rical values, of its co-ordinates, any one of the four -^''; ^=— «, y=—h will be distinguished from the others, and be unam- '^~ "' 2/=—* biguously indicated, as in the margin, by two equations. These are called the equations of the point. And it is plain that any one of these pairs of equations being given, the point algebraically represented by them — attention being paid to the signs of direction — can always be found, the fixed axes of co-ordinates being given. For instance, suppose the equations ^=5, ?/=— 4, in reference to the axes XX\ YY\ in the pre- ceding diagram, were given, to find the position of the point indicated by those equations. The first equation informs us that from the origin of co-ordinates A we are to measure the distance 5 (inches, feet, or whatever may be the assigned unit of measure), along the axis of abscissas, and as the 5 is plus, it must be measured from A towards X : we thus get AB. The second equation informs us that we are to measure 4, from the origin A, along the axis of ordinates ; and as the 4 is minus, it must be mea- sured from A towards Y^ : we thus get AC^ : it then remains only to draw parallels to the axes through B and C : the intersection P'^' of these is the point represented by the given equations. And thus it is with propriety said, that a point is given when its equations are given. Saying that its equations are given, is no more than saying that its co- ordinates are given ; in algebraically representing the position of a point, therefore, the formality of writing down two equations may be dispensed with ; it is sufficient to say that the point P is (a, b), the point P', (~a, b), the point P", {—a,—b), and the point P''' {a,—b), the general symbolical representation of a point being (x, y). A glance at the signs of direction, prefixed to the co-ordinates, is suffi- cient to enable us to pronounce at once in which of the four compart- ments about the origin A the point represented is situated ; thus the EQUATION OF A STRAIGHT LINE. 287 point (—2,-7), is in the same compartment as F", the point (—3, 2), in the same compartment as F\ and so on. But it may happen that the point is not in either compartment ; it may he upon one of the axes them- selves : it may, in fact, be at the same time upon both axes, for it may be identical with the origin A. If it be on the axis of x (as for brevity the axis of abscissas is called), the y of the point is 2/=0, and if the x of it be a, the point is {a, 0) ; if it be on the axis of y, and the y of it be h, the point is (0, 6), and if it be at the origin A, the point is (0, 0). 341. Equation of a Straight lane.— As before, let AX, AY, be any fixed axes given in position : it is required to express algebraically the position, in reference to these axes, of any proposed straight line. And first, let the proposed line pass through the origin A. From any two points B, D, in the line, let parallels BC, DE, to AY, be drawn, then AC, CB, are the co-ordinates of the point B, and AE, ED, the co-ordinates of D ; and since AC : CB : : AE : ED .-. -j^^-j^'y that is, whatever point be taken on A(y AHj the proposed line, the ordinate and abscissa of the point are always in the same constant ratio, so that calling this invariable ratio a, and, as before, putting a, y, for the co-ordinates of any point on the line, V we have generally -=a, /. y=ax [1]. X And this is the general equation of a straight line passing through the origin of co-ordinates. For any particular one of the innumerable straight lines comprehended in [1], the coefficient a must be given. 342. The beginner is here recommended to execute a few constructions of particular cases of the general equation [1] : let him take, for instance, the individual lines y=^x, y= — Sx, y=—4:X, &c., and proceed thus, the axes of reference being supposed given, and already drawn on the paper. Measure from the origin on the axis of x, towards the right, any arbitrary length: this length will be the abscissa of some point in the first line; and since by the equation the corresponding ordinate is double this, measure its double on the axis of y, above the origin A : draw the parallels through the points thus marked on the axes, and the point, whose co-ordinates are 1, 2, on the line, will be given by their inter- section : then a line through this point, and through the origin, will be the line i/=^x. Or, to show that this line mmt pass through the origin, find two points on the line, say the points (1, 2), and (2, 4) : draw a straight line through these, and it will be found to pass through A. In like manner, to construct the second line, take x of any lengtji, and make the corresponding y three times that length, measuring it from A downwards, and thus, as before, a point on the line will be founds, and the line itself will be that through this point and the origin. Or, find two points, as in the last ex., and a line through both will be found to 288 EQUATION OF A STRAIGHT LINE. pass through A . For the third line, the construction is similar to that for the second. DE The constant ratio ct=--p=, in the equation of a straight line, expressed . . . , . sin DAE sin DAE ,, ^ .. ,, « , , in tnffonometrical terms, is -: — -rFrFT'=-- — ttt^ » ^^ ^^^* " t"® ^^^^ ^'^g'© ° sin ADE sin DAY ° at which the axes are inclined be called «, and the angle at which the proposed line AB is inclined to the axis of x be called a', we may write the equation oi AB thus : sin a y=-^i — ^^ t^J- sin {a— a) If the axes are at right angles, a— a' will be the complement of a', so that then the equation will be y=tan a' . x [3]. Hence, when a straight line is referred to rectangular axes, the constant a, in the equation of it, always denotes the tangent of the inclination of that line to the axis of x. This constant, whatever be the axes, it is convenient to call the coefficient of inclination. If the coefficient of inclination be positive, then, whether the axes be oblique or rectangular, the part of the line above the axis of x will always lie to the right of the axis of y ; and if the coef. of inclination be negative, it will always lie to the left. This is only saying that if tan a' be plus in [3], a will be acute, and that if it be minus, a will be obtuse; and if a— a' in [2], be plus, ot>a.\ and if it be minus, oi!>a.. If a=2a', then coef. of inc. =1. 343. The general equation y=ax, considered above, belongs only to straight lines passing through the origin ; all others are excluded : let the restriction be now removed, and let the line sought to be represented by an equation take any position whatever, as the position Cxi/ in the annexed diagram. Let a parallel to CM be drawn from the origin A, and let the ordinate PE, of any point P in the pro- posed line, cut it in D : then, whatever point in CM, P may be, we must always have DP=JB, so that to any abscissa x^AE, the corresponding ordinate is y=ED-\-AB; but putting, as usual, a for the coefficient of inclination, ED=ax, as shown above: hence, the equation y=ax-\-h, where h is put for the constant length, AB, and a for the constant ratio Fi A DW An ""J^' ^^P^^sses the relation between the x and y of every point in the unlimited straight line CM, and is, therefore, the equation of that line ; and since CM is any straight line whatever, y=ax-\-l) [1], is the general equation of a straight line: it represents a particular straight line only when particular values are given to the constants a, h. These are called constants, because they never change in passing from point to point of the same straight line ; and since x^ y, do change their EQUATION OF A STRAIGHT LINE. 289 values, in passing from one point to another, these are called variables : the general equation of a straight line contains, therefore, two variables, and two constants, the constant b being always the ordinate of the line at the origin, or the value of 1/ corresponding to ^=0. Writing these constants with their proper signs, as indicated by the different positions of the line CM, in the following diagrams, the several equations of CM, or of CB, prolonged indefinitely, will be as below, each equation being written to the right of the line to which it belongs : — y=ax-^b. y=:— ax-f &, y=zax—b. y:=—ax—h, the coef. of inclination a being + or — , according as a parallel from A to BC, drawn above AX, would fall to the right or to the left of AY; and b being 4- or — , according as BC crosses the axis of Y above or below A. The preceding comprehend all the varie- ties of position of BC, except when it is parallel to one or other of the axes, as in the annexed diagrams : in the first, the equation is y=zO£c±:b, or y=:±:b, accord- ing as the line is above or below AX; that is, y is constantly the same, whatever he x: in the second, the equation of the line is x=±.c, that is, X is constantly the same, whatever be y. 344. As every straight line may thus be repre- sented by a simple equation, involving at most but two variables {x and y), so every simple equa- tion with only two variables is the analytical repre- sentation of some straight line. In other words, such an equation being proposed, we can always geometrically interpret it by a straight line drawn in reference to two fixed axes. Thus, every simple equation with two variables may be put in the form Mi/—Nx-\-T?, or .[1]. N P y=^«-f -jj^, or more sunply, y=Ax-^B. Now, assuming any axes AX, AY, of which the origin is A, make AB=^B, V 290 EQUATION OF A STRAIGHT LTNE. and AC =—, attending of course to the signs of these quantities, as pointing out the directions from A in which the measures are to be taken : through B, C, draw an unlimited straight line : this will be the geometrical representation of the equation. For the equation of the line thus drawn is y= — x+AB [2], AB the ratio —^ being the coef. of inclination (343) ; but by construction, -—^=B^— =A,aXao AB=B, 80 that the equations [1], [2], are identical; either therefore represents the straight line CB. 345. In the foregoing illustrations, the inclination of the axes has been taken at random ; but it is best, when the inclination is not over- ruled by other circumstances, to choose them rectangular, for then the coef. of inclination is simply the tangent of the inclination of the line to the axis of x. In the following examples, which the learner will do well carefully to construct, the axes may always be thus chosen. All that is necessary, in order to construct the line from its equation, is to get two points in that line : those most easily found are the points where the line crosses the axes : putting x=^0, in the equation, we get the y of the point where it crosses the axis of y, and putting i/=0, the x of the point where it crosses the axis of x, that is, we get the lengths, AB, AC: the line through B, C, is that required. Note. — The lengths AB, AC, intercepted between the line and the origin, are called the two intercepts ; and the line which any equation represents, is called the locus of that equation, being the line in which all the points [x, y) are placed. Ex. Required the locus of the equation yz=x—^ in reference to the given rectangular axes in the diagram. Put ^^ aj=0, then j/= — 3. Put y=^0, thena;=3: we thus have the lengths of the intercepts. Mea- sure, therefore, below A, AB=o, and to the right of A, AG=^: then the straight line through B, C, will be the locus required. The angle C is 45°, because the coef. of inclination, which, the axes being rectangular, is the tangent of the inclination, is 1. Examples for Exercise. Construct the locus of each of the following equations : — (1) y=2x-l. I (3) y-^x-l=0. I (5) Sx-y-^2=0. I (7) 2^+6 =ix-y. (2) y=:Zx-\-2. I (4) 2y-x+2=0. I (6) Ay-Q^x=0. I (8) 6x-Ay-\-d =0. Note. — If the general equation of a straight line, y=ax-j-by to which form all these are of course reducible, be written S+l- w THE STBAIGHT LINE SUBJECT TO CONDITIONS. 291 which is merely putting B for 6, and A for , the constants A, B, will be the intercepts on the axes of a; and y respectively, as we see by putting first y=0, and then ^7=0, in the equation. 346. The general equation of a straight line, as we have sufficiently seen, contains two constants — these are arbitrary constants — unless an assigned individual straight line is to be represented : that is to say, we may put for these constants any numerical values we please, and some straight line or other will always be represented by the equation : thus, in the foregoing examples, the numerical quantities have all been taken at random. We may therefore impose any two conditions on these con- stants, so that, by satisfying these conditions, they may become fixed and determinate in value, and thus represent only one straight line, drawn conditionally, to the exclusion of all others. Or, we may introduce but one condition, sufficient to fix the numerical value of only one of the two constants, leaving the other still arbitrary, or open to any numerical in- terpretation whatever, or else sufficient to fix a numerical relation between the two. There will then be represented, not a single exclusive line, but any one of a group of lines, infinite in number, seeing that one of the constants may take any one of an infinite variety of values: but this group of lines, in virtue of the fixed constant, or the fixed relation be- tween the two constants, will be exclusive of all other groups. We shall here give the different problems which thus suggest themselves. 347. The Straight Line subject to Conditions.— The notation (x, y) means any point, x and y being any pair of co-ordinates : if a particular point is exclusively meant, the notation is {x\ y'), or [af\ y'% &c. Problem I. To find the equation of a straight line passing through one given point. Let {x\ y') be the given point: then y=.ax-\-h repre- senting any straight line indifferently, so long as a and h are arbitrary, if one of these lines is to pass through {sd, /), the equation must be satisfied for these values of x and y\ that is, we must have y'=ax' -\-^y :. h=.'i/ —aaf . A relation is thus fixed between a and h : either may be anything we please, but once chosen, the other is necessarily fixed by the above condition. Substituting this value of b in the general equation, the resulting restricted equation necessarily excludes all lines not nassing through the point {x', y'). This equation is y=.ax-\-y'—aot:!, or, as it is usually written, y— y=a(a5— arf)...[l], from which we see b is eliminated ; but a may take any value whatever, showing that an innumerable variety of straight lines may be drawn through the same given point. If the point were on the axis of x, then y =0, and the equation would be y=a{x—x'') : if it were on the axis of y, then a;'=0, and the equation would be y—y'z=zax, or y—ax-\-y\ For ex. Suppose it were required to find the equation of a straight line passing through the point (—2, 3), in reference to assigned axes. Here y—y'—a(x—a/) is y— 3=a(a; + 2), the required equation; and whatever straight line be drawn at random through the point (—2, 3), some value of the arbitrary constant a exists which will render this equation the representative of that single line only ; but if a straight line be drawn not passing through the given point, then no value can possibly be u 2 292 THE STRAIGHT IJNE SUBJECT TO CONDITIONS. assigned to a, in the above equation, that will render it the equation of that line. Prob. II. To find the equation of the straight line which passes through two given points. Let the given points be (^', if), and [a/\ i/'). All lines passing through the first of these are included in the equation y~y'^=a{x — x'\ and the particular one here required must fulfil the ad- ditional condition 1/''— ?/'=«(:»''— ;y'); that is to say, for this particular line, a must have the particular value a=^^, — —„ or a——, — ^,. Sub- X —x x—x' stituting this fixed value of a, therefore, in the equation [I], we have for Equation of straight line through (as', 2/')> (^"> 1)")-, y—'i/—~j — ^/ (aJ— aj')...[2]. X — X Ex. Suppose the two given points are (—2, 3), (1,4): then the equation of 3—4 1 the line passing through them is?/— 3 = — - — ^,-(^+2), or y—2> = -^x-\-^), or 3?/— ll=;r. We see by the second form of the equation, that the par- ticular value which must be given to a in the last example, in order that, out of the group of lines passing through (—2, 3), that particular one which passes also through (1, 4), may be exclusively represented, is a=-. o If one of the given points (a;', /) is on the axis of x, then y'=0, and the equa. is y" y=- „_ , {^~^')i 0^ ^^ (^"> y") ^6 the point which is on the axis of x, the eqna. is v' y—y'— ^_ „ (^— ^')- If (^'> y') ^^ on t^e axis of y, then ^'=0, and the equa. is v" — y' y'-'i/=. — IT— a;; and if (x", y") be the point on the axis of y, the equa. is v'—y" y—y'=^ ^—{x—x'). If one of the points be upon both axes, that is, at the origin, «" then either x'=0, and y'=0, or x"=0, and y"=0, the equation being either y—— x, or x' V . . y" y —-x, the coef. of inclination being either — , or -, , these ratios being equal (341) x" X Note. — In the following examples the learner should always write down from memory the equa. [1], or [2], in the general symbols, and then put the particular values for the accented quantities. Examples for Exercise. Find the equation of a line, or of a group of lines, from the following conditions : — (1) Through the two points (1, —2), (—3,-4). (2) „ „ „ (0, 4), (-5, 2). (3) „ „ „ (3, 0), ( 0, -7). (4) Through the point ... (8, —3). (5) „ „ ... (0, -6). (6) Through the two points (0, 0), (-3,-4). Prob. III. To find the equation of a straight line passing through a given point, and also parallel to a given straight line. Let [x\ i/) be the given point, and let a' be the coef. of inclination in the given line ; then, since two straight lines must be parallel, provided they have the same coef. of inclination, the equation sought will be y—y'=a\x—x'). THE STRAIGHT LINE SUBJECT TO CONDITIONS. 293 Ex. Find the equa. of a straight line passing through (2, —3) and parallel to the line 5^/— 3^2; + 4=0. Putting this equa. in the form 3 4 3 y=ax-\-l>f it is y=.-x—-, .'. - is the coef. of inclination. g ^ Hence the required equation is y-\-S=-{x—2). o Prob. IV. To find the point where two given straight lines intersect. Let the equations of the two lines be y=ax-\-h, and y=a'x-\-h\ the con- stants being of course given, and both lines referred to the same assigned axes. The symbols x, y, it must be remembered, stand for the variable co-ordinates of the several points of the line y=ax-{-h\ into whose equation they enter, so that in different lines y=La'x-^V) they are differently related : the x and y of one line must, therefore, never be confounded with the x and y of another, except only at the single point in which they intersect. At this point, and at this only, the two equations are simultaneous, and we therefore link them together as in the margin, and find the values of x, y, common to both, as in ordinary algebra : we thus get for the point of intersection V-h ab'-a'b a{b'-b)-\-b{a-a') (b'-b) ^ x=. ;, y= ;-= =a y+o; a — a a — a a — a a — a so that, having computed the x of the intersection from the first of these conditions, the y of it is got by multiplying that x by a, and adding b, as indeed we should infer from the first of the original equations. Or if the X of the point be multiplied by a\ and then 6' added, the y of the point will equally be obtained, as the other equa. shows. If a=a\ the given lines will be parallel, and the co-ordinates of their intersection will be- come infinite, as they ought to do. If b=b\ then x=0, and y=b, showing that the lines intersect on the axis of y, their equations imply- ing that the intercept on this axis is the same for both. If a=a', and b=b\ the X of the intersection is -, any value whatever (191) : the lines then being identical, every point is a point common to both. Ex. Required the co-ordinates of the point in which the lines ^y— 3:c-f4 = 0, 3i/ + a;— 6=0 intersect. Here, proceeding as in common algebra, multiplying the first equa. by 3, and the second by 2, to elimi- nate y, and then multiplying the second by 3, to eliminate x, we find x=2^, y=l^, the co-ordinates required. 348. Since at the point where two lines intersect, the x, y in the equation of one, is the same as the x, y in the equation of the other, if we add these two equations together, or subtract one from the other, or do so after introducing any numerical factor into either, the resulting equation must be equally satisfied for the co-ordinates of the point ; just as in common algebra, though we add or subtract, in like manner, two simultaneous equations, the particular values of the unknowns, x, y, equally satisfy the resulting equation. In fact, every pair of simple simultaneous equations, with two unknowns [x, y), when solved, furnishes single values for the x and y, which are no other than the co-ordinates of the point in which the lines represented by the proposed equations intersect. Prob. V. To find the expression for the angle of intersection of two given straight lines. In all the foregoing problems the axes may be regarded as unrestricted as to their inclination : in the present problem 294. TUE STRAIGHT LIl^E SUBJECT TO CONDITIONS-. they will be assumed to be rectangular, because, unlike the former problems, it is not matter of indifiference here, as regards simplicity of result, whether the axes be rectangular or ob- lique. Let the equations of the two given lines CP, CP, referred to the rectangular axes AX, AYy be y=acc-\-b, and y=^a'a:-\-h\ in which a is the tan of PCX and a' the tan of PCX. Now, since the angle CPC is the difference of these two, if we call the tangent of it v, we shall have (242), .[1], l+aa' l-f-aa' according as this angle lies to the left or right of PC If a, a\ are so related that 1 +aa'=0, then the tangent v is oo, show- ing that the angle of intersection /* is a right angle : hence the condition of perpendicularity is aa'= — 1, or a'= , so that 7/=.ax-{-b, being the equation of any line, y=: — a; +6' will be the equa. of a line perp. to it. The perpendiculars are innumerable, since 6' may have any value what- ever: if one of these be fixed by the condition of passing through a given point (a/, «/')» ^^^^ (Prob. I.) the equation of this perp. will be .[2]. y—y'=- — (x— a/). Examples for Exercise. (1) Determine the point where the two lines 3?/— a;+l=0, 2/+3a;— 1=0, intersect, as also the tangent of the angle of intersection. (2) Required the equation to the straight line which passes through the point (—5, —3), and is parallel to the line 2y=— 3x4-4. (3) Required the equation to the straight line which passes through the point (—5, —3), and is perp. to the line 2^"=:— 3a; 4-4. Peob. VI. To find the equation of the straight line which passes through a given point, and which makes a given angle with a given straight line. Let the given line be y=ax-\-b, the tan of the given angle v, and the given point {of, y'), the axes being rectangular ; then the equa. of the required line will be of the form y—y'-=a'{x—ocf) in which a' is to be so determined that v= + :; ,. (Prob. V.), i-\-aa' ^ ' , a—v a-\-v 1-av* and the required equation either^ — y=— ^^^(a;— a/), l-\-av , , a-\-v , ,. or else 7/—7/=- (x — x') .[1]. There are thus two lines satisfying the required conditions, one forming the assigned angle on the left of the given line, and the other forming it on the right ; thus, the first of the above equations applies to the line JTHE STRAIGHT LINE SUBJECT TO CONDITIONS. 295 Pp ip being the given point), and the second to pP\ the given line being CD. But if the given angle is a right angle, then there is only one straight Hue that will fulfil the conditions : for the two coefficients of inclination, a—v a-^v /a \ /I \ /a \ /I \ ; , and :; , may be written ( 1 l-j-l — [-a ), and ( — hi j-rl « / 5 1-hay 1—av ^ \v / \v / \v / \v / — I zf M' "X and if v=tSLn 90°= oo , these become — -, and — , which are identical ; a —a so that the equation of a straight line through (a/, y'), and perp. to y:=aa!+b, is y—y'=. {x—x'\ agreeing with equa. [2] in last page. Prob. VII. To find the expression for the distance of two given points (co-ordinates rectangular). Let the given points be M{x' , y'\ and ^[^x'\ y"\ then if MP, parallel to AX, meet y" in P, we shall have MP=:x"—x'j and NP=y" —y', .: the distance or if one of the points {a/\ y'') be at the origin, that is, if x"=0, 2/"-0, then i>=V(»'H2/'-). If the axes had been oblique, MN would MNz=^{MP^-^NF^-2MP . NP cos MPN), art. (228), or since cos MPN=— cos M'PN=— cos A, .-. D=>y{{x"-xr-\-{y"-yy+2ix"-x'){y''-y') oos A] ; and if x"=0, y"=0, I>=^{:c'^-{-y'^+2x'y' cos A). Peob. VIII. To find the expression for the distance of a point from a line (axes rectangular). Let P{a;\ y^) be the point, and BC,y=ax-^b, the line. If PN be a parallel to the axis of y, the abscissa AD [—x') will be equally that of P and N. Hence, putting x' for x in the equa. of BCy we have DN=y=ax' -{-b, PN .'. PN=y' -ax' -b. Now P£=PN cos P= but P=180°-C; and tan C=c secP' .-. sec 2p=a2-|-l, As we know from the equation of the given line whether tan "'a is acute or obtuse, that is, whether a is + or — , there cannot be any am- biguity here ; for knowing whether the tangent is + or — , we know also whether the secant v/(a- + l) is + or — . If the point be at the origin, the distance, or Perp, = —f^ [2]. If a be the angle of inclination of the given line, then a=tan«, and the expression [IJ for the perp. may be written , y'—x'ta,na—b P='. -, that is. P=y' cos »—xf sin x—b cos a=(j/'—b) cos a— a;'sin « [3]. 296 PEOBLEMS REQUIEING THE EQUATION OF THE STRAIGHT LINE. This form for the length of the perp. from a point {x', y'), upon the line y—x tan cc-\-h, will be found useful hereafter. When the point is the origin (0, 0), the expression is P=-&cos« [4]. Note. — As the preceding problem is to determine the absolute length of a straight line which, in general, is not in either of the co-ordinate directions, but is usually oblique to them, the sign of direction is an appendage which we may, in general, disregard, more especially as in our original conventions we have not expressly provided for the directive signs of lines not in the co-ordinate directions. Neither, in the Trigo- nometry, did we provide formally for the directive sign of the secant of an angle, yet we found that the secant had a determinate sign, neces- sarily claimed by it as a consequence of the sign which our conventions had expressly given to the cosine. So here, the sign which algebra sup- plies to any line oblique to the co-ordinates, whether we want it or not, will always be found to be a necessary consequence of the co-ordinate signs, and to be in strict consistency with the co-ordinate conventions. Thus, in reference to the expression [4] above, suppose BN to turn about the point B, till it comes into a position of parallelism with the axis of x, the perp. from A will then be AB, which of course, being in a co-ordinate direction, ought to take the sign + . This, however, it would not do but for the minus prefixed to the general expression for a perp. from A above. For the angle a=iBCX, then becomes 180% the cosine of which is —I, and as [4J ought to apply to every value of the angle a, the necessity for the minus is obvious. As many writers on this subject have felt a perplexity about the direc- tive sign of a perpendicular from a point to a line, and have given erroneous interpretations of it, it may be well to add the following to the foregoing remarks. 1. The sign of the denom. in the expression marked [1] is never ambiguous ; for that denom. is the secant of the fixed angle the line makes with the axis of x. 2. The sign of the numerator is + or — , according to the relative positions of the line and point : there is no ambiguity of sign ; one or other appears of necessity, as an inevitable consequence of the co-ordi- nate conventions. The perp. being drawn, if the line be moved parallel to itself, till the point coincides with the origin, the proper sign will be ascertained from [4] above ; it will be the same sign that we should give to it if it were the secant of the angle which it makes with the axis of x ; we thus see how improper it is to prefix the ambiguous sign to the expression [1], or to speak about choosing the sign, so as always to make the expression positive : there is no choice offered to us : — the sign is definitely fixed in all cases. 349. Problems requiring the Equation of the Straight Iiine. — The following problems will show the geometrical application of what has been established respecting the straight line in the preceding articles. Prob. I. To determine whether perpendiculars from the three vertices PROBLEMS REQUIRING THE EQUATJON OF THE STRAIGHT LINE. 297 to the Opposite sides of a plane triangle all meet in one point. Let ABC be the triangle, and CD, AF, BE, the three perpendiculars. Assume the base AB for the axis of ic, and the perp. AY for that of y. Put a)\ y', for the co-ordinates AD, DC, of the point C, also put a/' for AB, the abscissa of the point jB, then the analytical representations of the three points A, B, C, will be (0, 0), (0, x"), {of, y'). And our object is to ascertain whether the ab- scissa of the intersection of AF, BE, is also the abscissa of C, that is, whether the x of this intersection is x'. The lines concerned are AC, passing through the origin and through the given point {x', y') ; BC, through the two given points [x", 0), and {x\ y') ; BE, through the given point {x", 0), and perpendicular to AC; AF, through the origin and perpendicular to BC. The equations of these four lines are, by the preceding articles, (See p. 292). Equa. of AC, y=-,x\ Equa. of BG, y=-y^,{x-x"). ^■' rip' lua. of BE, y=. — -{x—3ii')', Equa. of AF, y=.—- — r y y -X. (See p. 294). At the point where AFy BE, intersect, the ordinates must be the same for both lines hence, at this point —,(^- y •x") x^-xf' X, .'. x=x^, that is. the X of the intersection is x' : hence, AF, BE, intersect in CD. Prob. II. To determine whether perpendiculars from the middle points of the sides of a plane triangle meet in a point. Taking the rectangular axes as before, let the middle points be M, M\ W, and represent the points C, B, ^y C^''. ?/)' K'' 0), the point A being (0, 0). Now M being at the middle of AG, its co-ordinates are half those at the extremi- ties A, C ; and M" being at the middle of BC, its co-ordinates are half the sums of those at the extremities B, C, that is, the point M' is f — , |- Y and the point M" is ( "^ -, — Y And we have to ascertain whether the x of the intersection of perpen- diculars from these points to itC, BC^ respectively, is -— , the abscissa of til M, or not. Now (Prob. II., p. 292), Equa. of AC, y=^,x. Equa. of BC, yz=^~r—r,{pc-x"). X X —X And [2] (p. 294), Equa. of Perp. from M', 2/-^=-^(^-|-)' „ Perp. from M", y--L:=:—r-{^x ^ ). 298 PROBLEMS REQUIRING THE EQUATION OF THE STRAIGHT LINE. At the intersection of these perpendiculars their equations are simul- taneous, .'. 2x"x=-x"^, .\ x=-^, the abscissa of the intersection=^ J/. Prob. III. To determine whether straight lines from the vertices of a triangle to bisect the opposite sides meet in a point. Let ilf, M\ M" be the middle points of the sides. Draw CM, and for axes take AB and ^F parallel to MC. As before, let C be (x', y'), then B will be 5f (2^, 0): and as in last prob., M, W have to ascertain whether the line through A, and the second of these, and that through B, and the first, intersect in a point of which the abscissa is x-=.x'. 2 2 Equa. of AM'\ y=-^x. Equa. of BM\ y= {x-2x'). At the intersection, these equations are simultaneous, .'. «-«=-——», 2 2 2^ 1 1 The corresponding y of the intersection is y=o-=-ic'=-y' : hence PM=-CM. 3 , o o r Prob. IV. To determine whether the straight lines bisecting the three angles of a plane triangle meet in a point. Let AF, BE, bisect the two angles A, B: it is required to ascertain whether these bi- sectors cross CD, the bisector of the angle C, at the same point. Take DC, DB, for axes, and let AF cross DC in P, and let BE cross it, if possible, in some other point P', then (p. 290) the equations of AF, BE, are AF, The ordinates of the points where these cross DC are y=DP, and t/=DP\ and the inquiry is — are these equal ? PD=AD sin-^ 2 • ^ A dn-{A + C) cos- 5 ^ 2 ; P'D=DB- Bin-B sin-(5+C7) -=zDB sini^ IT- cos -4 2 PEOBLEMS REQUIRING THE EQUATION OF THE STRAIGHT LINE. 299 ' . T. . « sin--B cos-jB Now (Euo. 3. VI.), ^=5^, .-. ^D=— f ^^DB, sm-^ cos--d 2 2 sin-B which, substituted in the above value of PD, gives P2>= cos --4 2 DB=PD. Note. — The preceding investigation can scarcely be considered as an exercise in co-ordinate geometry, for except the first two equations, which in reality are not needed, the equations of lines do not occur. The problem has been introduced merely for the sake of completing the set of problems about three lines, connected with a triangle, all intersecting in a point. It may be solved much more easily as follows : thus, disregarding the line from C, let the bisectors AF, BE cross in P : then the two angles at A being equal, perpendiculars p, p^, from P, on AB, AC, must be equal. Again, the two angles at B being equal, perpendiculars p, p^, from P, on BA, BC, must be equal, .*. Pi=P2' Hence the line CP must bisect the angle C. Peob. V. To express the area of a plane tri- angle, in terms of the rectangular co-ordinates (in the same plane) of its three vertices. Let ABC be the triangle, and let the rect- angular co-ordinates of A, B, C, be (a?i, y^), (ajg, 2/2)' (^a' ^3)* Then from the three trape- zoids we have Area. ABC={ABFI)-\-BCEF-ACED). Area=- { (ya+yj {x^-^i) + (ya+ya) (^3 2?yi(^a- ^a)-(y3+yi)(^3-«i)}= ^3)+y2(-^3-^i)+y3(^i-^2)}- If A be at the origin, then a;,=0, 3/,=0, and the expression is Area= ' . Peob. VI. Any number of straight lines are given in length and po- sition ; it is required to determine the locus of a point P, such that wherever in that locus P be taken, and lines be drawn from it to the extremities of the given lines, the sum of the areas of the triangles thus formed may be constant. Any rectangular axes being chosen, let the inclinations of the several given lines to the axis of ic be a, a^ cc^, &c., and their intercepts on the axis of y, b, b^, b.^, &c. ; then if from any point P{£c, y), in the required locus, perpendiculars be drawn to these lines, we shall have for their lengths, [3], (p. 295), p=y cos a—x sin a—h cos a, Pi=^y COS a^ — x sin eti—b^ COS «,, p^y COS ct^—x sin »2~^2 ^^^ *aj ^^• Hence, calling the lengths of the given semi-bases of the triangles I, Zp l^, &c., we have for the sum of their areas, Ijy, l^Pi, l,;^P2, &c., {I COS «+^i COS a,-f ^a COS a.^'k---)y—{!' sin a+^, sin «,-f-^2 sin a^+---)^~« {lb cos as f ^jj, cos a,+?2*2 COS aj +...)= Constant, 300 THE CIRCLE — EECTANGULAR AXES. which is the equation of a straight line, the locus required. It is plain that if the areas be united together bj the signs + and — in any way, and the result is to be constant, the locus of the point which is the common vertex of all the triangles will still be a straight line. And the same is true if any multiples of the areas in like manner be united together, provided the result be constant. Examples foe Exercise. (1) Determine the equation to the straight line which passes through a given point {x', 2/'/> ^^^ makes equal angles with the given oblique axes. (2) At what distance from the origin of the rectangular axes is the line whose equa- tion is 2y—4a:+ 1—0? (3) Find the general equation of all lines passing through the intersection of the lines y^ax-^h, y=:a'x-\-b'. (4) Find the equation of that particular line in the last gi'oup that passes through the origin. (5) A triangle is right angled at A, the sides are produced through A till the pro- longation of the base is equal to the perpendicular, and the prolongation of the perp. equal to the base ; the parallelogram, whose sides are these prolongations, is then com- pleted. Prove that the line through A and the opposite vertex of this parallelogram is perp. to the hypotenuse. (6) A line is drawn parallel to the base of a triangle, and from its extremities lines are drawn, crossing each other in P, to the extremities of the base. Prove that the locus of P is a straight line from the vertex to the middle of the base. (7) Squares are described on the three sides of any triangle ABC, and outside the figure three lines are drawn connecting the comers of these squares. Prove that if per- pendiculars to the three lines be drawn from A, B, (7, they will all pass through the same point. (8) Squares are described on the base and perpendicular of the triangle A BC, right- angled at A. From the vertices B, C, lines crossing each other are drawn to the oppo- site comers of the squares. Prove that these intersect on the perp. to the hypotenuse from A. 350. The Circle— Rectangular Axes.—Let r be the radius OP of the circle whose centre is 0, and let AX, AY, be any rectangular axes in its plane. It is required to determine the equation in x and y, which shall be satisfied for every point P{sc, y), in the circumference, and for such points only. Let the co-ordinates AB, BO, of the centre be a, b, and those of any point P in the circumference, An=x, nP=y ; then drawing Om parallel to AX, we have Om=:x—a, Pm=y—b, and therefore since or a;2+2/^-2(ax+52/)+aH&2=r2 [1], which is the most general equation of the circle in reference to rectangular axes, and from which the following simpler forms are deduced : — 1. If the origin be at any point A^ of the circumference, equation becomes then a^+6^=r^ and the a;2^^2_2{cu;+6z/)=0 [2]. OBLIQUE AXES. 301 If the origin be at A.^, so that the axis of x passes through the centre, then a=r and 5=0, and the equa. is x^+y'^-2rx=0 [3]. But if it be at ^.,, so that the axis of y passes through the centre, then since b~r, and a=0, the equa. is x^+y^—2r7j=0 [4]. 2. If the origin be at the centre O, then since in this case a=0, and 5=0, the equation is simply a;2+2/2=r2 [5], which is the form most frequently employed. If in the general equation [1], we put ^=0, we shall have for the cor- responding values of y, that is, for the ordinates of the two points in which AY intersects the circumference (whenever the circumference is cut by that axis) i/ = 6± v/(r-— a'-^). If a=r, these two values unite in one, namely, y=b, for the axis of y then touches without cutting the circle. If a or, that the tangent through any point of a circle is perp. to the radius at that point. 353. Equation of Tangent from a Point,— We have now to find the equation of a tangent from a point F{a, b) without the circle. Put {x\ y') for the point of contact, then x'^^y'^=r^ [1], and the equation of the tangent being xa/-\-yy'=:r'^, on which [a, h) is a point, .-. a:c'+&2/'=r2 [2], CHORD OF contact: pole and polar. 303 In the equations [1], [2], sd, yf are the only unknown quantities, these therefore may be determined from those equations, and since [1] is a quadratic, there will be two values for of and two for y': hence tvco tangents may be drawn from P(a, 6) to the circle. Now without thus solving the pair of simul- taneous equations referred to, we may deter- mine the two points by a simple construction : for since equa. [2] must be satisfied by both values of x' and both values of y\ it follows that the two points of contact are both on the straight line ax-{-by=^r'^, which is there- fore called the equation of the chord of contact: if this line be con- structed, it must cut the circle in the two points required. Proceeding, for this purpose, in the usual way, we have for y=0, a;=AC=.—, and for ^=0, y=AB=-j- : then the line MM\ drawn through C and B, will be the chord of contact, M, M' being the two points where the tangents from P touch the circle. It is worthy of notice that since AC=—, an expression that remains the same for the same value of a, however h or VN may change, it follows that wherever on the line PiV, P be taken, and tangents from P drawn, the several chords of contact will all pass through the same fixed point C\ hence the following geometrical property of the circle, namely: — If from each of any number of points in a straight line, pairs of tangents be drawn to a circle, the chords joining the points of contact of each pair will all intersect in the same point. And it is a peculiarity of analytical geometry that unsought-for properties often spontaneously offer themselves to our notice in this way. 354. In the above diagram the proposed line FN lies wholly without the circle, and the intersection of the chords of contact is, in this case, always within the circle, since then AC=—r, and the point of intersection is therefore a ^ without the circle. In both cases this point is called the pole of the lino PN, and the line itself the polar of the point. It has been shown above, that wherever a point P[a, b), without the circle, be taken, the equation ax-\-b?/=r^ is the equation of the chord of contact in reference to tan- gents drawn from that point, provided the axes be any pair of rectangular axes whatever, originating at the centre of the circle. In like manner, Pj, P^, P:,, &c., being a series of points in the same straight line as P, the annexed equations all have place simul- « x-{-b y=r" taneously, since the lines they represent all pass through one a,a;+5,y=:r2 point («, (S), the pole of the line of which P P^...Pa is the ae^x+h^yz=r^ polar. Hence the equation ax+6/S=r^ holds for every : point (a, b) on this polar : in other words aa; + (3y=r'^ is the a,,x-{-b„y=ir^ equation of the polar of which (a, /3) is the pole. Every 304 LOCUS CONSTRUCTED. equation therefore of this form, the axes being rectangular, represents the polar of which («, i?) is the pole : — the chord of contact above, namely, ax-\-b7/=r-, is thus the polar of which {a, b) is the pole. And generally, if any point {a, 6) be on the polar of another point (a, /S), then will the point (a, /3) be on the polar of the first point [a, b). [For a very masterly discussion of poles and polars, the inquiring student is referred to the writings of the late Professor Da vies, of Wool- wich, as contained in the second volume of his " Course of Mathematics," and in his various papers in " The Mathematician," and in the " Lady's and Gentleman's Diaries," 1850-1. The elaborate researches of this dis- tinguished Geometer, on these and kindred topics, in the writings referred to, and the two volumes of Mr. Salmon on Conic Sections, and the Higher Curves, coutain a body of valuable information on the subjects of the pre- sent short treatise, which is unequalled in the English language].* 355. Locus of the 'Eq\xaXionx'-\-y- + Ax-{-JBy-\-C=0. We have seen that the general equation of a circle wljen referred to rectangular axes is of this form : it remains to be shown, conversely, that the locus of every equation of the above form is a circle referred to rec- tangular axes. Let AX, AY he any rectan- gular axes, and determine the point of which the abscissa ^M is — — , and the ordi- nate MO, — j. From as a centre, and with radius r^=A/\ — C [ , describe a circle: the circumference will be the locus of the proposed equation. For the equation of the circle just constructed is (a;--a)'+(2/— fe)'=r"% whicb, since A B J2_f_^2 a=— — , &=— — , and r^=. — (7, is the same as Z A 4 and this when developed, is x^ -\-y'^-\-Ax-\- By -{- C=0 : hence the circle, con- structed as above, is the locus of the proposed equation of the second degree. The student must pay particular attention to the peculiar form of this equation : he is to observe that the squares of the variables enter with equal coefficients: — these may be each made unit by division : he is also to observe that the product of the variables is absent. With these features in any equation of the second degree containing two variables, he may always pronounce that equation to be the analytical representation of a circle referred to rectangular co-ordinates ; but not when the equation is of any other form. If the general equation marked [6] at page 301, be developed, its form will be x^-\-y'^-\-^xy cos w+^;» + 5j/ + (7=0, where u is the oblique angle * The value of Professor Davies's volume, above referred to, is greatly enhanced by the beautifully-executed diagrams, all drawn by Mrs. Davies, now Principal of the Ladies' College, Southampton. EQUATION OF TANGENT FEOM A POINT. 305 at which the axes are inclined ; and it may he shown, as ahove, that every equation coming under this form represents a circle constructed in refer- ence to axes of which the angle of inclination is u. Here we see that the product of the variables does enter, but that the coef of that product must never exceed 2, because cos u can never exceed 1. The coefficients A, B, C, in the above equation, are, for brevity, put for the following ex- pressions, involving constants only, namely, J=— 2(a+6cos«), B=—2{b-\-aco8a), C=a^-{-P-\-2ahcosA>—i^, and since cos w is given, from the equation, it being half the coef. of xi/, these three conditions suffice for the determination of the centre {a, b) of the circle, and of its radius r. When we say, as above, that every equation of the second degree which comes under a certain specified form represents a circle, we mean that it has no geometrical representation but a circle. The coefficients of the proposed equation may be so related that the equation may admit of no geometrical representation at all, in which case we should infer that the existence of what the equation may have been supposed to represent is impossible. Thus, in constructing the locus of a;--{-^~-{-Ax-\-By-\-C=^0, we have seen that the radius of the circle conceived to be represented by //A^-\-B'^ \ A?-\-B^ it is r=^ / ( — C ). Now it may happen that — = C, and thence that r=0 : the inference in such a case would be that what we had expected to be a circle turns out to be only a single point, the point (a, h) which would have been the centre if the circle had had any finite radius : ^+— j -^-iy-^-^) =0, and two squares cannot give for their sum, unless each separately is 0, .-. x———, y= — — , the co-ordinates of the centre. In the case here supposed, therefore, the equation does not imply any geometrical absurdity, it merely denotes not a circle but a single point. If, however, the relation among the coefficients be such that A^-^B"^ — - — <(7, then r is imaginary, and there is no locus : and put whatever real value we may for x in the equation, the resulting value of y will always be imaginary. We shall illustrate what is here said by an example or two. 3 1 (1) To construct the locus of 2s(?-\-2f—Zx-\-iy—l=^0, or of a^-\-y'^—-x-k-2y—-=0. This is the form for a circle referred to rectangular co-ordinates, the centre (a, ft), being ( -, — 1 )) ^''^^ t^e radius r=^'| (-r) +l^+o \ 5 ^^^^ ^^ ^^^ ^^^^ given centre, and this given radius, a circle be described, it will be the locus of the proposed equation. 13 (2) Required the locus of a?-\-f—Zx—2y-\-—=zQ. This also has the prescribed form, the centre of the circle being (-, l) ; but as its radius is the point f -, 1 ) is the only point whose co-ordinates satisfy the proposed equation. 306 PROBLEMS REQUIRING THE EQUATION OF THE CIRCLE. (3) Let the equation be a^-fy'*— 3x4-2^+6=0 ; then since 324-2' — 6 is negative J we at once infer that no line or point can be represented by the equation : no real values of jc, y can satisfy its conditions. Suppose we take the equation and seek to obtain y in terms of x^ thus : 2/2+2y=3a;-a;2-6, .'. f-\-'2.y+l=Zx-x^-5, .'. y=-l±s/{Sx-x^-5)=~l±^{-\x^-Bx-{-5)], in which we see that the quantity under the radical is always negative, whatever real value be put for ^ (p. 107). It will be observed that for the locus to be imaginary, as here, the absolute term of the equation must be positive. is 356. Problems requiring the Equation of the Circle.-— We shall now give a few problems on circular loci. Prob. I. Given the base, and the sum of the squares of the sides of a plane triangle, to determine the locus of the vertex. Let AB be the given base, and put AC'-\-BC^=^m''\ Let 0, the middle of the base, be the origin of the rect- angular axes OX, OY, and put a for OB, the semi-base ; then (x, y) being any point C in the locus, we have y' -\-(a—xy =z BC', and y'^-\-(a-^xf=AC'^. The sum of these is 2/4-2x24- 2a2=m2 [1], which is the 'equation of a circle of which the centre, and the radius a / ( -— — a- J. If, therefore, with this radius and centre 0, a circle be described, and lines CA, CB, be drawn from any point C in its circumference to the extremities A, B, of the given base, the triangle thus formed will always have the sum of the squares of its sides the same constant quantity. Since 2(y'' + x')=0C^, this constant sum, as shown by [1], is equal to twice the square of half the base, and twice the square of the line from the vertex to the middle of the base. Prob. II. Given the base and vertical angle of a triangle, to find the locus of the intersection of straight lines drawn from the vertices to bisect the opposite sides. Let AB he the fixed base, and one position of the opposite vertex, and let CM bisect tlie base, P being a point in the required locus : then we know (p. 298) that PM=-CM: hence, taking the rectangular axes o AB, AY, calling the co-ordinates of the vari- able point C, X, Y, and those of P, x, y, the given base being a, we shall have Y=^2>y, and since Mc=3Mjp, ox=-a—X=^(-a—x\ X=3a*— < a. Now the locus of (X, r) is the arc of a circle passing through A, B, and containing RA.DICAL AXIS. 307 the given angle : hence X, Y are connected together by the equation XM-r^-2(aX+^y)=0, («, /3) being the centre of the circle (350). Substituting the above values for X and Y, we find that x, y are con- nected together by the equation (3a;-a)H(3i/y'— 2(3«;p-aa + 3jSi/)=0, or 9jj- + 9r-6(^ + <='>-6% + <^' + 2a«^=:0, which is the equation of a circle. This problem is given chiefly for the sake of the principle illus- trated in the solution of it : we see that when two loci are mutually dependent, how one may always be deduced from the other. The next problem, too, involves another principle of importance. Pkob. III. Two given circles {A\ [B) are cut arbitrarily by a third circle (C), and the chords EF, HG, in {A), {B), joining' the points of intersection, are prolonged till they meet in P: to prove that the locus of P is a straight line. Let 0, the point bisecting the dis- tance between the two centres A, By be the origin of the rectangular co- ordinates, the axis of x coinciding with OB. Put OB— a, and let r^ r^, be the radii of the two fixed circles and r that of the variable one, (a, 0) being the centre : then we have for the equations of the three circles {A)...f-\-x^-\-2ax=r^^-a\ \b).. .f-\-x'—1ax—r}—d^, ((?)...3/2+a,-2-2a;c-2/3?/=j'2_«2_/J-2. Now at the points where either of the two fixed circles are cut by the third circle, the corresponding pair of equations are simultaneous, that is, they are each satisfied for the same values (and for no others) of x and y, which values equally satisfy every equation deduced from them by com- bining the two original equations together. Subtracting, then, (C) from \A) and {B\ the resulting equations must be the equations of the lines EF, HG, that is, EF... 2aar+2aa;+2/3?/=(r,'-2-a2)-(r2-a2-/S2) [1], HQ...-1ax-\-1ax-\-1^y={r^-aF)-{r^-(t^-^'^) [2], inasmuch as the first is satisfied for the two points E, F, and the second for the two points H, G ; and, of course, every locus passes through all the points which satisfy its equation. These equations, again, exist simultaneously for the Xy y of the point P, in which they intersect : hence, subtracting the lower equa. from the upper, the result 46i^=ri^— r/ has place for every intersection P of the variable lines EF, HG : the locus of P is therefore a straight line PP^ perpendicular to OB, since the abscissa of every point in it is constant, namely, 2 4a =0D. .[3]. The fixed line PP' thus determined, is what is called the radical axis of the two fixed circles : its equation is found by simply subtracting the equa. of one circle from that of the other : for, subtracting (B) from (A), the result is ^ax=r^—r}. The principle adverted to above, and which this investigation illus- trates, is the following, namely • — X 2 308 EADICAL CENTRE. If two loci, whether fixed or variable, intersect, the sum or dif- ference of their equations (either multiplied bj a constant factor or not) will be a third equation representing a single line straight or curved, or a group of lines, passing through the intersections. Thus the equation [1], or the equation [2], resulting from the combination of a fixed locus with a variable one, represents a group of straight lines, upon one or other of which all the intersections of the original loci lie, for a, $, are constant only for a single one of these lines. Again, the equation [3], resulting from the combination of [1] and [Q], represents the line upon which all the intersections of the variable loci [1] and [2] lie, and it is found to be a straight line. Prob. IV. To prove that the radical axes of three circles, taken two and two, all meet in the same point. Let the equations of the three circles be as an- (y— 3/,)2+(a;— a;,)2=r,2 nexed. Tben the radical axes of (r^ 7*2), (rg, rj, and (y—y^^-\-{x—x^)^=r2^ (rj, r.) — denoting the circles by their radii — by what (y— 2/3)2+ (x—a;^2_y^2 is shown above, are 2Kyi-ya)-|-2x(a;i-ag=(V+yi2)-(x2^+y2=)-(r.2-r22) [1], ^yQ/u-y3)-\-^x{x^-x^=z{x,^-\-y,^-{x,^+y,^-(r,^-r.J^) [2], 2y(y3-y.)+2x(a:3-x,)=(a:32+y32)-(a:24.y,2)_(^^2_^^2) [3]. Now the sum of [1] and [2] is a line passing through the intersection of both : but this sum is [3] ; hence the three radical axes all meet in the same point, which point is called the radical centre of the three circles. Examples for Exercise. (1) From a given point in the circumference of a circle chords are drawn : required the locus of their middle points. (2) Required the equation to the straight line drawn from the point (1, 1) to touch the circle 3x^-}-Sy^=i. (3) Prove that the locus of a point P, from which the tangents to two fixed circles are equal, is the radical axis of those circles. (4) From a point A, without a given circle, two lines are drawn to cut it, and the points of intersection joined, so that an inscribed quadrilateral may be formed : produce these joining lines to meet in A\ and draw the diagonals of the quadrilateral ; then, if through the intersection of these diagonals a line be drawn from A', it will cut the circle in two points, to which lines drawn from A will be tangents. (5) Prove that the loci of the vertices of parallelograms of constant area circum- scribing a circle are also circles. (6) Two fires, of which the intensities are as m to 1, are at a given distance apart; it is required to determine the path which a person must pursue in order that he may always be equally heated by both fires. [The heat received, at different distances from the source of it, varies inversely as the square of the distance.] (7) If there be n given points {xy, y^), (xj, y^, ...{x„, yn), and straight lines from each to a point P be such that the sum of the squares of all of them is a constant quantity (P, prove that the locus of P is a circle whose equation is f+x^-l{x,-\-x,+ ...+Xn)x~(J/^-^y,+ ...-\-yn)y=l{o^) [2], as is sufficiently obvious from the diagram, where the dotted lines are two radii vectores (r), one on each side of the perp. (p). If a perp. from any point [x, y) of the proposed line, to a parallel to that line, through the origin, be drawn, it will of course be equal in length to AP, and (p. 295), the expression for it will be y cos a—x sin a. But since p, drawn from A to P, is opposite in direction, we have p=x &in a—y COS a, and since «=^'-)-90°, .*. sin a=cos ^, cosa=— sin^, .'. pz=x cos ^-\-y sin ^ [3]. The polar equation of a circle, the pole or origin being at any point, is easily written down, the centre (r^, 6^), and the radius R of the circle being given ; for two radii vectores being drawn, one to the centre C and the other to any point P in the curve, the constant R' will be given by the equation (228), r^■^r^^-2r^r cos {0c^6^=m [4]. If the pole be on the curve, and the diameter from it be the fixed axis, then r^—R, and 6i=0, and the equa. is r=2E cos 0... [5], and if the pole coincide with the centre, ri=0, and the equation is r=^R. SIO CHAKGE OF EECTANGULAR FOR POLAR CO-ORDINATES. The two following examples will illustrate the application of some of the preceding equations. [Note. — Cos (9 — 6') is the same as cos (9~G').] (1) To find the polar equation of a line j)assing through the two given points (r^, e^), {r^, fig). By equa. [2] we have for any point in the line,^=r cos {Q(^9), and .*. for the two given points, p=r^ cos (^if^O> 2'=*'2 cos i^-J^^)- Eliminating p from the first and second, and then from the secood and third, we have r cos ^— ri cos ^i+(?' sin ^— r, sin ^,) tan ^'=0, r cos ^—r^ cos ^2~l~(^ ^i^ ^— rj sin 6^) ^^ ^'=0. And now eliminating tan 9', there results for the equation sought • sin(^— ^,) sin(^,— ^j) sin (^— ^J_ ( sin (^— ^,) sin (^- -=0, or r .■/•2sin(^a-^,) risin^^^ (2) To express the area of a plane tri- angle in terms of the polar co-ordinates of its three vertices. Let the vertices A, B, C, be {rv ^i\ (r2> h\ (^3. ^sl the pole being at 0, and OX the fixed axis, then the areas of the three triangles AOB, BOG, CO A, are respectively as fol- lows (308) : -AO. OB sin. AOA=-r{r^siu (^,-^2), n^O . 00 sin B0C=^-r^r3Wx {fi^-fi^^ and -CO . OA sin COA: rirg sin (^3-^,), .-. ABC=AOB-^BOC-COA=-{r^r^Bm {6^-e.^-\-r^r^ sin {6^-6^-\-r^r^^ {6^-6)}. If the points A, B, C, are in the same straight line, the area is ; hence the expression within the braces, when equated to 0, will be the equation of condition that the points (r^, 9^), {r^, 9^, {r^, 9g), may all lie in the same straight line. ANALYTICAL GEOMETRY: Part II. The Conic Sections. 359. Besides the circle there are three other curves — and three only — which may be represented by an equation of the second degree between two variables (^, y). These are called the thi-ee Conic Sections, because, by cutting a cone iDy a plane, in different directions, these three plane curves — the Ellipse, the Hyperbola, and the Parabola, as they are respec- tively called — will be the sections. They are, however, here to be dis- cussed, independently of reference to the cone. EQUATION OP THE ELLIPSE. 311 the middle of F, /, put also PF-\-Pf=^a, 360. The Ellipse.— This is a curve, such that if from any point P in it straight lines be drawn to two certain fixed points F,f, called foci, the sum of the two lines will be always the same ; hence, if to the two foci F, f, the ex- tremities of a cord be fastened, and the cord be stretched into a loop by a pencil P, the motion of this pencil, thus confined by the stretched cord, will trace out an ellipse, seeing that the sum of the two focal distances, namely, PF-^Pf, is always the same. 361. Equation of the EUipsCp— Take for the origin of the rectangular axes OX, OY; and OF or Of=c, then representing, as usual, P by (x, y), we have the following conditions : — 2a=zPF^Pf. [1], y'^^{x^cf=PF [2], y''-\-{x-\-cf=Pf [3]. From [2], [3], by adding and subtracting, 2f-\- 2x2+ 2c2=Pi?2+ Pp [4], icx=Pf-PF^, .'. dividing by [1], —=.Pf-PF, .: [l\ cot* est c P/=a+-,Pi^=a--, .-. [4], y^\x^-^c^=d?^~x\ .-. aV+(a^— c^a^=a^(a^— <^, tlie equation of the curve. To find the points B, C, where it cuts the axes, put first y=0, and then iP=0 in this equation, and we have OB=^±:a, and 00= ± s/(a-—c')^=zt.h, showing that the curve cuts the axis of x at two points B, A, equidistant from 0, and the axis of y at two points C, D, also equidistant from 0. Since b'=0C'=a'^—c', we have, by substitution in the above equation, ay+h^x^=M^I^, or 2/2=^(a2-a^ [A] ; and this is the most simple form of the equation of the ellipse. If h=a, that is, if 00= OB, it becomes that of a circle of radius a, the origin being at the centre 0. The circle, therefore, is only a particular case of the ellipse, the two foci F, f, then uniting in the centre O, for when fc=a, c'^=^a'—b^=0. From the general equation [A] we have /=±-V(a'-a^, and x=±^^ib^-y^). •[B]J a ' ' " b hence, for the same value of x there are two equal and opposite values for y ; in like manner for the same value of y there are two equal and opposite values for x; the chord AB, therefore, bisects all the chords drawn parallel to CD, and the chord CD bisects all those drawn parallel to AB. If we put x=a, or x=—a, the corresponding value of ?/ is ; hence parallels to CD drawn through B and A never meet the curve again, that is, they are tangents at B and A. In like manner for y = ±b, the corresponding values of x are each 0, therefore parallels to AB through 312 EQUATION OF THE ELLIPSE. C, D, are tangents to the curve at those points. The curve is entirely comprehended within the rectangle formed by these parallels, for if x be taken either positively or negatively to exceed OB or OA, that is, if ar>a^, the corresponding y is imaginary; and if y'^>lr, the correspond- ing X is imaginary, showing that there is no point of the curve without the rectangle. The point is called the centre of the ellipse, for it bisects all chords drawn through it; thus, let y=^mx be the equa. of a line through 0, then substituting mx for y in [A], the result is a pure quadratic in x, so that the two values of a;— the x of the intersections — are equal and opposite in sign, the corresponding values of y are therefore equal and opposite in sign, and .*. OP'— OP. Every line PP' thus drawn through the centre is called a diameter of the curve : the greatest of these is AB^ and the least CD ; for the length OP of any semidiameter is which since a^— 6'^ is positive being =c^ is the greatest possible when x is, that is, when :B=a, the semidiameter P> being then =« : and it is the least possible when a:=0, i> being then =6. For these reasons AB\^ called the major diameter, and CD the minor diameter of the ellipse; when spoken of together they are called the principal diameters, or the princival axes of the ellipse ; hence the equation xV+62^=a262, or(^|y+(^^y=l is the equation of the ellipse related to its principal diameters, the origin being at the centre of the ellipse. Still retaining the major diameter for axis of X, if the origin be placed at its extremity A instead of at its middle ; then although every y will remain the same, the new x of any point will exceed the old x by a, so that if the changed x be written X, we shall always have X=x-{-a, or x=X—a : hence, to get the equation of the ellipse when the origin is removed to A, we have only to substitute a— a for x in the original equation [A], and we thus have aY-\-h^2(y^-2ab^x=0, or y^z=-(2ax-x^ .[C], for the equation of the ellipse when the origin is at the vertex of the major diameter taken for axis of x. There is also another form of the equation in reference to rectangular axes which is sometimes employed : it is deduced immediately from the equation [A], the origin and axes remaining the same, but e^ substituted ^2 52 g for 7—, or, which is the same thing, e for -; thus, since lr=a^—c^, equa. [A] is the same as f=(l-^^ {a'-x^, or y^={l-e'){a^-x^ [E]. — T— =-> is called the eccentricity of the ellipse, and the equation E is the equation of the ellipse as a function of the eccentricity. THE ELLIPSE IN RELATION TO ITS PRINCIPAL DIAMETERS. 313 362. The Ellipse in relation to its Principal Diam- eters. — The following are the chief properties of the ellipse, derivable from the foregoing equations : — Theorem I. In an ellipse, the squares of the ordinates are as the products of the segments into which they divide the major diameter. By [A], ■ ■ •; - -T=-7, .*. r '• {ci-\-x)[a—x) iib"^: a\ that is, y" and the product (a-^x){a—x) of the parts into which y divides 2a, are always in the same constant ratio. Suppose a semicircle to be described on the major diam- eter AB; then calling Y the ordinate of a point P' in the circle having the same abscissa a; as the point P in the ellipse, we have {a-\-x){a—x) a* =1, AE -=% that is, P'E'.PE'.'.jyO'. DO **'(a+a;)(a-x) W{a+x){a-xy' y V hence the chords DP, D'P\ if produced, will meet in the prolongation Y of BA : and because — is a constant ratio, the same is true wherever y D, IT, and P, P' may be. To find the length of the ordinate through the focus F, we have only to put xz=c in the equation [E] ; we then have ay=za?—c^z=zh'^, so that 2y= — =^^ — -'y that is, the double ordinate through the focus is a third •^ a 2a proportional to the major and minor diameters : this double ordinate is called the right parameter {p)y or the latus rectum. Introducing it into the equations [A], they become 2^=£^*'"'''^' ^^ 2/'=^^-£a;2. [1]. The semiparameter -p is always greater than once, but less than twice the focal distance of the nearer vertex B; for in the first of [1] put 1 1 a4-^ y—-Pf then 7:/?= (a — x), which is greater than a—x, because >1, but it is less than 2(a— a?), because <2, since xh, tan P is negative, .-. P is obtuse. The numerical value of tan P is the least possible when y is the greatest, that is when 2/=&; but the smaller the numerical value of the tangent of an obtuse angle, the greater that angle is : hence the angle at P is greatest when P is at the ex- tremity of the minor diameter. The expression for the maximum angle is a'—b' The investigation from which [1] above is deduced will equally apply to the second diagram, where the angle P is subtended by the minor diameter, if we write the equation of the curve a'^x^-rb-y'=a'^b'', instead of d-y'^-\-b'X-=a%''; for it is plain that by so doing we merely inter- change the axes of x and y, or, which is the same thing, turn the curve round into the position it takes in the second diagram. But the change spoken of is equally efiected by leaving the x and y in the equation un- touched, and merely interchanging a, b : hence, thus adapting [1] to the case where the minor diameter is the axis of x, as in the second diagram, we have for tan P in that diagram ta P— Z^^—- ^^^ which is always positive: the angle P, therefore, is always acute; and since an acute angle diminishes as the tangent of it diminishes, P is * The student may confine his attention to the first diagram till he arrives at the end of the investigation. TRANSFOKMATION OF CO-ORDINATES, ETC. 316 the least possible when y is the greatest possible, that is, when 2/=«- The expression for this minimum angle is therefore tan P- a~ — b' ; and since this differs from tan P in the first diagram only in its sign, we infer that the two angles together make 180°. [This is obvious, too, from the simplest geometrical considerations, for the figure formed by joining the extremities of the axes is a parallelogram (a rhombus), and therefore the two angles to which any side is interjacent make together two right angles.] From the circumstance that all angles in an ellipse subtended by the major diameter are obtuse, and all subtended by the minor diameter acute, it follows that if upon the major diameter there be described a semicircle or any greater arc, it will lie wholly without the semiellipse : but if a semicircle or any less arc be described upon the minor diameter, it will lie wholly within it : for in the first case the angles in the circular segment will all be less than those in the semiellipse, and in the latter case all greater. If the expressions for m, m', (page 314) be multiplied together, we shall have mm'-=.--^ — -, or [1], mm'-=i 5, showing that the product of X — a a the trig, tangents of the two angles formed by .lines, from the extremities }>" of a principal diameter, to meet in the curve, is constantly equal to 5, where a is the semidiameter subtending the angles. If we multiply each of these trig, tangents by 2a, we shall have the linear tangents of the same angles to radius AB : hence the following property : — If from the extremities of either principal diameter perpendiculars to that diameter be drawn, terminating in two lines from the same ex- tremities, and crossing each other on the curve, the rectangle of these perpendiculars will always be the same, namely, the square of the diameter to which they are parallel. [The perps. are tangents (p. 31-2).] The foregoing properties have been derived from considering the ellipse in reference to its principal diameters as axes : by viewing it in relation to other pairs of diameters, additional properties may be ob- tained. But to get the equation of the curve when the axes coincide with oblique diameters, it is necessary that we know how to pass from rectangular to oblique axes ha\-ing the same origin : the following article explains what changes are necessary for this purpose. 363. Transformation of Co-ordinates from Rectangu- lar to Oblique.— Let ^Z, A F, be the original rectangular axes to which any line is referred, and let AX\ AY' be the new pair of axes of reference. Then P being any point on the line, the original co-ordinates of it are and the new co-ordinates are x!=iAm, y'z=.'niP, 80 that between the old and new co-ordinates we have these rela- tions, namely — \ # "V' 316 ELLIPSE RELATED TO CONJUGATE DIAMETEES. x-=.A'p—rm,^ that is, x-=.x' cos «+/ cos a' ") p . -, ?/=:^»i+rP, that is, 2/=x' sin a.-\y' sin o' j Hence, in order to transform the equation of a curve, in reference to rectangular axes, into another referring it to any different pair of axes originating at the same point, we have only to write for x, y, in the original equation, the expressions for them in [A], and it is plain that the accents over the new co-ordinates need not be retained. If the new axes are to have a different origin from the primitive, then, calling the primitive co-ordinates of the new origin ^i and y^, the equations [A] would be x=.x^-\-x' cos »-\-y' cos «', y=y^-\-x' sin a-\-y' sin a!. 864. Ellipse related to Conjugate Diameters. — What conjugate diameters are will be seen presently : our object now is to deduce the equation of the curve when referred to any axes of co-ordinates whatever, originating at the centre. Substituting the expressions [A] for X and y, omitting the accents over the new co-ordinates, in the primitive equation a^y- -\-¥x^-=d^h'^, it becomes a^sin^ft' I 2/H2a2sin a sin a' I a^+a2sin2« 1 a?=a?V^ [1], &2 COS^ a I 262 QQg flj (jQg jj' I 52 (.Qs2 ^ I in which equation the axes of x, y make any proposed angles a, a with the primitive rectangular axes, that is with the semi-major and semi-minor diameters of the curve, and we may now choose these angles so as that certain prescribed conditions may be satisfied. Thus, without giving them any fixed values, we may establish such a relation between the two that in virtue of that relation the term in a-y shall vanish, and thus the form of the equation [1], become analogous to the form of the primitive. For this purpose we have only to assume _52 a^sin « sin a'+S^cos a cosa'=0, .*. a^tan et tan «' 4-62=0, .*. tan «'=— ^ -..[SI a- tan « Hence xy will be absent from the equation of the curve at whatever angle a the diameter taken for the axis of x is inclined to the major diameter, provided only that the inclination a.', of the diameter taken for the axis of 2/, to the same major diameter, fulfil the condition [2]. Subject, therefore, to the condition [2J, the equation [1] is {a? sin2 a'-f 62 cos2 a,')y'^-\- {a? sin2 a-f 62 cos2 »):i?=d?l^ [3]. Now divide both sides of it by a-6^ and for brevity put in the result T^ for the coef. of ?/% and —^ for the coef. of a?^, the form of the equation is then ly2+i. a^=l ... aV+6'V=a'26'2 [4], which is exactly the form of the primitive. And we shall find that just as in the primitive, a, b, were the semidiameters taken for axes of refer- ence, so here a\ b\ are the semidiameters taken for the new axes of reference : for if in [4] we put first x=0, and then y=0 for the purpose of finding the expressions for the semidiameters which lie upon the axes of y and x respectively, we find y=-±.b\ and x = ±a\ Hence the diameters ELLIPSE RELATED TO CONJUGATE DIAMETEES. 317 coincident with the oblique axes are 2a', 26', and it is shown exactly as at (3(51), that each of these bisects all the chords drawn parallel to the other. Pairs of diameters having this property are called conjugate diameters; and [4] is thus the general equation of the ellipse referred to any pair of conjugate diameters as axes : of these, the principal diameters is the only pair of conjugates which are at right angles to one another ; for, that the two diameters may form a right angle we must have tan a tan a'= — 1 7 2 (p. 295), whereas [1], tana tan a' = ^ which is — 1 only when a=6, or when the curve is a circle. It further appears from [1], that when one of the angles a is acute the other of must be obtuse, or when one of the two tangents is positive the other must be negative : consequently, according as the axis of x is above or below the major diameter, the axis of y must be to the left or to the right of the minor diameter. The major and minor themselves are called the principal conjugates, or the principal axes, the greater is the transverse diameter, the other its conjugate. 365. Since the properties established in (362) were all derived from the equation a'y'^ •\-h'x^z=.a^b'^, where a, 6, are the principal semiconjugates, similar properties are, of course, in like manner, deducible from the equa- tion a'-y4-6'^ic-=a'^6'^ where a', 6', are any semiconjugates whatever. Hence 1. Straight lines at the extremities of any diameter, and parallel to its conjugate, are tangents to the curve. 2. The squares of chords parallel to one of two conjugates are as the rectangles of the parts into which they divide the other. 3. If from the extremities of any diameter 2a' straight lines y=m(a? + a') 6'^ and y=m'{x—a') be drawn to meet in the curve, then mmf= -r,, where ' a^ m, mf, are the coefficients of inclination of the lines, referred to the con- jugates 2a', 26' as axes. These coefficients no longer denote the tangents of the inclinations a, of, of the chords to the axis of x, that is to a', seeing that the axes are now oblique ; the property, however, at page 315, still has place, that is if linear tangents be drawn at the extremities 2a' of any diameter, and these be limited by chords, from the same extremities, pro- longed beyond their point of intersection on the curve, the rectangle of these tangents will be equal to (26')'^ the square of the diameter to which they are parallel. For putting t, t', for the tangents, it is plain that t if y^ ttf m=-—, and m'= — -— „ so that mw'= ;-,= y. .'. tt'—Ab'^. Every 2a 2a a-^ 4a^ pair of chords thus drawn from the extremities of a diameter to meet in the curve is called a pair of supplemental chords. The consideration of supplemental chords suggests the following two additional properties. Theorem I. If supplemental chords be drawn from the extremities of a principal diameter, the diameters parallel to these will be conjugate ; and conversely, chords from the extremities of a principal diameter, drawn parallel to a pair of conjugates meet in the curve. b^ For from the condition [2] of a pair of conjugates, tan a tan «'= -, and it was shown at p. 315, that for a pair of supplemental chords from 318 TRANSFORMATION OF CO-ORDINATES, ETC. 7 2 the ends of a principal diameter, mm'^ ^, where m, mf, are the tan- gents of the inclinations of those chords : hence if ?7i=tan a, then must w'=tan cc\ and conversely ; that is, the inclinations which satisfy a pair of sup. chords from the ends of a principal diameter also satisfy a pair of conjugates, and conversely. Theorem II. Of all pairs of conjugates those parallel to the equal supplemental chords from the ends of the major diameter contain the greatest angle, and the conjugates themselves are equal conjugates. They contain the greatest angle because the chords to which they are parallel do, and they are equal because they make equal angles with the major diameter, or rather with +a and —a: the tangents of the angles they make with -\-a, are -, and — by [2]. When these equal conjugates are taken for axes of reference, the equation of the ellipse is if-\-x~=^d\ which would characterize a circle, were it not that here the axes are necessarily oblique. We shall conclude this article with a theorem which is independent of the foregoing deductions, and which is sufficiently interesting to deserve notice. Theorem III. If a^, b^, be any two semidiameters of an ellipse at right angles to one another, then — <, + 7-.=— -f-r^, that is the sum of the ^1 "i '*"' ^ squares of the reciprocals of two rectangular diameters is constant. The general equation [1] is that of an ellipse related to any axes what- ever originating at its centre, the angles a, a!, denoting the inclinations of these axes to the semimajor diameter a. Leaving a, a!, arbitrary for the present, if we puty=0, and then a?=0. in [IJ, we shall have for the squares of the semidiameters coincident with the axes cfi sin'^ «-}- 6'^ cos"^ a a? sin^ a' + 6^ cos^ al' Now let «, a', be subjected to the condition that these semidiameters con- tain a right angle; then a=a'— 90°= — (90°— a'), .'. sin^ a=cos" a', and cos2a=sin2a' ; hence We shall now show how by returning from the oblique conjugates as axes, to the original rectangular conjugates, other interesting properties will present themselves. 366. Transformation of Co-ordinates from Oblique to Rectangular. — In order to pass from oblique to rectangular co-ordi- nates having the same origin, we have only to find the expressions for x' , y\ in terms of x, y, from the conditions [A], and then to substitute these values for x, y, in [4]. Now from the equations referred to we have X sin a'—y cos a'=x'(sin a cos a— cos a sin a), y cos a —X sin a =y'(sin et cos «— cos «' sin «), TANGENT TO THE ELLIPSE. Bl9 therefore representing of— a, the angle between a\ h\ by [a', 6'], we have X sin et—y cos a' y cos a— a; sin a ~" sin [a', "6^ ' ^~ sin [a', 6'] Substituting these for x, y, in [4], the transformed equation is a'^cos^a I y^—2a'^ sin a cos a I xy-\-a'^sm^a 1 a;2=a'26'2 sin^ [a', 6'] 6'2 cos^ a' I — 2 6'2 sin a cos a' | 6'^ sin^ a' 1 Now in order that this equation may be identical with the original equa- tion a-y-+b'-x''=a'P, the coefficients must satisfy the following conditions, namely, — a'^ cos2«+6'2 cosV=a2 [1], a'2 sin2«+6'2 sin2 a'=62 [2], a'2sin « cos a+5'2sin «'cos «'=0 [3], a'26'2sin2 [a\ b']=a%^ [4]. The sum of [1] and [2] gives the remarkable property a'^-f fe'^=a^+6^ that is the sum of the squares of the diameters forming any pair of con- jugates is always the same, namely, the sum of the squares of the principal diameters. The condition [4] too furnishes another property equally deserving of notice, for it shows that 4aV sin [a\ 6'J=4a6, that is, that parallelograms circumscribing an ellipse, and having their sides parallel to a pair of con- jugates are all equal in surface (308) each being equal to the rectangle of the principal diameters (see diagram, p. 321). Problem. The principal diameters of an ellipse and the vertex of any other diameter being given, to find the length of that diameter and of its conjugate. Referring the curve to the principal diameters as axes, the distance of the vertex {a/, y'), of the semidiameter a\ from the centre is a'-^-=ix'--\-y''~f and since y"^z=il^--^x% .: a'^=lf'-ir'^^^—l)^^e^x'-\ But, as shown above, a'2+6'2_a2^52^ ... h'i=:a?-^x'\ .'. a'=^/(62+e2^'2)^ 6'=V(a2_e2a;'2) [i]. Cor. Since (361) the product of the radii vectores of {x\ y% ia fP.FP=a''-e^af\ it follows that fP.FP=:V^ [2]. 367. Tangent to the Ellipse.— In order to find the equation of the tangent at a point {a/, y'), of an ellipse, let us first consider a secant through {a/, y'), cutting the curve again in {x'\ y'^ as in the circle (352). We thus have the three equations, Subtracting the third from the second, a'%-+y")(/-y)=-n-'+.-)(^-."),.-. ^=i=_^. g^; j'2 x'-\-x" hence y—y'=.—— . - „ (a:— x/), the equa. of the secant through {x', y'), and {x\ y"). 320 TANGENT TO THE ELLIPSE. If these two points unite, then x"=a/, and y"^=}f, and the line becomes a tangent at {a/, y')^ of which the equation is 2,-2/'=-^^. ^(x-xO, or a'Yy\-h'^x'x=a%'^, or ^^+^^^=1 [1], the axes coinciding with any pair of conjugate diameters. But another form of the equation is sometimes useful, in which the point of contact {x\ 2/0» is not given, but m, the coef. of inclination of the touching line, instead ; from the first of the preceding forms we see that this coef. is 6'2 x' ^^ x' a'2 w= — - . -, so that— =— — m; a'2 y' y' 6'^ ' also from the equations of the curve and tangent, we have of Substituting in these the value of — , just deduced, they become Multiplying the first of these by 6'^, and squaring the second, to eliminate y\ there results y-mx=±^{7n?a'^-\-h'^ [2]. This is the form of equation which a line whose coef. of inclination is m, in reference to any conjugate diameters for axes, must have in order that it may touch the ellipse. We see, on account of the double sign, that such lines occur in parallel pairs, m being the same for both lines of each pair. Theorem I. If pairs of tangents to an ellipse intersect at right angles the locus of the points of intersection will be a circle. In the equation [2], the axes coincide with a pair of conjugate diameters : suppose now that these are the principal conjugates, and let us regard m (which is then the tangent of inclination) as open to any arbitrary condition. Removing the radical and arranging the terms according to the powers of m, we have (x2-a2)m2-2a^«i+y2_j2_o [3], where we see that for any point (.r, y) on the tangent, m has two values, that is, [3] has two roots, m, m\ so that two tangents pass through the point. Let the roots be so related that their product mwi'= — l : then since by the theory of equations (142) this product is also ■m2_J2 —. -„, •'. y^-h^=a^-x\ or y'^-^y?=a?+y^ ', x^—a^ hence the locus of the point (a?, y) common to the two tangents (these being by the condition ww'= — 1 at right angles) is a circle of which the centre is that of the ellipse, and the radius -/(a'-^+fc^). TANGENT TO THE ELLIPSE. 321 Theorem TI. Any pair of conjugates being taken for axes, the rect- angle of the ordinate at the point of contact, and of that at the point in the tangent where it meets the axis of ordinates, is equal to the gquare of that semidiameter (i') which is taken for the axis of ordi- nates. And the rectangle of the abscissa of the point of contact and that of the point where the tangent meets the axis of abscissas is equal to the square of the semidiameter {a^) taken for axis of abscissas. Thus, in the annexed diagram, 0B\ 0C\ being any semiconjugates, OT.MP=OC'\ and OB.OM=OB'K For in the second of the equations [1], put first ^=0, and then y=0, and we shall have yy=V', and a/x-=a''^, which establishes the theorem. Theorem III. If at the extremities of any diameter lines parallel to its conjugate be drawn terminating in any tangent, their rectangle will be equal to the square of the semidiameter {V) to which they are parallel. Thus, in the preceding diagram, A'T.B't^OC^. For in the same equation of the tangent put first x=a', and then x=—a\ and we get y=B't= — ^-j- — -, andy=^'T'=- ay The product of these is A'T . B'i— ay but from the equation of the ellipse, i/^-= — — j^— ^, .-. A'T . B't—h"^=:OC'^. Theorem IV. Diameters parallel to any pair of sup. chords are conju- gate. Let the conjugates A'B\ C'D\ be ^a\ 2&', these being the axes of reference, then A'M, B'M, being two sup. chords, their coefficients of inclination w, m', are such that fc'2 mw'=--,(365). Let a tangent at B^' [of, y^ be parallel to B'M, then 7ii' will equally belong to this tangent and to B'M: but from the equation of the tangent, m'= — ~ .'. wt=^, and since "^ 18 the coef. of inclination of 0B^\ therefore OB" is parallel to A'U and since OC", parallel to the tangent at B'\ is conjugate to it (365), there- fore 0B'\ 0C'\ parallel to the sup. chords A'M, B'M, are semiconjugates 32a SUBTANGENT, NORMAL, AND SUBNORMAL. The equation of a pair of conjugates, referred to any other pair 2a', 2&', as axes, are y=mx, y=m'x, in which mm'-=. -^, that is, the equations are M=m^, and w = 77—^- -The part 368. Subtangent, Normal, and Subnormal, MB, of the axis of x intercepted between the ordinate VM of the point of contact and the point 2?, where the tangent meets the axis, is called the subtangent; the perpendicular PN to the tangent, from the point of contact to the axis of a;, is the normal; and NM, the distance between the foot of the normal and the foot of the ordi- nate, is the subnormal: thus the tangent and subtangent lie on one side of the ordinate MP, and the normal and subnormal on the other side. The axes of reference being the principal diameters of the ellipse, the equation of the normal, since it is perp. to the tangent at the point of aW contact (a;', /,) is (p. 294), ^~y=---, {x—x'),ox, clearing fractions, and, as o~x at (361), putting c^ for a'^—b'^, the equation of the normal is &V2/-aVa;+cV2/'=0, or it is h^^-a'^^+c^=0. y ^ To get the expression for the length of the subtangent, we have only to put y=0 in the equation of the tangent, which will give for x the length OR, and then to subtract x\ or OM, from it. In like manner, for the length of the subnormal, we have only to subtract ON, which is the x of the normal corresponding to y=0 in its equation, and to subtract that length from OM=x' : we thus have Length T^ of the subtangent MRz c'2 52 — . Length iV, of the subnormal MN=-^' For the length N of the normal, we have PN=s/{MN^-\-PM-^), which, since by the equa. of the curve, PM^=zi/^=~{a^—x'^), is Length N of normal PiV=^|— a/ -\ — (a^— a/2) l= At the vertex B, where x'-=^a, the normal coincides with BO in direc- tion, but its length is still definite : putting x'=^a, in the second of the POLE AN'D POLAR. S'SS above expressious, the length of the normal at the vertex i^ is — , the a semi parameter of the principal diameter. As to the length of the tangent Pi2=-y/(subtan^+2/''), it is Length T of Tangent PiJ=^/|('^!i;^'y+^(a2-a/2)|. Theorem I. From any external point (R) two tangents may be drawn to an ellipse, and the chord of contact will be parallel to the conjugate to that diameter which passes through the point (see fig. p. 321). From the way in which the expression for the subtangent has been deduced, it is plain that if oblique conjugates {a', h') had been employed for axes of reference instead of the rectangular pair, that expression would have had the same form, namely T, = — -j — , which we see is independent of the of sign of y\ showing that whether the point of contact be [x\ y') or {a/,—y'), the tangent at the point meets the axis of x in the same point, the dis- tance a/ -\-T^oi the centre from that point being the same : two tangents therefore may be drawn from the external point, and since the two points of contact are {a/, y'), {a/,—y') the straight line FF' passing through them is parallel to the axis of y. It is further obvious, since T^ is independent of 5, that though innu- merable ellipses, all having a principal diameter 2a in common, be described, all the points having the same abscissa x\ will have the same subtangent, a truth which might indeed have been inferred from the pro- perty at (362). The equation of the line joining the contacts (x\ y), {x'\ y") of two tangents, from any external point {x^, y^), is ^+^=1 [11. For the equations of the two tangents themselves are y'y x'x 2/> , «"'« and (xp 2/i) being a point on both, both equations are satisfied for x^x^, and y=yi ; therefore [1] is satisfied for both x=x\ y=-y\ and for x=x'\ y=y", so that [1] passes through both {a/, y'), and {x'\ y'% that is, it is the equation of the chord of contact. Theorem II. If from any number of points in a straight line pairs of tangents be drawn to an ellipse, the several chords of contact all pass through the same point. (See diagram to Theor. II.) Let the axis of y {b') be parallel to the straight line BR, and {x^, y^ any point on that line : the chord of contact [1] cuts the axis of x [a') in the point x=. — , which point we see remains the same whatever be y^) in other words, x^x^ is the equation of a line from any point of which, if tangents be drawn, the several chords of contact will all intersect in the / <*'^ \ point f — , j, which point (M) is called ihejpole, and the line x-=x^^ the polar of it. And we see that the polar is always parallel to the diameter conjugate to that (or that produced) which passes through the pole : the y 2 Siii POLE AND POLAR. pole can never be at the centre, because then the chords of contact would be diameters, and tangents at the extremities of a diameter are parallel. 369. The general equations, in reference to any conjugate axes, of the tangent, the chord of contact, and the polar of a point, being all alike in form, it begets confusion when the constants in these equations, which are of course all different, are represented by the same symbols, as is frequently done : it may therefore save perplexity to the student to have these three distinct equations put thus : Eq. of tangent through the point of contact (x', y'), '-j--\ — -=1 [1]. Eq. of chord of contact of two tangents from (a;,, y,), ^ +-77=1 [2]. Eq. of polar RE of any point M (a, jS), ^+^=1 [3], That this is the equation of the polar is obvious, since (a, /S) being a point If R X X on the chord of contact, we have [2], ^-j--^ = 1, therefore [3] is tlie " a~ locus of (^p 2/i)- [Observe : («, /5) may be without the circle as well as within if]. Theorem III. If any point {x^, y^ be taken on the polar of (a, ^), then the polar of the point {x^, y^ passes through the point (a, j9). For the polar of the point (»'»„ 2/,) by [3] is -^■\ — 72 = 1, which is a chord of contact passing through (a, /5). Theorem IV. If the focus be the pole, the polar will be a perpendicular to the principal diameter at the distance - from the centre. e For taking the principal diameters for axes, the distance of the pole from 2 2 the centre is c=— , .-. x.=—, and c=ae (p. 312), .-. a;,=-: hence x=- ajj ^ c ^^ ^ ^ e e is the equation of the polar of the focus, 2a, 2&, being the axes of co- ordinates. The polar of the focus is called the directrix of the ellipse : as there are two foci, there are two directrices. Theorem V. The normal at any point bisects the angle between the radii vectores of that point. Putting y=0 in the equation of the normal PA", we have OJV=e'x', .'. fJV=(a-\-ex% .'. FN=2ae-fN={a-ex')e, therefore (361), fN\FN::Pf: PF, .'. (Euc. 3. VI.) PN bisects the angle P, and consequently if fF be pro- longed, the tangent FF^ must bisect the angle FFG. [Because FF,fF, make equal angles with the perpendicular FN to the curve at P, rays of light or heat issuing from Fj and striking the curve, will all, after reflexion, unite again at /. It is on this account that the points F,f, are called foci, or burning points,] PEEPENDICULARS FROM FOCI ON TANGENT. 325 Theorem VI. The perpendicular from either focus on any tangent to the curve will meet the tangent in a point P', the locus of which is a circle described on the diameter AB. For if PG be made —JPF, the line PF' bisecting the angle P must also bisect FG. As, therefore, P' is the middle of FG, and ihe middle oifF, the line OP' will be the half of /(?, or oifP + PF=JB : hence OP' is constant, and =0B: the locus of P' is therefore a circle on the diameter AB. Otherwise. The general equation of the tangent [2], page 320, is y=mx-{-y/{m\'^-{-b^, or y—mx=i^{m^a'^-\'lP) [1] ; and the equa. of a line through (±c, 0), that is, through (y^a^— 6'-, 0), perp. to this, is y= (x-Va2-62), or my+x=^(a^-l^) [2]. Squaring [1], [2] to eliminate m, we have, by adding the results, {!/^+x^){m^+l)=aP{7nP-\-l), .: y^+x^=a^ : hence, the locus of the intersections of [1], [2], is a circle of radius a. Theorem VII. The rectangle of perpendiculars from the foci upon any tangent is equal to the square of the semiminor diameter. The length of the perp. from either focus (^a^—b\ 0), upon the line [1] above, is (p. 295) /, •> , -. T 1 a^d *lie product of these two expressions is m--{-l w'-^-f-l Note. — It will be observed here that the double sign is not given to the radical in [1], because we are dealing with a single tangent; but the point {^a'—b\ 0) being either of the two points (±c, 0), the radical takes the double sign. Cor. The two perpendiculars FP\ fp\ from the foci on the tangent, evidently form, with the two radii vectores, two similar right-angled triangles, so that FP'_/p' ^ / FP' \^_ FP' . fp' _ / 5 \ 2 ^ FP'_b FP ~fP' '' \FP ) ''FP . fP\ b'J ' ''' FP~b" /b \2 J2 a property that will be found useful hereafter. Also FP'^=( -jjFP ) =z—{a—exy : but (p. 319),6'2=a2-eV^.•./''P'2=&2^^^=^', .-.from the theorem, //2=:62^±^'. ' a-^ex ' ^ a— ex The perp. Oja from the centre being - {FP'-\-fp'), we have u ah "Examples for Exercise. (1) Of all pairs of conjugates the sum of those which are equal is the greatest, and of those which are rectangular the least. (2) The rectangle of the subtangent and abscissa of the point of contact is equal to the rectangle of the parts into which the diameter taken for axis of abscissas is divided by the ordinate. 3 "2 6 POLAR EQUATION OF THE ELLIPSE. (3) The rectangle of the radii vectores of any point is to the square of the normal, as the square of the major diameter is to the square of the minor. (4) If a tangent be drawn at any point of an ellipse, the square of the semidiameter conjugate to that from the point of contact, will be equal to the rectangle of the two parts of the tangent intercepted between the point of contact, and any pair of conjugates whatever. (5) The tangent at the extremity of the latus rectum cuts from the tangent at the vertex of the major diameter a part equal to the distance of the focus from that vertex. (6) If from the foot of the normal at any point P a perp. be drawn to either radius vector of the point, the part of that radius intercepted between P and the perp. will always be equal to the principal semiparameter — . (7) A circle is described on the major diameter of an ellipse : normals are drawn from those points P, P\ in the two curves which have the same abscissa : prove that the locus of the intersections of these normals is a circle. (8) Every chord drawn through the focus is perpendicular to the line joining the pole of that chord with the focus. (9) If any rectangular semidiameters of an ellipse be drawn, and a perpendicular from the centre to the chord joining their extremities cut that chord in P, the locus of P wiU be a circle. 370. Polar Equation of the Ellipse.— First let the centre O be the pole, OB being the fixed axis; then for any point P, OP=r, and POB=^; and p substituting t/=r sin 0, and x=r cos 6 in ay + ^ — ^A n. h'x'^=d^b^, we have /^ y^ l'^' ^\ r\a^ sin* 6\W cos2 ^=a%\ X—/^ — 11 ^srA^ Now 62=a2(l_g2)^ ,., a2j2=„4^1_g2). ^^ <> ^J hence »'2a2{(l-cos2 ^)+(l_g2) cos2 ^}=a4(l-e2), \„^^^^^^^ .'. j-2(l_e2cos^)=a2(l-e2) ... ^2=-^^^^- 1— e^cos^^ which is the polar equation when the centre is the pole. Next, let the focus/ be the pole: then fP=r=a-^ ex, anda?=r cos 9—ae, .'. r=a-\-re cos fi—ae, .'. *'=r-^ is the polar equation for pole/. X € COS V If F had been the pole, x would have been =r cos &+ae, so that .. a(l-e2) l-f-e cos^ is the polar equation for pole F. If we put for the constant a{l—e^), the semiparameter -p, to which it is equal (p. 313), we may write the polar equation thus, -=2jo(lqi cos 6), the upper sign applying when / is the pole, and the lower when F is the pole. Also, whichever sign applies, the opposite to it must be taken for the opposite radius vector fP\ If r be prolonged, completing the focal 1 1 r+r' chord PP', we shall have -+— =4», or — t- = 4:» : hence focal chords r r rr are to each other as the rectangles of the parts into which they are divided by the focus. CENTRE AND BADIUS OF CURVATURE. 327 371. Curvature of the Ellipse. — The circle is the only plane curve of which the curvature is uniform throughout ; and since a circle may be described of any length of radius, we may get a circle of any degree of curvature we please. It is thus well adapted to exhibit throughout its entire circumference the curvature that any other plane curve may have only at a particular point. The circle whose curvature is thus the same as the curvature of another curve at a particular point, is called the circle of curvature at that point. If P mark the point at which the circle of curvature is required, we may conceive that circle arrived at thus : let two other points on the curve, in the vicinity of P, be taken, and let a circle be described through all three : this circle will the more nearly approach in curvature to the curvature at any one of the three points, the closer those points are together ; and if by causing the three points to approach nearer and nearer till they at length all coalesce in P, and by watching the continuously- varying radius of the circle pass- ing through them, we could detect its exact length and position when the coalescence occurred, we should know the magnitude of the circle of curvature at the point P. Or the centre of this circle would be dis- covered thus : let the equation of the normal at P be found, and combine this with that of another normal in its vicinity ; we shall thus get the point where the two intersect, and we have then only to find whereabouts on the fixed normal at P this intersection settles, when the second normal, by continuously approaching nearer and nearer to the first, coalesces with it. It was somewhat in this manner that the equation of a tangent was deduced from that of a secant. 372. Centre and Radius of Curvature. — Let (ic', y') be the proposed point on the ellipse, and {a/\ if) any other point in the curve, then if the two normals at these points intersect in the point (a, /3), we shall have for their equations (p. 323), h"x' ^—ary' a-\-(?x'y' =.0, b^x"li—a-y"a-\-c^x"y"=0, which solved as a pair of simultaneous equations, we get for a, and ^, ^_ c^x'x"{j/-y") chjy"{x!-x") d\xy—xy")'' ¥{y"x—y'x") Now af'y' —cc'y" =a/{y' —y'')-^y\x' ^a/'), .'. dividing num. and den. of a by y^—y'\ •*• ** sin [«', b ]=-7= the semiparamet^r of 2a', of a sin [a', 6j a ^ ' the diameter of the ellipse at the point (x', y"). 373. Chord of Curvature. — The chord of the circle of curva- ture through the point of contact [x\ y% and which passes through the AREA OF THE ELLIPSE. 329 centre 0, of the ellipse, is called the central chord of curvature, and that from the same point through the focus is called the focal chord of cur- vature. The angle [a\ b'], ahove is that between the semidiameter a' at the point of contact, and its semiconJQgate 6' ; and since the tangent at the vertex of «' is parallel to b\ sin \_a', b'], is the same as the sine of the angle between «' and the tangent, or as the cosine of the angle between a' and the normal, and this cosine multiplied by r is half the chord (of the circle) upon which a' is situated ; hence, -'"'"^ ...[1], (page 319). 6'2 2(a2 The central Chord of Curvatiire=2— =-^^ r a a In like manner any other chord of contact is found by multiplying 2r by the cosine of the angle that chord makes with r. If the chord pass through the focus F, its length is 2r cos NPF (see fig. at p. 324), but NPF=PFP', FP' h h'^ and cos PFP'=-=-;:=^-r, (p. 325), and as shown above, r=— . FP b' 0,0 Consequently the focaJ chord of curvature=2— = . 374. Area of the Ellipse. — Let AB be the major diameter of a semiellipse ACBy upon which diameter let a semicircle be de- scribed. Divide the semidiameter ^ into any number of equal parts, at the extremities of which let ordinates be drawn both to the ellipse and to the circle, and complete the rectangles as in the figure. Then, since rect- , p angles of the same base are to each other as their altitudes, we have m rect. Qp : rect. Q'p : : Qq : Q'q : :h : a (p. 313) ; ^ hence the rectangles PA, P'A, the rectangles Qp, Q'p, &c., are always to I each other in the constant ratio b\a. ^ And since one antecedent is to its consequent, as the sum of all the antecedents to the sum of all the con- sequents, it follows that the irregular polygon Amno...CO : A'mn'o...DO : b : a. And this is true, however numerous the subdivisions Ap, pq, &c., may be, that is, however small be the bases of the rectangles ; but the smaller these bases are, the smaller is the difiference between the surface of each polygon and the curve connected with it ; and since there is no limit to the smallness of Ap, pq, &c., so is there no limit to the smallness of this difference ; hence the curvilinear spaces must also be to each other as the polygons, that is, as b to a. Consequently the semiellipse ACB is to the semicircle ADB as b to a, 330 THE HYPERBOLA. .-. Semiellipse=- times the semicircle =- X- aV=- ahst. ^ a a 2 2 .'. Area of ellipse =a6*=2ax2&X •7854. Hence, to find the area of an ellipse we have only to multiply the pro- duct of its principal diameters by '7854. 375. The Hyperbola.— This curve is such that if from any point P in it two straight lines be drawn to two fixed points F, /, called the foci, their difference will always be the same. As in the ellipse, the lines PF, Ff, from any point P to the foci, are called the radii vectores, or focal dis- tances of that point. 376. The definition of the curve suggests the mode of determining any number of points in it, and of thus discovering its form by observing the track of the series of points. The foci F, f, being given in position, and the constant length of the difference, between the focal distances of any point, being known, let a circle of any radius be described from F as centre : then let the radius be lengthened by the given difference, and with this increased radius, let another circle be described, from centre/. The two circles will intersect in two points : one as at P, above /F, and the other at an equal distance below fF, provided the first radius be not assumed too small to render intersection possible. By repeating this operation, with two new radii, the second still ex- ceeding the first by the constant difference, another pair of points on the curve will be determined : and this determination of pairs of points may be continued till the track of the curve becomes sufficiently apparent. In the above, the smaller circles are considered as described from F^ and the larger from/; but if on the con- trary the smaller be described from /, and the larger from F, it is plain that a distinct series of points will be marked out by the intersections, so that the curves passing through each series will be entirely detached ; but as both satisfy the definition, the two are regarded as merely branches of the same curve. The annexed diagram will sufficiently illus- trate this method, by points, of describ- ing either branch of the hyperbola. The line fF is drawn, joining the two given foci : if the branch is to pass through an assigned point B on this line, then from Bf cut off a part BA, equal to the proposed constant difference, and in BF prolonged, take any number of points 1, 2, 3, &c., the first being either at or beyond F. Then with F and /as centres, and with the respective radii Bl, A\ ; B'2, A2, &c., let the intersecting arcs be described as in the diagram, and the branch of the hyperbola may be traced through them. EQUATION OF THE HYPERBOLA. 331 Or the branch may be described by continuous motion thus: Let a ruler /E, be fixed to one focus /, and be at liberty to turn round that poinc, in the plane whereon the curve is to be described : then having assumed, or having given, the other focus F, connect it by means of a cord, shorter than the ruler by the given difference, to the extremity R. A pencil P, keep- ing this cord always stretched and pressed closely to the edge of the ruler, will, as the ruler revolves round /, describe an arc of the hyperbola of which /, F, are the foci; for Rf^ RPF=Pf—FF, the constant difference. 377. Equation of the Hyperbola.— Let 0, the middle of /F (first fig. above) be taken for the origin of the rectangular axes OB, 00, and for the constant difference Pf—PF, of any point {x, y), put 2a, as also c for OF, or Of: then we shall have the three following conditions, namely, P/-PJP=2a...[l], y2+(^+c)2=P/2...[2], f+{x-cf=^PFK..[Zl and from these equations we have to eliminate Pf, PF. In order to this, take the sum and difference of [2], [3], and we have 2{y^-^3(?^a, .*. ic4-a>2a, .'. ~p'^2{x—a). In the equilateral hyperbola, where h=a, /)=2a=the transverse diameter. By removing the origin of the rectangular axes from O to B, that is, substituting a+^ for x, [7] becomes ay-ly^aP-2ab^x=0, or f=-{aP-^2ax)...[ll]. which [C], at p. 312, also becomes when b- is changed into —P, and a put for —a. In terms of the parameter, the equation under this change of dfcgin, is I 2^-^^'+^^ tl2]. 380. The student will observe that there is no departure from analogy in placing the origin at B in the hyperbola, and at A in the ellipse : in both cases the curve, for positive abscissas, proceeds from the origin to- wards the right. He will also do well further to notice, as instanced above, that in passing from one curve to the other, the changing of h'^ into — 6S is equivalent to changing 1— c^ and d-—x'\ into e- — 1, and x^—a^. And wherever b"^ is left unchanged in sign, x—a must be written for a—x (as in next article), and ex— a for a—ex, as at p. 336. It is necessary to observe these particulars, in passing from the ellipse to the hyperbola, because in many of the expressions deduced from the former curve, 6^ does not explicitly enter. 381. Properties of the Hyperbola related to its Prin- cipal Diameters.— 1 . Changing Ir into — i- in (362), or leaving 6^ unchanged, and putting ^— a for a—x^ we have (^»fcj=«^' ••• ^^ ^ (-+»><-»)- *^ -'. that is, the squares of the ordinates are as the products of the parts of the prolonged principal diameter, included between those ordinates and the two vertices of the curve. If fc=a, then y^—^x-^-a'j^x—a)', so that in the equilateral hyperbola. 334 THE HYPERBOLA RELATED TO CONJUGATE DIAMETERS AS AXES. the square of any ordinate is equal to the product of the parts of the principal diameter (prolonged) between the ordinate and the vertices of the curve ; a property analogous to that of the circle. 2. It was shown at p. 314, that an angle P at any point of an ellipse, and subtended by the diameter 2a, had for its tangent the value — In the Ellipse, tan Pz=—t — — -, .'. in the hyperbola, tan P-=-- which expression, being independent of the sign of X, must be the same for the point {—X, y) as for {x, y). And since it is always positive for every point P above the axis of x, it follows that all the angles at the curve, subtended by the transverse axis, are acute : they diminish as y increases, becoming U when y is infinite : they thus range between 0° and 90°, or rather between 90° and 0°. The lines from A, B, meeting in P, as in the ellipse, are called supplemental chords : their tangents of inclination to the axis of x being w, m', we have, (p. 315), mrnf^z—^, which being always positive, it ct follows that the angles, which two supplemental chords make with the axis of X, are either both acute, or both obtuse : it is plain that they are acute when the point is on the right-hand branch of the curve, and obtuse when it is on the left-hand branch. If 6=a, then mm'=l. .*. m'=— : m hence, if the hyperbola is equilateral, the angles which a pair of sup. chords make with the principal axis, are together equal to a right angle. 382. The Hyperbola related to Conjugate Diameters as Axes. — We have seen, at p. 315, that by a transformation of the axes of reference from rectangular to oblique, the equation of the ellipse preserves its original form, provided that, in the transformed equation, we put « for . „ — -— — » and 6 2 for — 5—7 . a-* sm^ a 4-0 CDS'* a a^ sm^ u'-\-¥ cos'' a Hence, changing 6^ into —6^, and putting »'. to „ . -"'f! . , and -V^ for -"* -62 cos^ a a? sin^ a —V^ cos- a the equation of the hyperbola becomes the same as that at p. 331, when h' is changed into h"^\ that is, the equation, in reference to the new axes, is whenever a, «', at which the new axes of x and y are inclined to the primitive axis of x^ namely to the diameter 2a, are such that tan a'=-n • «■* tan a In [1], for a:=:0, we have 3/=/^/— 6'^=±6'/>/— 1 ; and for ?/=0, we have .r=:±a'. THE HYPERBOLA RELATED TO CONJUGATE DIAMETERS AS AXES. 335 It is impossible, therefore, for the curve to meet the new axis of y ; but it crosses the new axis of x, at the extremities of the diameter ^a\ Although the new axis of y never meets the curve, any more than the primitive axis of y, yet, as in the former case, a length of this axis, equal to 26', the centre being at the middle of this length, is marked off for a second diameter to the corresponding transverse 2a'; so that the semidiameters a\ h', enter the new equation exactly as the semidiameters a, 6, enter the primitive : and therefore, as in the ellipse, analogous properties are de- ducible from each. Thus from the equa- tion [1], as at p. 317, we infer the following particulars : — 1. Each diameter 2a', 26', bisects the chords parallel to the other : such diameters, as in the ellipse, are called conjugate diameters,* so that a'hf-lf^x^=-a'"h'\ oriy2_i_^__l [2] is the equation of the hyperbola related to a pair of conjugate diameters. And, of every pair, one only is a transverse diameter, that is, one only has its extremities in the curve : the other is a second diameter, that is, it never meets the curve however far it be prolonged. 2. Straight lines, at the extremities of any transverse diameter, and parallel to its conjugate, are tangents to the curve (p. 332). 3. The parts of a prolonged transverse diameter, between the ordinate at any point, and the vertices of that diameter, are such that their pro- duct is to the square of that ordinate as the square of the transverse to the square of the conjugate (p. 333). 4. Since tan a tan a'= — , and that, as already proved (p. 334), mm'=-—, Cb a" it follows, as at (365), that diameters parallel to a pair of supplemental chords, from the ends of the principal transverse, are conjugate; and conversely. Therefore (381), the angle included by a pair of conjugates may be of any magnitude from 90° to 0°. 5. Again : from the above relation we also see that if y=mx represent any diameter of an hyperbola referred to its principal axes, then will 2/=-^-^ represent the conjugate to that diameter. And moreover, as at (365), when any pair of conjugate diameters 2a', 26', are taken for axes, still m7w'=-7„; where m, m', are the coefficients of inclination, to the dia- a^ meter 2a', of the pair of conjugates referred to (2a', 26'), as axes. * In the figure, OB', 00\ are a pair of conjugate semidiameters. 336 THE ASYMPTOTES OF THE HYPERBOLA. 6. Still referring to the former investigations, the equations at p. 319 upon changing b~, h'\ for — fe^,— 6'^ become a'26'2 sin2 [a', h']=a%'^, and a^ojh'-^=a^(Kjl\ SO that the parallelogram constructed on any pair of conjugates is equivalent to the rect- angle on the principal axes of the hyper- bola ; and the difference of the squares of any pair of conjugates is equal to the differ- ence of the squares of the principal dia- meters. 7. In a similar manner, changing 5^, 6'^, into — &^ — ft'^, in the problem at p. 319, we have a'=V(eV2-J2), and h'=^{^x'^-a^, and .-. h'^={ex'-\-a){ex'-a)=Pf . PF, where V is the semiconjugate to the diameter at P. This property we might at once have inferred from the corresponding one of the ellipse, by merely writing eV^— a- for ar—ex'-^ (p. 319). 383.— Tangent to the Hyperbola. — The tangent to tho ellipse at any point {x\ y'), being S'2 x' For the ellipse, y—y'= — - . —(x—x), or a'Yv+^'^x'x, or y—mx=s/{a'^mP+h'^X a y .'. For the hyperbola, y—y'=:— . — (x— a;'), ox ahj'y—h'^xx, or y—mx=:.s/{a'Hi^—h''^. And the equa, of the normal, therefore, isy— 2/= ^{x—x'). We might now proceed to give the expressions for the lengths of the tangent, subtangent, &c., as at p. 322, and thence to deduce properties analogous to those already established for the ellipse. But enough has been done to convince the student how easily the conclusions arrived at, in reference to the ellipse, may be adapted to the hyperbola ; we shall, therefore, pass on to the consideration of certain properties peculiar to the curve now under consideration. 384. The Asymptotes of the Hyperbola.— It has already been seen that certain diameters of the hyperbola never meet the curve at any finite distance from the centre, or at any finite values for x, y. Of every pair of conjugate diameters, one of that pair — the second diameter, as it has been called— is always in this predicament. It has been seen, too, that the conjugate to the principal transverse makes with that transverse an angle of 90"* ; and that as the angle which an oblique transverse makes with the principal increases, the angle it makes with its conjugate di- minishes, till at length, conceiving the transverse to have arrived at its ultimate position, meeting the two branches of the curve at an infinite distance, the conjugate, which all along has been approaching, actually THE ASYMPTOTES OF THE HYPERBOLA. 337 unites with it, and the two coalesce, the angle between them being then (p. 335). These peculiarities constitute a marked difference between the hyperbola and the ellipse, and imply that, in reference to these peculiarities, the former curve must have properties entirely distinct from those which are common to both : we have now to investigate these. From the general equation of the hyperbola, in reference to its princi- pal diameters, 2/2=_ (a;2— a2), we have y=±-'s/{x^—a^), or y=±-xj(\ — 5); a^ a a \ x / and since as x increases, the fraction -- diminishes, it follows that for these increasing values of or, the corresponding values of y go on continuously approaching towards ±:-a, which extreme value, however, is never actually reached till x becomes infinite, ren- dering the fraction zero. If then through the origin two straight lines KM, LN, be drawn, making angles KOX, LOX, with OX, whose tangents are re- spectively -I--, and — , these two a a lines will continually approach nearer M" and nearer to the curve the further they are prolonged, and yet can never meet the curve within any finite distance from the origin, or for any finite value of a. Tor the equations of these two lines, or the values of y for any value of x being y=- x, a and y= — x; and a for the same x the y of the curve being ?/=- a;^( 1— — Y yz= — ^\/( 1 — § )> it is plain that the difference (PP') diminishes continually as « increases, and vanishes altogether when x = cc, but not till then. These two lines, KM, LN, are called the asymptotes to the hyperbola ; and it is plain that they separate all the transverse diameters from their conjugates ; the former being all situated within the angles KON, MOL, and the latter within the angles KOL, NOM: and moreover, that the asymptote is the limiting or ultimate position of a tangent to the curve ; that is, it is coincident with the tangent when the point of contact is infinitely remote. In fact, if we substitute for y' in the equation of the tangent, its value y'=-x'y/{l—^\ that equation, namely, ayy-b^x'z=.—a^i^f becomes a^y-x'^ i^~~^) -^'^^'^= -a^^'^ 338 THE ASYMPTOTES OF THE HYPERBOLA. or, ay.^(\ — -\—hxz=——, which, when a:' = go, {%y^lz=-x, .'. y=+-x, the equation of the asymptotes. a Two tangents to a branch PBP of an hyperbola, always meet within the angle KON of the asymptotes which embrace that branch : for if they could meet without that angle, as within the angle LOK, then might a tangent be drawn at less inclination to OB than the limiting or ultimate tangent, though the inclination of this is the least possible ; for in the equation of the tangent just written, the general expression for the tan- gent of inclination of the line is — '-./( 1 ), which is necessarily greater than - for all finite values of x. From any point, therefore, within the upper angle LOK, only one tangent can be drawn to each branch; and, in like manner, can only one to each branch be drawn from a point within the lower angle MON. The following two properties of the hyperbola are also readily deduced from the equations of the asymptotes : — 1. If from the focus F, a perp. Fp be drawn to the asymptote, then will Op be equal to the semitransverse OB=za, and Fp equal to the semi- conjugate h. For the angle pFO, being the complement of pOF^ its tangent is -7, and its secant y/ ( I+T2 /' Now, OF divided by this secant is Ff, and since OF'^=:.Jimxz=+2^mx. The first expression shows that the curve lies entirely to the right of the axis of y, since a; is always positive : this axis is also a tangent to the curve, since for x=^0, we have y=0, and no other value ; showing that (0, 0) is the only point common to the axis and curve. The second equation shows that for each abscissa there are twc equal ordinates, on opposite sides of JX; so that yiX bisects all the chords parallel to AY: it moreover shows that y is always possible, so long as a; is positive, however long x may be ; so that the curve extends without limit to the right of AY, and both above and below AX, which line is called the principal axis of the parabola; and ^Fis the principal second axis. These axes are both unlimited in length : but the former, since it bisects all chords parallel to the latter, is called a diameter — the principal diameter. As in the other curves, the double ordinate through the focus is called the latus rectum, or principal parameter of the curve : the equation of the parabola, in terms of this parameter 2?, is easily obtained from [IJ ; for we have only to put m for x in that equation, and we have y'=4m2, or yz=.2m=.-Pi •*• y^=-px [2] is the equa. in terms of the parameter. And since 2w=-2?, we see that the semiparameter Fp is equal to the dis- tance FK of the focus from the directrix, as is indeed also evident from the construction of the curve. The equation [2] shows that the abscissas of different points of the curve are to one another as the squares of the ordinates. 389. The Parabola referred to Conjugate Axes.— As in the other curves, we shall now inquire what must be the peculiarities of those oblique axes, in reference to which the equation of the parabola pre- serves the same form. In order to this, we must substitute, in the equation [2], the expressions at p. 316, and thus transform that equation from rect- angular to oblique co-ordinates; and we shall, at the same time, remove the origin from (0, 0), to a point {a, b), on the curve ; that is, we shall put a-\-x cos a-\-y cos a for x, and b-\-x sin a+y sin a' for y, in [2] above, and we shall thus get, after transposing, the equation y2 sin'' a'-\-2ocy sin a sin a'+x^ sin^ et-^l^—ap-\- \ (2h sin a— p cos et)y-\- {2b sin a— p COS a)x ) ^ ■'* And that this may have the desired form y^-=p^x, we must have sin a sin a'=0, sin^ a=0, 26 sin u—p cos a'=0, 62_^^-_o j-2j^ the last implying merely that the new origin (a, 6) must be on the curve ; but this point, not being restricted to any particular position, may be any point [A!) on the curve whatever. THE PARABOLA EEFEREED TO CONJUGATE AXES. 343 The second condition necessitates the first, and shows that since a=0 the new axis of x is parallel to the primitive AX\ and from the third, we learn that the in- clination a', of the new axes to each other, is in tan a'=.-- : moreover, since sin a=0, .'. cos a=l. 26 Under these restrictions, the equation [1] is p y^=-rr, — x, or y^:=p'x [31; sm^ «' and since, as already noticed, the origin may be anywhere on the curve, and therefore h of any value, positive or negative, and conse- quently, since tan a'=^-T-26, the axes may have any inclination whatever, it follows that there are innumerable systems of co-ordinates, in reference to which the equation of the parabola is of the form [3J. And we further see that, since tan of diminishes as h increases, the more remote the origin is placed from A, the less becomes the a,ngle of at which the axis of y is incliued to that of x, or to the principal diameter AX. Equation [3] being of the same form as [2], at p. 342, we draw from it similar inferences, thus : — 1. The axis of x bisects all chords parallel to the axis of y, which latter axis is always a tangent to the curve at the origin. Lines thus bisecting parallel chords are always called diameters: hence all parallels to the principal diameter are themselves diameters. The coefficient p' in equa. [3], which is of course fixed for any particular diameter taken for axis of X, is called the parameter of that diameter. And any diameter, with the tangent at its vertex, form a pair of conjugate axes. 2, The abscissas, measured on any diameter, are as the squares of the corresponding ordinates, measured parallel to the tangent at the vertex of that diameter. p / In order to find the value of the paiameter . ^ , =^p , independent of trig, quantities, we have only to observe that, since tan' «'=—-, .•. sec^ a=l-\-~. 1 sec2*' 462_|-_232 Aa-\-p p a/ ,P\ a/ , \ .-. -—-,=—-;—= -— = by [2], .-. -H|-,=4a+jp=4( a+7 )=4(a+'/»). sin^a tan^ a' p^ p sm^a' \ 4/ Now by the definition, a + m is the distance of the point A' in the curve, of which the abscissa is a, from the focus : hence, putting for this dis- tance, r, the radius vector of the origin of co-ordinates, the general equa- tion of the parabola, in reference to any conjugate axes whatever, is y^=4rx [4]. The parameter />' of any diameter is thus always equal to the double ordinate corresponding to the abscissa x=r. We shall now show that this double ordinate necessarily passes through the focus. Let A'M^=r, and let pp' be a chord through M\ parallel to the tangent A'Y\ Refer the curve to the principal axes originating at A : then the abscissa of A^ being a, and its ordinate b, we have for the abscissa of M% ii?=a-f r=2a-hm, and for its ordinate b. Also for the tangent of the inclination u, of pp\ we have tan ei=-— : hence the equa. of pp is y—b=— {x—2a—m). 26 2o 344 TANGENT, NOEMAL, ETC. At the point wliere pp' crosses AXy y=0, for whicli value the equa. gives x=2a-\-m =i2a-\-m—2a=7n:=AF. P Thus the chord through the focus is always equal to the parameter of the diameter bisecting that chord. 390. Tangent, Normal, &C. — To find the equations of these lines, we proceed exactly as we did in reference to the former curves : the equa. of a straight line through two points {a/, y'), {x'\ y"), on the curve, is y—y'z=— -{x—x), where i/^^irx', and y"^:=irx", X — X v'—v" 4r ••• W-^y"){y' —y")=^r(x! —x"), .-. , „ ■=- r. ; hence the equa. is 4r y—y'-=. j '{x—x)j which, when the points coincide, becomes 2r y— 2/'=— («—«)> or yy=2r{x-\-xf) [5], y the equation of the tangent at the point [a/, t/). The second form is got from the first by multiplying by y', and then putting for y^ its equal 4:raf. As the normal is a straight line through the same point perpen- dicular to this, we have for its equation, in reference to the rectangular conjugate axes, y—y'=z —^(x —x) . . . [6], Equa. of the Normal. 2m Putting 2/=0 in [5], the resulting value of X, namely AB, is x=AR=—x'=—JM, .'. the subtangent MR=^AM, whatever di- ameter AX may be. The subtangent MR at any point F is therefore twice the ab- scissa of that point. In like manner, putting y=0 in [6], we have x—x'=:2m=MN, the subnormal. For the length Pi2 of the tangent, we have PI!^=MP^+MR^=y'^+ix"^=^nix'-\-x% and for the length of the normal, PN^=MP^+MN-2=y'^+Am^=4:{mx-^m^). At the vertex A, where a;'=0, the length of the normal is '2m ; so that, as in the former two curves, the length of the normal at the vertex of the principal diameter, is equal to that of the principal semiparameter. The equation of the tangent [5] may be put in another form, from which the co-ordinates x\ y\ of the point of contact, shall be eliminated, and in which the coefficient (n), of the inclination of it to the diameter, shall appear instead. For, writing that equation thus, «= — x-\ , where — is now to be replaced by w, we shall have, 1/ y y' 2r y' from the equation of the curve, — =—-;=», y 2x ^TX T T ,\ !!—-=-, /. «=:%fl5-| — is the equa. of the tangent [7]. y n n PBOPERTIES OF THE PARABOLA CONNECTED WITH TANGENTS. 345 From the foregoing expressions the following properties are at once deducible : — 1 . The expression for the subtangent being independent of the constant r, it follows that for all parabolas, having a pair of conjugate axes in common, the subtangent, measured on the common diameter, is always the same for the same abscissa. 2. And, in like manner, from the expression for the subnormal, it follows that, for every point in the curve, the subnormal is constant, and equal to the distance of the focus from the directrix. 3. As in the ellipse and hyperbola, two tangents from P, P', the ex- tremities of any chord, meet in the same point R of the diameter bisect- ing that chord ; for the expression for MB remains the same, whether «/' he plus or minus. [See fig. p. 343.] 4. The tangent at A\ the vertex of the diameter, being parallel to the chord PP', the part of it, intercepted between the two tangents at P, P', is bisected at the point of contact A', the same as in the other curves. We shall now proceed to develope some properties requiring a little additional investigation. 391. Properties of the Parabola connected with Tan- gents. — The last form [7], given to the equation of the tangent, enables us very readily to solve problems concerning the parabola analagous to those already solved for the ellipse, when a similar modification of the common form for the tangent is employed ; thus, Prob. I. To determine the equation of the tangent to a parabola drawn from an external point {x^, y^. The point [x^, y^ being on the tangent, we have from [7] above, y,=:na;^^ — , .*. n^——n-\ — =0, giving two values for n. n Xy X\ Hence, through the same point two tangents may be drawn to the curve, T T y their equations being 2/=wa; + -, and2/=w'a?H — ,, where (w-{-/i')=— , and T nn'=—. If the two tangents intersect at a right angle, then, regarding the axes of reference as rectangular, and therefore putting m for r, we must have the condition w»'=—=-—l, .*. a?, = —m; hence the locus of ^1 the point, from which a pair of tangents are always at right angles, is the directrix of the parabola. Prob. II. To find the equation of the chord joining the two points of contact of a pair of tangents from [x^, y^). Taking the other equation [5] of the tangent, the points of contact being (a/, /), and {x'\ y"), we have, since (x^, ?/J, is a point on each tan- gent, y,'!/=2r{x,-\-x'), and yy=2r{x^+x"). Hence the two points (a;', t/'), {a/', y'% must lie upon the straight line » y^='2r{x^-{-x)t the equation of the chord of contact. 346 PROPEETIES OF THE PARABOLA CONNECTED WITH TANGENTS. This line (as in the ellipse) is the polar of the point {x^, y^}, and it may- be shown, in precisely the same manner as at p. 323, that if from any point (ajp 2/1), in a straight line, tangents be drawn to the parabola, the several chords of contact intersect in a point, which is the pole of that straight line, the line itself being the polar of the point. The polar of the focus, for which ^=m, and y—0, has for its equation a;=—m, the directrix, and we have just seen that the pairs of tangents from this always intersect at right angles. The tangent, through an extremity of the latus rectum, is called the focal tangent. Theo. I. Any rectangular ordinate to the focal tangent is equal to the radius vector of the point, where that ordinate cuts the curve. Referring the curve to its principal axes, when the abscissa of the point of contact is x'=m, we see, from the expressions at (390), that the length of the tangent is equal to that of the normal, and that the subtangent is always equal to the subnormal. The equa. of the tangent is y—2m=—(x—m), or y=x-\-m, that is, Mp=:FP. Let a;=0, then y^^m, that is, AT=:AF. Theo. II. The radius vector, and the diameter at any point of the parabola, are equally inclined to the tangent at that point. To MN=2m, add FM=af—m, .'. FN=x'-\-m=FP, .-. FPN=FNP=NPXt. [Rays of light or heat, issuing from the point F^ will all be reflected in a direction parallel to the principal axis AX, since incident and reflected rays always make equal angles with the normal at the point on which they impinge. Also rays, striking the curve in directions parallel to XA, will all be reflected to and unite in the point F: it is on this account that F is called the focus, or burning point.] CoR. Since (p. 344), FB=x'-\-m, .-. FE, FN, FF, are all equal. Theo. III. A perpendicular from the focus on any tangent always meets that tangent on the principal second axis, or tangent at the princi- pal vertex. Let FQD be the perp. to the tangent PR ; then since FF=FD, Q is at the middle of FD, and A is at the middle of FD', .-. Q is on the parallel AY, to B'D, CoR. Hence FQ^=FR . FA=FF . FA, and FA is constant, therefore the perp. from the focus on any tangent varies as the square root of the radius vector of the point of contact. Theo. IV. The part of the tangent, between the point of contact and the directrix, always subtends a right angle at the focus : in other words, the perp. to a focal chord from the focus always meets the tangents at the extremities of that chord at their point of intersection. PROPEBTIES OF THE PARABOLA CONNECTED WITH TANGENTS. 347 Let FD' be drawn to the point where the tangent at P crosses the directrix ; we shall then have the two sides PD, PD\ and the included angle, equal respectively to PF, PD\ and the included angle, .*. the angles FDD', FFD\ are equal, .-. FIX subtends a right angle at F. Otherwise. The pole of PFP\ that is, the point of concourse of the two tangents from P, P', is some point D' on the directrix (391); the co-ordi- nates of it will therefore be (— w, yj. The equation of FF' is y^y=^m[x—m\ or y= — (a;— m), Vx the axes being rectangular, and the equation of a perp. to FF' from F, or (7«, 0),isi/=-^(a;- •jw). To find where this meets the directrix, put x——m, then y—y^'y hence it meets it in the point (— m, j/i), that is, in T>' ; therefore a perp. from the focus to a focal chord meets the tangent on the directrix, the point of meeting being the pole of that chord. Theo. V. A perpendicular to any radius vector, from the foot of the normal, cuts off from that radius a part, measured from P, equal to the principal semiparameter. FO The part Pn cut off will evidently be PiV cos FPN^PNsm FPQ^PN-^. But (390), PN=z2^(mc'+m% and (p. 346), ^^=^=-^; hence the part cut off is Pn=z2^(mx'-\-rnF)<^/--r-^ — =2m=the semiparameter, the same as in the central curves before discussed. Theo. VI. If two tangents be drawn from any external point, and a third tangent be intercepted between them, then the circle, circumscribing the triangle thus formed, will always pass through the focus. Let FEG be the triangle formed by the intersections of the three tangents. The perpendiculars on these tan- ' gents from F will meet them in points A, P, C, all situated on the principal second diameter ; and it is plain, since FG subtends each of the right angles B, C, that a circle may be described through F, B/C,G\ hence the angles CGB, CFB, in the same segment are equal. In like manner a circle will pass through F, A, B, E, .-. FBA = FEA. Now FBA =FCB -f CFB=FOB + CGB=FGC=FEA; but FEA-\-FEF= 180°, .-. FGC+FEF =180"; hence the points F, F, P, G, alllie on the circumference of a circle. CoR. From whatever point in a tangent to a parabola two lines be drawn, one to the focus, and one to touch the curve, the two will always include the same angle, for, as here shown, FGC=FEA = FBA . 348 POLAE Equation of the parabola. 392. Polar Equation of the Parabola.— Taking the focus for the pole, and the principal diameter for the fixed axis, we have r=.m-\-x, and xz=m-\-r cos 6, .'. r=2m+r cos fi, .*. r=- • 1— cos^ is the polar equation of the curve. By referring to the value of p, the principal parameter, in each of the three curves, we find that In the EUipse, -^=a(l— e*). In the Hyperbola, ■xp=a{i^—l). In the Parabola, -jp=2m. 2 Hence, by substituting - jt for these values, in the polar equations of the respective curves, the focus being the pole, all become included in a single equation, namely, 1 2^ .[1], I— e cos ^ where, for the parabola, e=l. That this is really the value of e in the parabola will appear from considering the meaning of the symbol in the other curves. The distance of the focus from the centre, or point in Q which all the diameters intersect, is denoted by c, and - is e. Now in the a parabola the diameters, being parallel, can be said to meet only at an infinite distance from the focus; hence in this curve, c=qo , and a-=.c^m is also infinite, "a c+m" m~l+0~ c 393. The three curves may all be comprehended in a single equation, not only when polar co-ordinates are employed, but also when rectilinear co-ordinates are used, their origin being at the vertex of a diameter. For in both the ellipse and hyperbola the semiparameter of any diameter a, is — ; so that all the three curves are included in the equation, 2/2=pa;+2a^^ [2], where 5=— —in the ellipse, -f — in the hyperbola, and in the parabola. These names seem to have been given to the curves either because in the first, e < 1 , in the second, e > 1, and in the third, e= 1 , or because in the first the distance of any point in the curve from the ^cus is le8% than its distance from the directrix, in the second greater, and in the third the distances are equal. It appears, from what is here said, that the properties of the three conic sections might be investigated by discussing either or both of the general equations [1], [2], giving to e, p, and 5, the values proper to each AREA OF THE PARABOLA. 849 curve in the results. Or, just as in the preceding articles, much that con- cerned the hyperbola was deduced by slightly modifying the conclusions arrived at for the ellipse, so might the principal properties of the parabola have been derived, in like manner, from those of the same curve, a para- bola being regarded as an extreme ellipse, or as one whose opposite vertices are infinitely distant from each other. Thus, it was shown in the ellipse that perpendiculars, from a focus to a series of tangents, had all their intersections with those tangents in the circumference of a circle, the diameter of which was the principal diameter of the ellipse. Had the ellipse degenerated into a parabola, by the removal of the centre to infinity, the radius of the circle spoken of would have become infinite ; but a circular arc of infinite radius must be a straight line, perp. to that radius at its extremity : and hence the property of the parabola at p. 346. 394. Area of the Parabola. — The parabola being referred to any system of conjugate axes, let Ax, anj, be the abscissa and ordinate of any point in the curve. Divide ay into any number n of equal parts, from the extremities of which let parallels be drawn to the curve, as in the figure ; then Fp will be equal to one of these parts (fe), and the ordinates Fp, Qq, &c., of the points F, Q, &c., will be k, 2k, U, &c [1], and since y'-=:4:rx, the abscissas of the same points will be 4r' 4r' 4r' ^''•' and consequently the distances Ap, pq, &c., will be P 47' 17' ^^ t^J- Now the chords AF, FQ, &c., being drawn, after the triangle on the base Apz=—, there will be a series of trapezoids on the bases [2], and the area of each trapezoid will be found by multi- plying half the sum of the two parallel sides by the base, and by the sine of the angle a, at which the axes are inclined, for the product of the base by this sine evidently gives the perp. distance between the parallel sides. Hence, commencing with the second of the ordinates [1], and adding each to the preceding, we have for the sum of the trapezoids (including the triangle AFp) the expression Area of ins. polygon =—{1 + 324-524-. .. + (2)i—l)2}sin«, where ^=-, and .'. •— = — - — =— , since y^^irx. Now by (198), or (199), l+32-f 524-... + (2m-1)2=— g— , .•.Area=(?a:y-|',)sin«. The greater the number n, the closer does the polygonal area approach 850 EADIUS OF CURVATURE. to that of the curvilinear space Ayx, and the two coalesce when n=co , Hence the parabolic area is -an^ sin a, or - of the parallelogram Bx ; there- fore any segment of a parabola is two-thirds of its circumscribing paral- lelogram. 395. Radius of Curvature. — The point (a, 0), where normals at {^, 2/0' {^'' yO> two points on the curve, intersect, will be determined from the equations of those normals, when a, /3, are put for x, y, that is, from the equations 2m^-\-y' a—2my' —xy=^Q^ and 2mli-^y"u—2my"—x"y"=^0. _ . 27n(y' — y")-[-xy'—x"y" ^ . x'y—x"y" By subtraction, «= — ^-^ — ^^^ —=2m-^ — ^, f-. y-y y-y But a;y-a;y=y(a:'-a;")+a;"(y'-/), /. «=2w+a:"4-y ^P^;but(890), -=r—^ : hence when x'=x'\ and y'=y\ a=2»i-|-a:'-|-^=2m+a/+— — , y —y 4m 7 ^ ^ ^ 2m 2m x'y that is, a=:2m+3a;', .*. substituting in the first equation, ^=- . These are the co-ordinates of the centre of curvature of the parabola at the point {a/j y') : and for the radius of curvature, we have \ m / w> \ iih^/ 4m2+47Jia5' , 4 {m-\rx )2=- {m-^-xy, 2 , , .1 2^^^, , A (Normal)3 ,^^^^ ••• '=7^<-+^) -^(»+^) -h^f^^o)- At the vertex of the principal diameter, where a;'=0, 2/'=0, we have r=2w, the semiparameter, as in the ellipse. And generally, the circle of curvature at any point, cuts from the diameter of the curve at that point, a part equal to the parameter of that diameter. For the perp. from the centre (a, iS) of the circle, to this diameter of the curve, must bisect the portion of it cut off by the circle, since this portion is a chord of the circle. The semichord will therefore be- the radius of the circle multiplied by the cosine of its inclination to the diameter of the curve referred to, which cosine is the sine of the angle between the normal and the ordinate of the point, and is therefore MN expressed by the ratio — — . (See fig. at p. 344.) and this, multiplied by the above expression for r, gives 2(wi+a;')=the semiparameter of the diameter at («', y) [p. 343] : hence the diametral (we cannot call it the central) chord of curvature is always equal to the parameter. The semichord of curvature through the focus is, in like manner the product of the radius by the cosine of the angle between the normal and LOCUS OF THE EQUATION OF THE SECOND DEGREE. 351 radius vector of the point {a;\ y'), or by the cosine of that between the radius vector and the perp. on the tangent, which cosine is expressed by FO the ratio ^ (p. 347). and this, multiplied by r, gives the same expression as before, so that the diametral chord of curvature is always equal to the focal chord of curvature. In the parabola, therefore, 4i?P=the focal chord of curvature. ^^^^ Km) =(-^) ' ••• '={fq) '"^=(?Q ) • 2^» as in the other curves : and since in the parabola FQ^=zFP . FA^ .*. r=2 -=p-. 396. Locus of the Equation of the Second Degree.— We have now to show that every equation of the second degree with two variables is the algebraic representation of a conic section, whenever it is any locus at all. In the more simple cases, where the equation proposed is in one or other of the forms My'^+Nx^=P [1], or y^^Qx [2], it is easily shown that the locus of the first is either an ellipse or an hyperbola, and that the locus of the second is a parabola. For let a^ be P I* F P written for — , and 6^ for — ; then we shall have iNT-^-r, and M=-ri» so N M a* 0^ that [I] is converted into which we know to be the equation either of an ellipse or of an hyperbola : the former when the constants a-, b'^, are both positive, and the latter when one of them is positive and the other negative ; that which is nega- tive applying to the second diameter of the other, which is the corre- sponding transverse. Either may of course be coincident with the axis of ^ ; in other words, the hyperbola may take either of two conjugate positions. If, however, P alone is negative, then [1] has no geometrical interpre- tation : the locus is imaginary, for since, in this case, whatever real value be given to x, the corresponding y will be imaginary, showing that no real locus exists. But if P is zero, and M, N, both positive, [1] represents merely a point, which, with the circle (M=N), are varieties of the ellipse. If, P being zero, one of the other two coefficients is negative, N for instance, then [1] gives the asymptotes of the hyperbola —c^^—r^y^^'^' 352 LOCUS OF THE EQUATION OF THE SECOND DEGREE. Thus the two straight lines passing through the origin, and of which the equations are denote a variety of the hyperbola [1], iV being negative and P=0. The form of the equation [2] shows at once that the locus is a parabola, the parameter of the curve being the constant Q ; we should regard this parameter as positive, even though Q were negative ; for the equation would then merely imply that the curve is situated wholly to the left, instead of to the right of the axis of y ; the abscissa of each point in it being negative. The smaller Q becomes, the nearer do the two opposite portions of the curve approach towards each other ; and they actually coalesce when Q=0, and become confounded with the axis of x\ hence a single straight line — the axis of x, is a variety of the parabola. There is also another variety of this curve : suppose the origin of the axes to be removed to- wards the right, or, which is the same thing, imagine the vertex of the curve — still remaining on the axis of x — to recede towards the left, to a distance U from its original position : the equation of the curve will then aj-H^ j, where Q and K ai'e arbitrary constants. Let now ^=0, then we shall have y=^K, .*. ?/= ±5', which denote two straight lines, parallel to and equidistant from, the axis of X. 397. Let us now take the equation of the second degree in its most general form, namely, Af-\-Bxy-\-C3?-\-Dy-^Ex+F=0 [1]. And first we may remark that, whatever this equation may be, the form of the locus cannot be altered by altering the direction of the axes to which it is referred : we may consider at the outset that these axes are rectangular, for if they were originally oblique we could change them for rectangular by a process similar to that at page 318, and it is plain that the transformed equation could not be more general than [1], since the form of this is the most general possible. Now to convert this equation into another, referring the locus to a new system of rectangular axes, we have but for the primitive x and y to put (p. 316), x=.x COS a—y sin a, y=^x sin a-\-y cos a, and we shall find, for the coef. of xy, the expression, 24 sin a cos u-^-B cos^ cx,— B sin^ «— 2(7 sin a cos a; consequently we have only so to determine a that this may become zero, in order that the term in xy may disappear from the equation of the locus; that is, we have only to put 2{A — C!) sin a cos a4-J5(cos^ a— sin^ a)=0, or {A —C) sin 2a-\-B cos 2«=0, and we get the condition, tan 2a= — [2], A — O which being fulfilled for «, the angle which the new axis of x makes with the primitive, the resulting equation of the locus must be of the form My''+Nx^-{-Ey-\-Sx-^F=0 [3]. LOCUS OF THE EQUATION OF THE SECOND DEGREE. 353 Hence the term in [I], containing the product of the variables, becomes removed by merely turning the axes round through an angle a. 398. We shall now see that, by simply shifting the origin of these new axes, without disturbing their directions, we may next remove the terms containing only the first powers of the variables. In order to shift the origin from the point (0, 0) to the point {a, b), we have merely to put in [3], a;=iu-\-a, and y=:y-{-b, when that equation will become My''-^Na;'-\-{^Mb-\-B)y-^{^Na-t S)a;—F=0, where — P is written for the quantity independent of the variables. Now, that the first powers may disappear, we have only to determine the constants a, b, so as to fulfil the conditions 7? Sf 2Mb-\-R=0, and 2Na+S=0, .: b=—--, and a=— — . Hence, by fixing the values of the otherwise arbitrary constants a, a, b, so as to conform to the three conditions, an ^«-^_^, «- 2ivr' ^- 2M' we see that the equation of the locus [1] becomes reduced to the form My''-\-Nx'=F, 399. In the foregoing reasoning, it is assumed that neither My~ nor Nx^ has disappeared from the equation [3]. Should, however, either of these, as Nx\ be absent, then the coef. of x, in the subsequent trans- formed equation, would be simply S, the same as in [3] itself; and this, of course, we could not equate to zero. If, therefore, the term Sx be not absent from [3], as well as Nx^, it cannot be removed from the trans- formed equation: we may then determine a so that the final term, — P, shall disappear: this final term is Mb''-\-Rb-\-Sa-\-F, which, equated to zero, fixes for a the value a^=— ^^ ; and with this value for «, o and the above value — ;r^, for b, the equation of the locus takes the form o My'^ + Sx^^Q, or the form y'^=Qx, where Q is put for — ^ And, in like manner, if My'^ had been supposed absent from [3], instead of iVa?^ the reduced form would have been, Nx--{-E2/=0, or TO x'-^Q'y, where Q' is put for — — . If M and N were both zero in [3], the equation would no longer be one of the second degree : but if the terms containing the first and second powers of the same variable, as, for instance, the terms Sx, Nx'^ be absent from [3], then the equation, though of the second degree, will contain but one variable ; so that the locus will be a pair of parallel straight lines : for, Mf^RyJrF=% gives y=-~±^s/{R^-^FM), and these can exist only so long as R->4:FM. If D and E be both absent from [l],R and 8 must both be absent from [3] : the axes .-. must then originate at the centre, if the curve have a centre (396), It thus appears that the locus of the general equation [1] is always a conic section, or else a variety of one or other of the curves thus called. A A 354 FOBM OF THE LOCUS. It remains for us now to inquire how we may find out which of the three curves any given equation represents. 400. Form of the Locus. — Writing the general equation [1], p. 352, thus, and solving it as a quadratic in y, we have for y the expression y=-?^±±^{(B^-AACf)x^+2iBD-2AE)x+I^-iAF} [1], which will be real or imaginary, for real values of a;, according as the quantity under the radical is positive or negative. It will not interfere with the sign of this quantity if we divide it by ai^, since x^ is positive, whether ^ be -f or — . Hence the sign of the irrational part of [I] will be the same as the sign of {E^-iAC)-h2(BD-2AE) 1^{L^-AAF)\. X ar Now (194) a value so small may be given to - as to cause the sign of this expression to be the same as the sign of its first term (B^—4:AC), for this small value, and for all smaller values of -, whether positive or negative. In other words, there exists some value for x such that for it, and for all numerically greater values, positive and negative, the value of y in [1] will be real if B^—^AC is positive, and imaginary if it be negative. But the only one of the three curves — one or other of which, as already shown, must be the locus of [1] — of which, after a certain pair of values positive and negative for x, the ordinates are all real, how- ever great d-^ may be, is the Hyperbola. And the only curve of which, after a certain pair of values positive and negative for x, the ordinates are all imaginary, is the Ellipse. Hence, if {B^—4:AC)<^0, the locus is an Ellipse. if (JS2_4^(7)>0, „ „ „ Hyperbola. And.-., if {m-iAC)=0, „ „ a Parabola, inasmuch as, in this latter case, it can be neither of the other curves. If, however, here BD—^AE vanish, as well as B'—^AC, then [1] will represent, not a curve, but two parallel straight lines, namely, the straight lines y=-^^^s/iI^'-^AF), and y=-?^-^{D--iAF), which are parallel because they have the same coef. of inclination. And we know that a pair of parallels is a variety of the parabola : the two parallels merge into one straight line if D~=4:AF. There is also a variety of the hyperbola, namely, two intersecting straight lines (396), and these the locus will be, whenever the roots of the irrational part of [1], equated to zero, are equal: for, calling each root otp [I] will then be Bx-\-D^^(B^-iACf)^ y= — 2Z~^ 2I ^^-"'>' representing two straight lines which intersect, because the coefiBcients of X are unequal. Hence the form of the locus may always be ascertained DETERMINATION OF PARTICULAR LOCI. 355 by examining the given coefficients of the equation. If either Ay^, or Ca?, be absent from the equation, Bxy being present, the locus is always an hyperbola, as shown above : it is equally so if hoth be absent, Bxy being still present. The form of the equation in this latter case being which shows that 1/ is real for every real value of x ; and that, for every such value, there corresponds but one value of y : hence the axes, to which the hyperbola is referred, must be parallel to the asymptotes. If we had solved the equation [Ij for x, instead of for y, the only difference in the result would obviously have been that x and y would have been interchanged, as also A and C, and D and E ; that is, we should have had x=-?^±^^{{B^-iAC)f+2{BE-2CD)y+E^^-iCF} [2]. And it is plain, omitting the irrational part of both [1] and [2], that Bx-\-D ^ By-\-E represent two diameters of the locus : for the first of these lines bisects all the chords parallel to the axis of y, and the second all the chords parallel to the axis of x. We say this is plain, because, calling the omitted irrational parts X and Y, respectively, if any point M, in the first of these lines, be taken, and we prolong the ordinate through M, and then cut off parts MP, MF', each equal to Y, the points P, P", must be in the locus of [1] ; so that as the line thus bisects every chord PF^, parallel to the axis of y, that line is a dia- meter. And in like manner is it shown that the other line is also a diameter. The centre of the curve, therefore — when it has a centre — must be at the intersection of the lines [3] : solving then [3], as two simultaneous equations, we have for the co-ordinates {x^, y^ of the centre, in the central curves, _ 2AE-BD _2CD-BE and these each become infinite when B^—4:AC=0, that is, when the curve is a parabola. 401. Determination of particular Loci- —We shall now give an example or two of the practical application of the preceding theory. (1) To determine the locus of the equation y'^—^xy+^x''-\-^y— 4a;-3=0. Since here -B'— 4^C<0, the locus must be an ellipse: its position is determined as follows : — Solving the equation for each of the two variables, y:=x-\±^-{2x^-2x-i)...\ll «=|(y+2)±iv/-(2/+23/-13)...[2]; AA 3 356 DETERMINATION OF PARTICULAR LOOT. or, replacing by their factors the irrational parts of these, y={x-l)±s/-{x-2){x-^l) [1], From [1] it appears that the locus is wholly comprised within the limits x= — \, and x=2; for it is only within this range that [1] is real. From [2] it appears that the locus is also wholly comprised within the limits y=- 1+n/27 and y:. ■l + x/27 for it is only within this range that [2] is real. Consequently the ellipse is circumscribed by the parallelogram BT formed by parallels to the axes AX, AY, these parallels being distant from the origin by the values of x and y just deduced. The four points of contact are at the extremities of the diameters y—x — \, and aj=-(?/ + 2), since o the X, y, of each point causes the radical in [1] or [2] to vanish. The first of these diameters cuts the axes in the points x=l, y= — 1, so that if through these the line PJP be drawn, it will be a diameter ; and its extremities P, P, will be two of the points of contact. The second cuts the 2 axes in the points a;=-, ?/=— 2, so that the line QQ through these o will be the diameter, at the extremities of which the other two points of contact will be situated ; and the intersection of these diameters will be the centre of the curve ; that is, it will be at the middle of each : its co-ordinates, therefore, are ■l+x/27 \ ^, _ 1 1 — j, that IS, x=-, y= . X=: -1+2 1/ l + x/27 2 Or the '4(- of the centre may be found by putting - for x in y=x—l. A parallel from the centre to the tangent TT, that is, to the axis AY, will be in the direction of the semiconjugate to OF : the length. Op, of this semiconjugate will be found by putting - for x in the irrational part of [1 J, since the rational part for this value gives 2/=mO : hence the length of Op is ^_(a;-2)(,r+l)=i2^, and thus knowing the lengths and directions of a pair of semiconjugates {a\ ¥), we may find the lengths and directions of the principal diameters &8 follows : Calling the given angle [a\ b'], a, and putting 9, 6', for the angles which a\ h', make with the major semidiameter a, we have (366) a'2+6'2=a2+62^ and 2a'&' sin a=2a6, .-. a'2±2a'i'sina+6'^=(a±6)-, which determines a-\-h, and a—h, and consequently a, and b. And since 6=6-\tt, we have (364) tan 6 tan {6-\-a)=. — :, which determines t. PROBLEMS ON LOCI. 057 (2) Required the locus of the equation t/'^—S/z??/— 3^-— 2?/ + 7a?— 1=0. Here, since B~—4:AC>0, the curve is an hyperbola: to construct it, the easiest way will be first to find the asymptotes, and then a single point on the curve : we may then determine as many more points in it as we please, and thus readily trace the locus. Solving the equation for y, we have Now the greater x is, the closer does this curve approach to the straight lines t/=a; + 1 ± ( 2^"~7 ) > ^^^ when a!= oo , they coalesce : these straight lines are therefore the asymptotes, and may easily be constructed in the usual way. And a point in the curve being determined for any assumed value of w, as, for instance, for ^=0, as many points in the locus as may be necessary to trace out the curve may be afterwards found (386). 402. Problems on Loci.— (l). The extremities B, C, of a straight line of given length, move along the sides AY, AX, of a given angle A : required the curve described by a given point P in that line, or in its prolongation. Let PB=a, PC=h, and cos A=c; then {x, position, y) being the point P in any we have (228) h^=MCP-2MC. cy-\-f ; but a : h X : MC=.-Xj a ... »^4>-^%+^, the equa. of the locus. And since the curve is an ellipse, the centre of which is at the origin A^ because the first powers of the variables are absent (399). If ^=90°, then c=0, and the equation is a23/2+JV=a2j2 . hence is suggested an easy method of tracing an ellipse by continued motion, when the centre and the principal diameters of it are given. The instrument employed for this purpose is called a Trammel. In the annexed figure, representing the trammel, the two grooved bars, usually of wood, are fixed at right angles, their intersection being over the point intended for the centre of the ellipse, and the bars themselves lying in the directions of the principal diameters. On the cross bar BCP, are moveable pins, or sliding guides, B, C\ and at the point P in this bar is fixed a pencil, so that the distance BP may be equal to the semitransverse diameter, and CP equal to the semiconjugate. If the guides B, C, are allowed to move freely along the grooves, the pencil will trace the required ellipse. 358 PROBLEMS ON LOCI. (2) Given the base and altitude of a plane triangle, to determine the locus of the intersection of perpendiculars from the vertices to the oppo- site sides. Putting a for the altitude CD, and c for the base AB, we know that the locus of G will be a straight line parallel to AB. Hence, taking AB, AY, for rectangular axes, and putting {x^, y^), for any point in the locus of C, the equa. of that locus my^=a; and the equa. of BC, passing through the two points (c, 0), (x^, y^}, is also the equa. of a perp. to this from A is c—x. .[1]. At the point P, where this perp. intersects CD, we must have x:=x^', hence, substituting in [1], X for x^, and a for y^, we have for the locus of P the equation ay=cx—x^, or ac^+ay— ca;=0 [2], which is that of a parabola, since 5^— 4^C=0. When ^=0, we have 2/=0, .-. the curve passes through A, but does not again meet the axis of y, which, prolonged, is therefore a diameter of the curve ; but for 2/=0 we have the two values a;=0, and x=c : hence the curve passes also through B. The chord AB is thus perp. to the diameter at A ; hence the principal axis of the curve bisects AB at right angles. To find the principal vertex F, 1 c2 put «=- c in [2], and we get y= j- ; (1 c* \ - c, ;2~ ) ^^ *^® vertex F. If we remove the origin to this point, that is, in [2], therefore the length {a) of the parameter 1 (? if we put x-\-- c for x, and y-\--7- f°^ the equation will be x'^=—ay is CD. (3) Given the base and the difference of the sides of a triangle, to find the locus of the centre of the circle touching one side [BC), and the two prolongations of the other sides {AB, AC). Let P be the centre of any one of the circles, and p, p\ p", the three points of contact ; then since Take the origin M at the middle of AB, and let MX, MY, be the rectangular axes ; then ■^z=zAp-~AB- ■^{AG-\-BG)=x. Now AC(^CB being constant (=2a), the locus of C (x^, y^, is an hyperbola, namely, a^y''-bW=-a%^ [1], \ ••, PKOBLEMS ON LOCI. 359 and in this curve -(AC-{-BC):=exi=.x, /. ic,= — [21, since e=-. Also P must be always on the line bisecting the angle BCp'\ that is, on the normal through (o^j, y^, to the hyperbola. If in the equa. of this normal we put for x^ its value — , it becomes 62y -ary,+aVi {a—c)G c a-\-c Hence, substituting the values [2], [3], for x^^ y^, in [1], we have {a-\-cfy^—h^x^=z—h'^{^XYcos[X,Y]}\ so that (68) the only real pair of values for X and Y, that satisfy this THE POLYGON OF FORCES. 367 necessary condition, must be X=0, F=0, as before. Such then are the analytical conditions which must be satisfied in order that a set of forces, acting in the same plane, upon a point, may keep that point at rest. The corresponding geometrical conditions are determined as follows : — 412. The Polygon of Forces-— Let the forces P^, P^, &c., act upon the point A, as in the margin ; then re- placing them by their representative straight lines, we know (409) that if P,B, equal and parallel to AP^, be drawn, AB will be the resultant of Pj, Pg, and may therefore be substituted for these two forces. Compounding, then, this resultant with P.^, by drawing BC equal and parallel to AP^, AC will be the resultant of P^ Po, P.y Continuing thus to draw parallels, as many as there are forces — each successive parallel from the extremity of the preceding — the line AR, from A to the extremity R of the last of these, must represent the resultant of all the forces Pp Pg, P.^, &c. If this resultant be zero, that is, if P^, Pg, P.^, &c., keep A in equilibrium, then the last of the parallels spoken of must close the polygon ^Pj^C...v4, the point terminating this last parallel then coinciding with A, the final parallel (J^A) being, in fact, equal and opposite to the last force— the force P5 in the preceding diagram. The polygon thus formed is called the polygon of forces: there may, however, be as many such polygons as there are individual forces, since any one of the forces Pp Pg, &c., may be taken for the first force, or for the first side of the polygon ; the last side, however, will always be equal and opposite to the line representing the hist force. And as everything here said holds, whether the original forces act in one plane or not, the crooked line AP^BC.A will always be a closed polygon, but a plane polygon only when the original forces all act in one plane. 413. Applications of the Preceding Theory.— In the foregoing exposition, in order to avoid unnecessary complication in our diagrams, we have uniformly regarded the forces concerned as pulling forces, for a pushing force may always be replaced by a pulling force of equal intensity, acting in the opposite direction. In the following examples, a pulling force will be conceived to act on a point through the intervention of a cord or string, regarded, in itself, as without weight, and incapable of extension : the tendency of the cord to stretch is of course the measure of the pulling force, which is thus the force of tension of the string on the point. The efifect of a pushing force is pressure, or thrust. 414. There are two principles implied in the theory which has been established above — the formal enunciation of which we have thought it prudent to reserve for this place — that demand the special attention of the student, previously to his attempting the solution of statical problems : they are the following, namely : — 1. If three forces acting at any points in a rigidly-connected system of lines, keep that system at rest, the lines along which these forces act, must, when prolonged, all meet in one point. For (supposing first that no two are parallel) two of them must meet in a point, and these meeting lines being regarded as rigidly connected 368 APPLICATIONS OF THE PRECEDJNG THEORY. with the system, the two forces may be transferred to that point, and acting there must have the same effect, which effect is counteracted by the third force, since the system is at rest: the direction, therefore, of this third force must pass through the point. Hence, Imowing the positions of the three points on which the forces act, and also the directions of two of the forces, the direction of the third will be found by joining the point on which it acts, with the point where the other two directions meet. Similarly, if any number of forces act on as many points, and if the directions of all but one are known to meet in a point, then the direction of that one must pass through the same point, in order that the system may be kept at rest. If two of the three directions of the balancing forces considered above, are parallel, all three must be parallel, since if two meet, the third must pass through the same point. Note. — It is also useful to remember that since three equilibrating forees are represented by the three sides of a triangle drawn in the directions in which they act, and that sides are as the sines of the opposite angles, each force may be represented by the sine of the angle between the directions of the other two : thus, if P, Q, R, equilibrate, then P : Q : : sin [Q, B] : sin [P, K]. 2. If any number of forces in the same plane act on as many points rigidly connected, and keep the system at rest, then the same forces, acting in the same directions on a single point, would keep it at rest. For we may resolve each force in the same two rectangular directions, and thus replace the whole by two sets, each set consisting of parallel forces acting in opposite directions ; and these opposite forces must balance, because the system is at rest. Hence the original forces, if trans- ferred to a single point, will satisfy the conditions [1] at p. 866, and will therefore keep the point at rest. [Note. — These conditions prevent the system from moving bodily in any direction, but they are not sufficient to prevent a twist, or rotation: another condition is necessary for this. See (415)]. Ex. 1. A weight W hanging oy the string AW, tied in a knot at A, is supported by two forces acting along the strings AB, AC, which make given angles a, ^, with the vertical line AY: required the tensions T, T, of these latter strings. From any point B in AB draw a parallel BD to AW, meeting the pro- longation of CA'm D ', then the sides AB, BD, DA, will represent the three balancing forces in magnitude and direction (409), so that we shall have T siii/3 T' sin a T _AB T_AD J W~BD' W~BD'' ^^ Tr""sin (a+^)' TF~"sin(«+/J)' sin/3 ■W, T'=. W. sin (a 4- /3) sin (a -|- 13) Otherwise. Resolving the three forces along the two perpendicular axes AX, AY, we have r'sin/3— 7'sina=0, T' QO&^-\-T co&a=W. Multiplying the first by cos /?, the second by sin /3, and subtracting to eliminate T\ we have sin /3 T(sin a, cos /S+cos a sin /3)=prsin /3, .*. T=. sin (a+/3) W. Similarly r= sin (a-|-/J) W. APPLICATIONS OF THE PEECEDING THEORY. 369 2. A cord ACB is fixed immovably at the ends A, B, and a weight W is at liberty to slide, by means of a ring C, freely along the cord : at what point of the string will the ring rest, and what will be the tension of the string ? Since the string passes freely through the ring, its tension must be the same throughout. C is thus kept in equilibrium by the two equal forces acting along CA, CB, and the weight W acting vertically downwards. Let CB represent one of the equal forces or tensions ; prolong AO to meet the vertical BD in Z>, then DC, CB, ^ ^ BD, represent the three forces in magnitude ^ and direction ; that is, DCB is the triangle of forces, and it must be isosceles, DC being z=CB; so that the perp. CF bisects BD, .-. for the tension DC, or CB, we have T=-pr-r-cosZ>. In order to find the angle D, we have AE=a, the horizontal distance between the fixed points, and also AC+CB=AC+ CD=AD=l, the length of the string, so that sin D=-. Otherwise. Having found D or B &s above, we know its equal BCY: resolving, therefore, the two equal tensions along CY, calling the known angle BCY=AGY» ^, we have rcos/3 + rcos^=TF, .-. T=-Tr-7-cos^, as before. To determine the position of the point C, we have ED=^(l'^—a'^), and EB, the distance of B from the horizontal line AE is known : hence BD is known, and .-. BE, the half of BD, and 5^ tan ^=JjV: the point C is thus determined. 3. A weight Q is suspended from the extremity B of a rigid line (or rod) AB, which can turn freely about a hinge at its other extremity A ; a string, acting perpendicularly to it, at its middle point D, sustains the system in equilibrium. What is the magnitude and direction of the pressure on the hinge at A, when the rod makes an angle of 30° with the horizontal line AX, and what is the tension of the string ? Produce ED, and BQ (if necessary), till they meet in C, and draw A C, which will be in the line of direction of the pressure on the hinge at A, or of the resistance P to that pressure acting from A towards C (see 1, p. 367). The point C, therefore, would be kept at rest if this resistance and the other two forces w^ere applied to it. Now, since CE bisects AB at right angles, it must also bisect the angle C: hence CjB, the direction of the resultant of the forces P, Q, bisects the angle between the direc- tions of those forces, .-. P=Q, whatever be the angle the rod makes with the horizontal line. Again : since AB makes an angle of 30° with the horizontal line AX, .*. B=60°, so that the triangle ABC is equilateral. Replacing forces by their proportional lines, since CB, GA, are equal and opposite to two of the balancing forces at C, the third force ^GD, (p. 364) is the resultant of B B 370 APPLICATIONS OF THE PRECEDING THEORY. CB, CA ; that is, it is =9Q sin B=Q\/S, which is .-. the tension of the string BE. Lastly, since BA (7=60°, .-. the direction of the pressure at J, namely, the direction CAp, is inclined at the angle CAX of 30° to the horizon, so that the inclination pAX is 150 \ Otherwise : suppose the three forces, namely, the vertical force Q, the force T perp. to AB, and the resistance P acting at A, all to act on the point D (See 2, p. 368), and let them be resolved into their components acting along the axes AB, EC. Then, calling the unknown angle which the direction of the resistance makes with AB, (p, we have, siflce the forces along AB destroy each other, Q cos 60°— P cos &c., being to the right of A, B, &c.,) we must have A(cot /3+cot iS,) = Tr„ /i(cot Ai+cot li.^=W^ A(cot /3^+cot /3^ = 1F3, &c., cot/3+cot /3, _TFi cot/3,+cot/3^ _Tr2 ' ' cot /3,+cot i32~lr3' cot /3^+cot jSa—^Fg' ^ ^ ^' If the angles which the several sides of the polygon make with the horizontal components be called a, a^,, a.,, &c., then since h=iT cos a=7', cos a, =^2 COS a^, &c., .'. T=h sec «, ^,=A sec «,, 7^2=7* sec a^, &c., .-. the tensions of the several sides are as the secants of the angles (all measured in the same direction) which they make with the horizon. The fixed points a, b, evidently sustain the whole of the weights ; the vertical pressure on these points is, therefore, the sum of those weights, while the horizontal strains balance each other. If, then, the extreme sides of the polygon be prolonged till they meet, and the sum of the weights be affixed to the point of meeting, the tensions and directions aA, bC, of the extreme sides must remain the same. But to determine the form of the polygon, the horizontal distance ac, and the vertical distance be of the points a, b, as also the weights and the lengths I, l^, l^, l.^, &c., of the several portions aA, AB, BC, &c., of the cord being given, we may proceed thus: — Imagine a horizontal line drawn through the lowest knot (or knots, if there be two equally low), and terminating in the two verticals from a, b : this line de is given, it being =ac ; also the difference be — ad is given, it being =6c. Now the several portions of de, intercepted between the verticals from a. A, B, &c., are I sin /?, l^ sin /J^, l^ sin /^g, ^3 sin ^^, &c., 80 that I sin /3+Z, sin /Sj + Z^ sin (i.^+h sin /3j+--- + ^» sin (in=de [2J. In like manner I cos )3+Z| cos /S. + Zj cos )3;i+^3 cos lij-{-...-^l^ cos |3,j=&c [3]. Now there being n weights, the equations [1] are n—l in number, so that these, with the two just given, make n-{-l simple equations, in which |S, iSp /Jg, $.^,...^„, are the only unknown quantities ; these, therefore, may be found, and thence the form of the Funicular Polygon, as it is called, ascertained. It is worthy of notice that it would be impossible for any forces, how- ever great, applied at a, b, to stretch this polygon into a straight line, however small the weights may be, and however few in number. For conceive the polygon to be stretched into the straight line joining a, b, the weights, or only a single weight, hanging vertically, then the components of the forces at a, b, in a direction perpendicular to the cord are each zero; but the components of the weights, or of the single weight, in the same direction, have a certain intensity, which there is nothing to oppose, consequently, it is impossible that the line ab, to which any weight, however small, is suspended, can be a straight line, however great the stretching forces, unless, indeed, the points a, b, are one ver- tically under the other. And the same is true of an unloaded material line, since every such line has weight. Examples for Exercise. [Note. — In resolving forces acting at different points, in the directions of fixed axes, the student will often find it convenient to take a point B B 2 872 THE PRINCIPLE OP MOMENTS. distinct from the figure ; to draw from it the axes, and the forces, in their proper directions, and then to resolve the latter along the former. All perplexity, as to the signs of the components, will thus be avoided.] (1) Two forces represented by 2 lb. and 3 lb. act on a point at an angle of 60° : re- quired tbe magnitude of the resultant. (2) Two forces expressed by 26 lb. and 127 lb. act on a point at an angle of 76° : required the resultant. (3) Two weights, each equal to 10 lb., balance each other when hanging at the ends of a string which passes over three fixed points in a vertical plane, the lines joining these points forming a triangle, of which the base is horizontal ; the vertical angle of this triangle is 90°, and the other angles 60° and 30° respectively : required the pressures on the fixed points. (4) If three forces act perpendicularly to the sides of a triangle at their middle points, and if the forces be proportional to the sides on which they act, prove that they must be in equilibrium. (5) In the figure at p. 369, (ex. 2), the forces at ^, B, are 3 lb. and 4 lb., and the weight W, 5 lb,; the distance ^jB is 6 feet, and the inclination of ^^ to the horizontal line AE is 30° : required the lengths of the two parts AC, BC, of the string. (6) A weight of 10 1b. slides freely along a cord 8 feet in length, attached at its extremities to two hooks 6 feet apart, and in a line inclined at an angle of 30° to the horizon : required the pressure upon each hook when the weight is at rest. (7) A cord 12 feet in length is tied to two hooks 8 feet apart ; the cord passes through a ring upon which a force of 60 lb. acts in a direction, making an angle of 60° with the line from one hook to the other : required the pressure upon each hook. (8) A cord 14 feet long is fastened to two hooks 12 feet apart ; to this cord another is firmly tied at a part dividing the first into two lengths of 9 feet and 5 feet ; a force of 60 lb. is then applied to the second cord, and it acts in a direction making an angle of 70° with the line joining the hooks : required the pressures on the hooks. (9) In the figure to ex. 2, p, 369, the string AC, with a weight attached, is fastened to the hook A ; another string, CBD fastened to (7, passes freely over the fixed point By with a weight attached to D, the string CB makes a given angle a with the horizontal line GF : what angle 6 must -^ ^ a ^ "R make with the vertical CY, in order that W may be kept at rest by half its weight at Z) ? (10) The rigid rod AB (without weight), but with the weight W attached to its extremity B, is kept from turning about the hinge A, by the supporting rod DC, also without weight, and having hinges at C, D, as in the diagram : re- quired the strain on the hinge A when ^ J5 is horizontal [AB=a, AC=b, AI)=cl % -I'J 415. The Principle of Moments.— In what has preceded the acting forces, all in one plane, have been regarded either as applied to a single point, or else as so applied to a system of points connected by rigid lines, that their directions all concur in a single point. In this latter case the point of concurrence may obviously be considered to be rigidly connected with the system by these concurring lines, and since the con- ditions of equilibrium keep this point at rest, the entire system is pre- vented from advancing bodily, or as a whole in any direction ; or, as it is technically called, there cannot be any motion of translation, for no such motion could take place without every point of the system partaking of it, and yet the point of concurrence just adverted to remains fixed by the con- THE PKINOIPLE OF MOMENTS. 373 ditions. But it would still remain fixed, though the rest of the system rotated round it ; there is, therefore, nothing in the prescribed conditions to prevent this motion of rotation. Consequently, that a rigid system may be kept from all motion by forces applied to different points of it, these forces must satisfy not only the conditions (410) necessary to pre- vent translation, and which are sufficient for a single point, but also some other condition or conditions necessary and sufficient to prevent rotation. We had an instance of the necessity for a distinct condition in ex. 3, at p. 369 ; this necessity suggests the introduction of a new term— the term moment, which is thus defined. The moment of a force, in reference to a given point, is the product of that force by the numerical value of the perpendicular from the point upon its direction. Theorem. — The moment of the resultant of any number of forces acting in a plane upon a point is equal to the sum of the moments of the component forces, wherever in the plane the point of reference be taken. Let A be the point to which the forces Pj, P,' &c., are applied, then drawing two rectangular axes from A, and projecting (411) the several furces upon them, we have for the sums X, Y, of the rectangular components the values [1], p. 366, or putting R for the resultant, we have X=B cos a, F=/2 sin a, where a is the inclination of R to AX. Let C be the point of reference, or centre of moments; call AC, c, and the angle CAX, S ; then multiplying the expressions for X, Y, at p. 366, by c sin 6, c cos 6, respectively, we have Re cos a sin 6=zP^c cos as, sin 6-\-P^c cos et.^ sin ^4-...+P,iC cos a,i sin ^...[1}. Re sin a cos 6=.PiC sin a^ cos 6-{-P^c sin u.^ cos 6-\-...-\-P^c sin «„ cos ^...[2], Subtracting [1] from [2], and putting sin {a—i) for sin a cos ^— cos » sin 6, Re sin {a-^=P^c sin (a.-^ + PgC sin («^-^)+...+p„c sin (a,,-^. Now P, in the diagram, being any one (as PJ of these n forces, we have c sin (ai— ^=c sin (7^^=perp. Cp, on the direction of that force. Hence, calling the several perpendiculars from C, p^, 'p^,...p,„ and the perp. c sin (a— &) on the resultant R, r, we have, as the theorem affirms, J^r=P,p,-VP^P^+P,p,-\-...+PnPn [3]. Now if the lines along which the applied forces Pj, P^, &c., act, be sup- posed to be rigid, as also the perpendiculars connecting them with the point C, these perpendiculars however being at liberty to obey any force which would turn them about C, as about a pivot there : — under such con- ditions, it is plain that the single force P in the diagram, if acting alone, would turn the system about that point ; and similarly of any other one of the forces. In order, therefore, that no rotation may ensue, the efforts to produce it in one direction must just balance the efforts to produce it in the contrary direction. That the intensities of these efforts are cor- rectly represented by what has been called the moments of the forces will appear from what follows. 374 THE PRINCIPLE OF MOMENTS. Let two forces P, Q, act at the extremities M, N, of the rigid lines CM, CN, and let it be required to estimate the efforts of these forces to turn the lines about the point C, regarded as a fixed pivot. Prolong the directions of the forces, to meet in D, and draw CD, then a force equal to the pressure on C acting along CD will keep G at rest, though the pivot be removed. Now in order that not only G but the ex- tremities M, N, may also remain at rest, the point D must remain at rest, and therefore MD, ND, must retain their positions, and this is the same as saying that the perpen- diculars CE, CB, continue invariable. Draw the parallels CF, CG, to the fixed directions ND, MD, then CG : CF : : CB : CE, since the angles at G and F are equal ; but P : Q : : CG{=FD) : CF{=DG), .\ P : Q : : CB : CF, .-. P . CF=zQ . CB, which is the condition necessary in order that there may be no rotation about C; that is, the moments of P, Q must be equal; and conversely, if these moments are equal, the foregoing proportions have place, and .-. D, as well as C, is at rest, and consequently there can be no rotation about C. The moment of a force, therefore, in reference to a given point, fitly represents the effort of the force to cause rotation about that point. In order that there may be no rotation about the point, the sum of the moments [3], regarding the tendency to turn in one direction as +, and the tendency to turn in the contrary direction — , must be zero ; and this it may be, although the point in which all the forces concur have a pro- gressive motion, that is, although the resultant R of those forces have a finite value : for wherever upon B the point be chosen, the perp. r must beO. The conditions, then, that are necessary and sufficient to keep at rest a rigidly-connected system, acted upon by forces whose directions, all in one plane, meet in a point are, 1st. ^=0: this forbids progressive mo- tion; and 2nd. P^p^-\-P^p.,-\-...-\-P„p,^:=0, which forbids roiafori/ motion. It is common to employ the symbol Z to represent the algebraic sum of a set of like quantities, such as these last, so that writing the expressions [1], at p. 366, in this more abridged form, the following are the conditions which are necessary and sufficient to keep completely at rest a rigidjy- connected system, acted upon by concurring forces in a plane : — 2(Pcosa)=0, 2(Pcos/3)=0, ^{Pp)=:0 [4]. And since, when this complete equilibrium has place, R in [3] must be zero, it follows that 5:(Pjo)=0 for every point C in the plane of the equilibrating forces, or whatever be the lengths of p^, p.2, &c. Referring for an instant to Ex. 3, p. 369, taking D for the centre of moments, and putting 2a for the length of the rod, we have, for the perp, from D on AC, P{=^cL sin (p, and for the perp. from D to BC, p^=a sin 60°, .*. Pa sin (p—Qa sin 60''=0, .'. P sin (p=Q sin 60°, the condition at p. 370. PARALLEL FORCES ACTING IN A PLANE. 375 Again : take A for the centre of moments ; then T being the force in the direction BE, we have Ta-Q .2a sin 60°=0, /. T=2Q sin 60', as at p. 370. The last of the foregoing three equations of condition necessarily has place whether the resultant R be 0, or, the resultant being finite, if r be 0; all that we can infer .-. from that condition alone, is, either that E is 0, and .-. the system completely at rest, or that the system moves, without rotation, in the direction of AC, C being the centre of moments: but if the same condition equally hold for a different centre of moments C\ out of the direction of AC, then there can be neither progressive nor rotatory motion, the latter being forbidden by either one of the conditions holding, and the former because there are not two directions for R. If, therefore the third condition [4] hold for three centres of moment C, C , C", not in the same straight line, since one of these, at least, must be out of the direction of R, whatever this direction might be supposed to be, the system must be completely at rest. Hence, instead of the three conditions [4], if the third alone holds for three distinct points C, C\ G", not in the same straight line, there must be complete equi- librium. And if it hold for two points not in the same straight line as the point where the forces meet, it will be enough to establish complete equilibrium. 416. Parallel Forces acting in a Plane. — Hitherto we have treated only of forces whose lines of direction all meet in a point ; we have now to consider parallel forces, or such as applied to different points of a rigid system have their lines of direction parallel to one another. Let Pj, Po, be two parallel forces, and let AB be any straight line con- necting the lines AF^, BP^, along which they act. The object is to find the resultant of P^, P^, that is, to determine the intensity and direction, as also the point of application in ^P of a single force equivalent to the two. To the extremities of AB, conceived to be rigid, let any two equal, but opposite forces (Mj, MJ, be applied, the one acting from A C ' towards AM^, the prolongation 7!i^ <" y^ — >TO2 of BA, and the other towards Pilf 2 ; these, as they destroy each other, can have no effect on the system. Let the force P^ be the resultant of the two now acting at A, and R^ the resultant of those acting at B : our two parallel forces will thus be replaced by two obhque forces Pj, P^, having the same effect on the system. Let C be the point in which the directions of these meet, and to which, the lines being rigid, P^, Pg, may be transferred. Resolve them, when thus trans- ferred, into their original components m^, m^, equal to each other and to JWj, Mg, acting in opposition, and in a line parallel to AB, and Pj, Pg, acting along CO, parallel to AP^, BF^ ; the system of forces is thus again reduced to two : the two conspiring forces P„ P^, both applied in the direction CO, or at in the line AB ; the resultant .-.of the original 376 PARALLEL FORCES ACTING IN A PLANE. parallel forces is equal in magnitude to their sum, that is, B^P^+P^, and is the point in ^5 to which this force must be applied to produce the same effect. Its situation is determined thus : — The three forces 3fi, P,, i^,, are proportional to the three sides of the triangle AOC, and the three forces at B, proportional to the sides of the triangle BOG. P, CO . P, CO . ^^ ^ P, OB ^^^ '-'Mro-A^ ^^^ wroB' ^-*^-^^ •'•p=0A w- Hence the resultant of two parallel forces is itself parallel to those forces, and equal to their sum, and its direction divides the line A B between them into two parts inversely proportional to their intensities, that is, P^-\-P^=R, and P^ : P^ \ : OB \ OA [2]. The forces P^, Pg, are here supposed to act both on the same side of AB^ if Pj act on one side and Pg on the other, as in the annexed diagram, similar reason- ing applies ; but will here be in the pro- longation of AB, and P,, P.,, will of course take opposite signs, so that we shall have P^—Pi=B. Consequently when the two parallel forces are equal, and act in opposite directions P=0, and the point 0, which in other cases would be the point of application of the resultant, is infinitely distant, for AB^ then bisects the angle at A. Rud BB., bisects the equal angle at B, so that BC is parallel to AC, that is, C is infinitely remote, and .-. 0, in the line bisecting the angle at G, is infinitely remote. Or thus, since [I] Pa OA OB+BA , ^^ ., „ „ , , BA Hence no single finite force, acting at a finite distance, can be equivalent to two equal and opposite parallel forces : we see, indeed, that the forces y^j, i?2' ^y ^vhich these have been replaced, are themselves also two equal and parallel forces. When, however, the forces are unequal, then, since the proportion [2] involves their magnitudes only, independently of their directions, we see that however the directions of the same pair of parallel forces may vary, the point of application 0, of their resultant, is invariable. From what has now been proved, it follows that any two unequal parallel forces can be counterbalanced by a single third parallel force, determinable in magnitude and in point of application: but that no single force can ever counterbalance two equal parallel forces, if they act in opposite di- rections. A pair of such forces is called a statical couple : we infer, therefore, that a rigid system cannot be kept in equilibrium by any three forces, two of which constitute a couple. 417. In all the preceding articles the directions of the forces in each case have been regarded either as all uniting in a single point — which we have seen may be considered as the point to which their conjoint action is applied — or else as being parallel, and the forces themselves acting at PARALLEL FORCES ACTING ON A RIGID BODY, ETC. 377 necessarily different points. These points, however, may be understood to be points in some rigid or solid body, or points connected together in one rigid system : this condition of rigidity is necessary, in order that the forces may not alter the form of the figure with which the points are con- nected. But the body, or the rigid framework, thus acted upon, must always be regarded as destitute of weight; for weight itself is a force, and in determining the condition of a body as to rest or motion, all the forces, to which that condition is due, must of course be taken into account. It is easy to see that, in the problems hitherto given, this could not have been done ; for every particle of a material body has weight, or is acted upon by force, and as yet we have treated only of forces acting in one plane, and if not at a single point in that plane, yet (as in last article) at only two points, separated by an interval. In some of the problems ad- verted to, we have indeed employed weights, but this has been only for the purpose of transmitting the forces of those weights, through the inter- vention of strings, to certain points : the strings themselves we have been obliged to consider as without weight, so as to pass from one point to another without flexure, that is, in a perfectly straight direction, which no material string, unless it hang vertically, can do (p. 371), because it has weight. We shall now consider bodies as we really find them at the surface of the earth, that is, as an aggregate of heavy particles, cohering together in one solid mass; confining our attention, at present, to that force alone which, acting upon the constituent particles, causes what we call weight. 418. Parallel Forces acting on a Rigid Body: Centre of Gravity. — Experience shows that all bodies, within our reach, tend towards the earth, to the surface of which, if abandoned to themselves, or left unsupported, they would fall, in a vertical direction. The reason why smoke and vapour do not, in general, thus fall, is that they are not left unsupported, being borne up by the air, in a similar way that a piece of wood is borne up by the water in a vessel. The force which thus solicits all bodies towards the earth, we call the force of terrestrial attraction — or the force of gravitation — or simply Gravity : it is a force that acts upon all bodies alike, so that if a piece of cork, and a lump of gold, be both let fall at the same time, from the same height, in a place exhausted of air, they will keep side by side, during the whole of the descent, and will both reach the surface of the earth at the same instant. This fact has been proved by many experiments, and justifies the conclusion that each con- stituent particle of every body is equally acted upon by the force under consideration. If, as in gold, the particles are more closely packed together than in a lighter substance, as cork, more force of attraction must be expended upon the gold than upon the cork, to keep them together during their descent, when simultaneously dropped from the same height, because a greater number of particles are to be brought down : — it is owing to this greater number of particles that a lump of gold would strike an object opposed to its descent with greater force than an equal bulk of cork, and that the pressure of the gold on any support is greater than that of the cork. The line of direction of the force of gravity, acting on a particle, is the line from that particle to the centre of the earth ; at least, its deviation from this direction (arising from the earth not being a perfect sphere) is 878 PARALLEL FORCES ACTING ON A RIGID BODY, ETC. insensible. The distance, too, of this centre, from the surface is so great, and, comparatively with the bulk of the earth, the bodies which we have to consider in mechanics so small, that the lines from their several par- ticles, thus in reality converging to the earth's centre, differ in their directions insensibly from parallelism : we have only therefore, in ex- amining into the effects of gravity on a body, to regard the particles of that body as acted upon by so many parallel forces, all operating in the same direction. LetPj, Pg, P3, &c., be parallel forces applied at the points A^, A.-^, A,^, &c., of a rigid body: it is required to determine the resultant of these forces, or its contrary, namely, the single force which, acting in the opposite direction, would prevent progressive motion. The most obvious way of finding the re- sultant is this : — first find the point of applica- ^ ->^ tion in the line A^A^, of the resultant of two of /" ^ the forces P,, Pg, that is, divide AiA.^ in a, so / /jt^^-r^^\ that P^: P^\ : A^a : Aya (416) : then a single / /\\I(^ \ ) force z=P^^\-P^, applied at a, will be equiva- / ^/ "^z A ' / lent to the two forces P,, Pg, and may be sub- f ''j . y/ ; j / stituted for them. Imagine, then, that the \^^^| "^ '/^^^ forces P„ Pg? ^^^6 withdrawn, and that the force t^-*-^_j-H|'^^ 1 Pi + Pg, acting at fl^, in the same parallel direc- J 11^ tion, is introduced : join a, A.^, and find in like ' p 1 manner the point of application h, in this join- * ' ing line, of the single parallel force equivalent to Pj-f-Pg acting at a, and P.^ acting at A.^: then the single force PiH-Pg + ^.s' acting at fc, may replace the three forces Pj, P^, P3. Pro- ceeding in this manner, all the forces will at length be compounded into a single resultant, and the point of application (as G) of this resultant will become known : this point G is called the centre of the system of parallel forces : it is the point to which, if the united energies of all the individual forces were applied, the effect on the body would be the same. And it is plain, that however the directions of all the parallel forces be changed, provided only that these directions continue to be parallel, and the intensities of the forces remain unaltered, the position of the centre will continue undisturbed ; for in determining this centre, the magnitudes and points of application, and not the directions of the forces, alone enter into consideration. If the forces P^, Pg, &c., be those of gravity, or in other words if they represent the weights pulling A^, A^, &c., downwards in a vertical direction, then G is the centre of gravity of A^, A^, &c., regarded as so many heavy particles, having the respective weights IFj, W^, &c., replacing Pj, Pg, by these symbols for weights ; and it is the point into which all the individual weights may be considered to be concentrated, for as just seen, it is the point upon which, if all the weights W^^\-W^^\-... act, the effect is the same, and it is obvious, from the way in which this point is determined, that it is the only point of which the same can be said. The figure, roughly sketched above, is intended for the outline of any solid body, every particle of which has weight; A^, A^, &c., are merely detached points in that body, at which points alone we have assumed weights to act : but suppose that the point tI, is the centre of gravity of one portion of the body, A^ the centre of gravity of a second portion, and so on, A^ being the centre of gravity of the nth or last portion ; then it is plain THE CENTRE OF GRAVITY. 379 that G, determined as above, would be the centre of gravity of the body: the following problems are thus suggested. Prob. I. To find the centre of gravity of a material straight line : — of a metal rod, for instance, so slender that its thickness may be left out of consideration. This physical line is composed of weighty particles, one of which is at its middle M ; and pairs of particles, one of each pair on one side of M, and one, at an equal distance, on the other side, make up the line of particles: hence, whichever of these pairs we take, the re- sultant of their weights must pass through M: consequently M is the centre of gravity of the entire line, which will balance about this centre in whatever position it be placed (p. 378). Prob. II. To find the centre of gravity of a thin triangular plate. Let ABC be the triangle : draw AD bisecting BC, and hdc, pai'allel to BG: then, by similar triangles we have Ad'.AD::dc\ DC, and Ad:AD :\ld\ BD, but BD=DC, hence hd=dc, and .-. the line be will balance about d in all posi- tions ; and similarly will every other parallel to BC balance about the point where AD cuts it : and since the whole triangle is made up of these lines of particles, the whole -must balance about the line AD, which therefore passes through the centre of gravity of the triangle. In like manner it may be shown that the triangle will also balance about BE, the line bisecting another side AC; that is, that the centre of gravity lies in BE, as well as in AD : hence G, where the two bisectors of the sides intersect, is the centre of gravity of the triangle ABC. It was shown at (p. 298) that DG=-AD, and EG=-BE, so that we o o have only to draw from one of the vertices of the triangle, a straight line to the middle of the opposite side, and to mark a point G on this line at one-third of the whole length of it from that middle, to get the centre required. By dividing a polygon into its component triangles by means of di- agonals, we may find the centre of gravity of each triangle, as above ; and thence the centre of gravity of the quadrilateral made up of two of the triangles, thence that of the pentagon made up of three, and so on. Prob. III. Three equal weights are suspended by strings at the three vertices of a triangle : whereabouts in the plane of this triangle must an opposing force be applied to keep it at rest ? The weights may be regarded as heavy particles placed at A, B, G. Bisect AB Sit D; then will D be the centre of gravity of A, B, so that the effect, as far as these two weights are con- cerned, is the same as if both were removed and placed at D. Join DC: then it remains only to find the centre of gravity of 2 A and C, that is, to divide CD, so that CG : GD : :^ : I; hence CG^^DG, or DG=~CD, so that the 3 centre of gravity of three equal weights at the vertices of a triangle is the same point as the centre of gravity of the triangle itself: this centre 380 THE CENTRE OF GRAVITY. .-. remains unchanged, whether the vertices be equally loaded, or not loaded at all. Prob, IV. To find the centre of gravity of a thin plate in the form of a parallelogram. Draw the diagonals AC, BD, intersecting in G: then the centre of gravity of the triangle ABV is at g, where Gg=-GA. In like man- 3 ner the centre of gravity of BCD is at /, where Gg'=-GC; and GC=^GA: hence the equal weights of the two equal o triangles may be considered to be concentrated in the points g, g\ equi- distant from G, which is .*. the centre of gravity of the parallelogram. Prob. V. A triangle is divided into two equal parts by a line he parallel to its base BC : required the centre of gravity of the trapezoid Be. Let G, g\ be the centres of gravity of the whole triangle, and of the triangle cut off; and let g be the centre of gravity sought. The weight of the whole acts at G, and the weights of the two equal parts act at g\ g: hence the weight at G is the resultant of two equal weights at /, g, .*. g\ g, are equidistant from G, and g is in the same straight line with g'Gy .'. Ag-AG=AG-Ag\ .: Ag=2AG-Ag'=^{2AD-Ad). o Now the similar and equal triangles ABD, and Abdy being as the squares of their like sides, we have Aiy'-.A^-.-.^-.l, ,. Ad=^^, ,. Ag=l(2-±)AI>=.l(2-l^2)AI>. Prob. VI. A body of known weight, and of which the centre of gravity is known, rests upon three points: required the pressure upon these points. Let A, B, C, be the points of support, and G the centre of gravity of the body: then the weight W will act in the vertical Gg, g being the point where this vertical meets the plane of the triangle ABC. Now P,, Pj, P3, being the pressures (or weights) at A, B, C, we have P,+P2 + P3=Tr: and the two pressures at C, B, are equivalent to a single pressure at some point D, such that P^ • BD^B^ . CD. In like manner, the single pressure P2 + P3 now supposed at D. and that at A, are equivalent to the single pressure Pj + Pa + P.,, or TF, acting at some point in the line AD : but the resultant of all the pressures is in the vertical through g : hence AgD is a straight line, and F, . Ag = {P^ + Ps)Dg. Consequently, the sides of the triangle ABC being given, as also the point g, .'. CD, BD, Ag, and Dg, are known or determinable, and we have the three simple equations just given to find the values of the three unknowns Pj, Pj, P3. THE CENTRE OF GRAVITY. 381 Prob. VII. To find the centre of gravity of a triangular pyramid. Let ABC be the base and the vertex of the pyramid. Bisect a side BC of the base in D; draw AD, OD, and make DE=z-DA, arid DF=-DO ; then E is the centre of gravity of o the triangle ABC, and (by similar triangles) the centre of gravity e, of any section abc parallel to ABC, is in the line OE. Similarly : the centre of gravity of every section parallel to the triangle OBC is in the line AF: hence, conceiving the pyramid to be made up of thin triangles parallel to ABC, the centre of gravity of the pyramid must be in OE. Similarly : conceiving it to be made up of thin triangles parallel to OBC, the centre of gravity must be in AF: hence the inter- section G of these lines must be the centre of gravity sought. The distance OG may be found thus : — DE DF 1 Draw EF, then - , — - are each =-, .-. DE .EA :: JDF : FO, LA JbO 2 .-. EF is parallel to AO (Euc. 2. VI.), .'. OGA, EGF, are similar triangles, EG EF DE 1 OG+EG ,.14^^^. OG 2, ^^ 3^., .*. — =1 — =: — =-. .'. ■ =:1-^ — =-. that IS, — =-, .*. OG=-OE. OG OA DA 3' OG ^3 3' ' OE 4' 4 Hence, having found the centre of gravity of one of the triangular faces of the pyramid, let a line be drawn to this centre from the opposite vertex of the solid : the point G in this line, which is three-fourths of its length from the vertex, or one-fourth of its length from the opposite face, will be the centre of gravity of the pyramid. It is obvious that if a plane parallel to the base and distant from it =- the height of the pyramid, cut the solid, it will pass through the centre of gravity. If the base be any polygon, we may by drawing diagonals divide it into triangles, and the polygonal pyramid may be regarded as made up of the pyramids on these triangular bases : a plane parallel to the base, and at the distance from it of - the height, will pass through the centre of gravity of each triangular pyramid, and therefore through the centre of gravity of the combination. Now this centre of gravity of the polygonal pyramid must lie in the line from the vertex, which passes through the centre of gravity of the polygonal base; for the pyramid, as in the former case, may be regarded as made up of thin slices parallel to the base : hence the centre of gravity of the pyramid is that point in the line joining the vertex with the centre of gravity g, of the base, which is at one-fourth the length of that line from g ; so that the height of G above the base is - the height of the pyramid, whatever rectilinear figure the base may be, and however numerous the sides of it : hence what is here established for the pyramid, equally applies to a cone, of whatever base. 882 the centee of gravity. Examples for Exercise. (1) Two ■weiglits of 5 lb. and 7 lb. are attached to tbe ends of a rod without weight, 6 feet long : required their centre of gravity ; and also how far towards the smaller weight the centre will shift, if an additional 1 lb. be added to that weight. (2) If the three sides of a triangle ABC^ be bisected in the points i>, E, F, prove that the centre of gravity of the perimeter of the triangle will be the centre of the circle inscribed in the triangle DEF. (3) Three weights of 1 lb., 2 lb., and 31b., are applied to the vertices of a triangle, each side of which is a : find the distances of the centre of gravity of these weights from each vertex. (4) What must be the altitude of a triangle which, when cut out of a square one side of which (a) is the base of the triangle, may be such that the vertex of the triangle shall be the centre of gravity of what remains of the square? (5) From a circle whose radius is r, the smaller circle described upon r as a diameter is taken away : at what distance is the centre of gravity of the remaining part from the centre of the original circle ? (6) If weights proportional to the lengths of the opposite sides are placed at the vertices of a triangle : prove that their centre of gi*avity will be the centre of the inscribed circle. 419. From the nature of the centre of gravity— the point in which the entire weight of the body may be regarded as concentrated — any point beneath the body, in the vertical line from this centre, may be taken as a fixed point of support on which, the line being rigid, the body will rest ; but it will not rest on any single point out of this line of direction : in like manner, any point of the vertical through the centre of gravity, above the body, may be taken as a point of suspension, from which the body will hang at rest. If in the former case the body be disturbed ever so slightly, so as by displacing the position of the centre of gravity to bring it into a new vertical, then, since in this new vertical there is no point of support, the body must overset : if, however, instead of being supported on a point, the body stand on an area or base, it will not upset so long as the vertical from the centre of gravity does not fall beyond the limits of the base ; for at whatever point within these limits the vertical falls, that point is a point of support in the line of direction. If instead of the body resting on an area, it be supported only on points, which being connected by straight lines would inclose an area, it is just the same ; for in falling, the centre of gravity would proceed down the vertical, and motion in this direction is as much prevented by the isolated points, or legs, as if the area were filled up. If we incline the body over so as to move the vertical from its original position, till at length it just passes through the edge or point on which the body is then supported in the inclined position, it will rest in that position ; but the slightest dis- turbance will either upset it, or cause it to return to its original position : this is said to be a state of unstable equilibrium ; but when the state ia such that a slight disturbance cannot upset the body, which recovers its first position when the disturbing cause is removed, it is in a position of stable equilibrium. But if when disturbed, the body rests in its new posi- tion as firmly as in its original position, that original position was one of neutral equilibrium : a cylinder, on its side, or a rolling stone, furnishes an instance of this kind of equilibrium. GENERAL EQUATIONS OF EQUILIBRIUM OF PARALLEL FORCES. 383 420. In all the preceding problems, except in that one concerning the pyramid, the bodies have been regarded as merely thin plates, discs, or lamince of matter, of insensible thickness : they have been so regarded in order that the point in each, which is the centre of gravity, might be exhibited. But when, as is usually the case, the centre of gravity of the body is within the solid, that point is practically inaccessible ; and it is but seldom that more than the vertical which passes through it is re- quired to be found. In the case of the plates adverted to, it is evident that this line of direction, when the surface is horizontal, will be the same whether the plate be thin or thick; as a thick plate may be re- garded as made up of equal horizontal layers, in each of which the centre of gravity is in the same vertical : the horizontal plane, which divides the thick plate into equal plates of half the thickness, will evidently cut the vertical in the point which is its centre of gravity. It is plain, too, that the centre of a circle is equally the centre of gravity of the circle itself, and of a rim of that circle : that the centre of a sphere is also its centre of gravity, as likewise the centre of gravity of a spherical shell of uniform thickness. In fact, whatever be the shape of the body, if it only have a centre of figure, that is, a point which equally divides every line passing through it that is limited in length by the exterior boundary of the figure, that point must be its centre of gravity ; because, being the common centre of gravity of each of these diametral lines of particles, it must be the centre of gravity of all. We shall now investigate the general equations of equilibrium for any system of parallel forces. 421. General Equations of Equilibrium of Parallel Forces. — For convenience, and as including all the most useful cases, we shall suppose the forces to act vertically, like gravity, upon so many distinct points or particles. The directions of these forces pierce the horizontal plane in points at which the forces themselves may be re- garded as applied, the lines of direction, connecting these projections of the points with the points themselves, being assumed to be rigid. Let AX, AY, be two rectangular axes in the horizontal plane — the plane of the paper, suppose — and let A^, A^, A,^, &c., be the points in this plane to which the vertical forces P^, Pg, Pg, &c., may be re- garded as applied : the co-ordinates of the points ^p A^, &c., will be represented by (^1' 2/i). (^2' 2/2)' <^c. Now if C (Zj, rj be the centre of gravity of the forces P^, P^^, acting at A^, A^j we know (p. 376) that but A^C :A^0:: x^-X^ : X,-a;,, x^-X, : X.-x, : : P, : P„ .-. {P,+P,)X,=P,x,-^P,x,, .'. Z,= Proceeding in a similar way with the force (Pi + P^), now supposed to act at C, and the force P.„ and calling the centre of gravity of these (Z^, Y.J, we have, in like manner, 884 GENERAL EQUATIONS OF EQUILIBRIUM OF PARALLEL FORCES. (P,+Pa+P3)Z2=(Pj+P2)Zi+P3a?3» =PiX,+P^x^-\-P^x^, by substitution, Xo=- P, + P,+P3 And continuing this process till we have included the last force P„, and putting (X, Y) for the centre of gravity thus determined of the entire system, we have the following general expres-sions for the co-ordiuates of the point where the resultant of that system meets the horizontal plane, the second expression being got by merely substituting the axis of y for that of a:. p,x,^-P,x^+P,x,+...+PnXn y_ P{yx+P^y,+P^yz-^ ■■■+P^,yn p^i ^- P, + P,-^P3+...+P, ' P. + P,+P3+... + P„ •••'--'' the denominator of each fraction being i?, the resultant of the system. If the forces equilibrate i?=0, .-. i?Z=0, and Er=0, .-. using the notation at p. 374 the equations of equilibrium are 2(P)=0, X(Pa:)=0, 2(P?/)=0 [2], where i:{Px) expresses the sum of the moments of the forces with re- spect to the axis AY, and ^(Pi/) the sum of the moments with respect to AX. Or, if we do not include the resultant E, acting in the opposite direction under 2(P), these equations are then ^{P)=R, t{Px)=RX, ^{Py)=RY [3], where X, F, as above, are the co-ordinates of the point where the re- sultant, or line of direction of the centre of gravity, pierces the horizontal plane. It is obvious that although in the above reasoning we have regarded the forces Pj, Po, &c., as those of gravity, the lines of action being vertical, yet everything would remain the same if the lines of action of the parallel forces were other than vertical, the plane of projection being, as above, perpendicular to the lines of action, though not then horizontal. Those of the forces which tend to move the body in opposite directions, take, of course, opposite signs, in the foregoing general equations. If the points of application, instead of being projected upon a hori- zontal plane, had been projected upon a vertical plane, a plane to which the directions of the forces are all parallel, as, for instance, on the vertical plane passing through ^Fand perpendicular to AX, what above is the x of either of the points A^, A.^, &c., would evidently be the perp. distance of the particle itself from this vertical plane : this perp. distance, multi- plied by the force, is the moment of that force in reference to the plane, so that the sura of these moments would be equal to the moment of the resultant in reference to the same plane. When a force equal and oppo- site to this resultant is introduced, that is, when the body is in equi- librium, the sum of the moments — taking the moment of the opposing force with opposite sign — is therefore zero. Should the plane pass through the body, so that particles are on each side of it, the moments on one side must of course be taken with opposite signs to those on the other. If the sum on one side be equal to the sum on the other, the moment of the resultant must be zero, that is, the direction of the re- sultant must be in the vertical plane. An experimental way of deter- mining the centre of gravity is thus suggested. If the body be placed on a horizontal table, and be pushed more and more over the edge till it EQUATIONS OF EQUILIBRIUM OF PARALLEL FORCES, 385 just balances itself, the centre of gravity will be in the vertical plane through the edge of the table: by help of this edge a line may be marked round the body ; then turning the body into another position, we may, in like manner, mark on its surface a line through which a second plane will also pass through the centre of gravity : this centre will therefore be in the straight line joining the two points where the lines marked round the body intersect. If it be suspended by a string at either of these points, and then left to itself, it will hang at rest without any oscillation. If it be suspended at any other point, it will turn round till the centre of gravity comes into the vertical line under the point of suspension. 422. Returning now to the equations [1], [2], it is plain that if the points P,, F21 &c., all lie in a straight line, this line may be taken for the axis of X, and we shall then have F=0, the first of [1] giving the x of the centre of gravity, and the conditions [2] of equilibrium being 2:(P)=0, l(P^)=0. As in the former investigations, the parallel forces are sup- posed to act upon particles of the body separated by an interval ; but as there is no limit to the minuteness of this interval, the particles may be regarded as actually in contact, and forming in their aggregate a solid body: the number of terms in numerator and denominator of each of the fractions [1] is then infinite, as gravity acts upon each particle ; but the sum of these terms is in each case finite ; the denominator being the finite body itself, and the numerator being also finite, inasmuch as X and Y are finite. But to determine the finite expression for the numerator, consisting as it does of an infinite number of elements (like a decreasing geometrical series), requires the aid of the Integral Calculus, under which head the subject will be resumed. We shall here, however, give one or two applications of the preceding theory to cases in which the numerators above referred to, consisting of a finite number of terms, common algebra sufiices for their discussion. (1) Required the centre of gravity of the perimeter of any polygon, the sides of which are heavy, uniform, slender bars. Let the weights of the n sides of the polygon be TFj, W^, ...U^n, re- spectively, and let the vertices be the points \x^, y^), {x^, i/J, ...(x^, y„): then the co-ordinates of the middle points will be 1111 -(a;,+irj), -(y^+y^ ; ^(x^+x^, ■^(jt/^-\-y^ ; &c., which points are the centres of gravity of the several sides, and .'. the weights of the bars may be regarded as concentrated there, each weight in the centre of the bar having that weight : hence [ 1 ], 2{W^-\-W,+ ...-\-W„) 2{W,-j-W,-^...-\-Wn) It is plain that it is quite optional whether, as here, we represent the uniform bars by their weights, or by their lengths ; since, being uniform, their lengths are proportional to their weights : — the ratio l^^,H-(PFi + TF2+... + TF„) is the same as the ratio l^^{l^ + l^-^...^l,^), where I is put for length: for if Wi-\-W.-^+...-\-W„ is m times TF"„ then of course the length of the n bars, united in one, must be m times the length Zi of the bar weighing IFp Hence the lengths or weights of the c 386 EQUATIONS OF EQUILIBRIUM OF PARALLEL FORCES. sides being known, and the situations of the vertices, the position {X, F), of the centre of gravity, is given by the above expressions, putting Z,, l^, &c., for TF„ TFg, &c. If the polygon be regular, it is obvious that the centre of the figure, that is, the centre of the inscribed circle, will be the centre of gravity, and Wi, W^, ...TF„ will all be equal, so that then r J(^'+y'+^-+y'\ ... nT=y,+y,+ ...+y^ which expresses this geometrical property, namely : — If from the n ver- tices of a regular polygon perpendiculars to any straight line in its plane be drawn, the sum of them all will be equal to n times the perp. from the centre of the polygon. Instead of perpendiculars, it is clear that the lines from the vertices may be all inclined at any constant angle to the given straight line ; for each of these parallels, multiplied by the sine of the constant angle, will give the corresponding perpendicular. Suppose the polygon to consist of but three bars l^, l^, l.^, forming a triangle, and let the base Z.^ be taken for the axis of x, the axis of y originating at its extremity: then 0:^=0, yi=0 ; also for the other ex- tremity of the base, x.y=.l.^, ^3=0, the vertex being {x^, y^. Then we have If the three bars be equal in length (=y, these become X=-L, Y=-y^^ because then x^ is half the base, that is, x.^^=-l^, so that the centre of gravity of the perimeter of an equilateral triangle is the same point as the centre of gravity of the triangle itself, that centre being the centre of figure, as noticed above. (2) A body of given weight W, and whose centre of gravity is also given {X, Y), rests on three given points [x^, yj, {x^, y^, (a?3, y.^, on a horizontal plane : required the pressures on those points. (See Prob. VI., p. 380.) Calling the pressures P,, P^, Pg, the conditions [3] for determining them are P^+P^J^P-W, P,x,+P^^^P^,= WX, P,y,+P^,-^P^,= WT; and when the given numerical values are put for all the quantities except Pp Pg' ^3' these may he easily found from the three simple equations here represented. But if the question had been to find the pressures upon four, or a greater number of points, it could not have been answered, without additional equations, since three equations are insuf- ficient for the determination of four or more unknown quantities. [In reference to the above fact, writers sometimes say that the pressures upon the four legs of a rectangular table — the centre of gravity of which is in the middle — are indeterminate ; but this is a mistake : the pressures on the legs are equal, each being one-fourth of the entire weight. If any fourth condition be introduced in reference to the pressures upon four points, the problem is always determinate.] (3) Two isosceles triangles ABC, ABC, are on the same side of the same base : required the centre of gravity of the portion left of the larger, when the smaller is removed. Draw CC'M, which will be a straight line bisecting AB at right angles. ANALYTICAL CONDITIONS OP EQUILIBRIUM IN GENERAL. 387 Let y^=Mg^, be the distance from M of the centre of gravity of the smaller triangle, Y=MG, the distance of the centre of gravity of the larger, and y—Mg, the distance of the centre of gravity sought : then TFp TFg, TFg— TFj, being the weights of these re- spectively, we have W Y—Wv Now, putting H, h, for the heights of the two tri- angles, we know (p. 379) that Y=-H, and y^=i--h: o o also the weights of the triangles are as their areas, and these again are as the altitudes H, h, •■ "= — un — =3if=r=3(^^+'''- Hence the distance of g, the centre of gravity sought, is - the sum of the heights of the two triangles. Having now established the general equations of equilibrium for a system of parallel forces, acting upon a body at different points, the way is effectually prepared for the consideration of forces acting at any points, and in any directions whatever, as will be seen in the following articles. 423. Anal3^ical Conditions of Equilibrium in General. — The line of action of every force must always be situated in some vertical plane; for if any vertical plane whatever be conceived to cut that line in a point, the plane, still continuing vertical, by being turned round will at length pass through the entire line, as is obvious. Imagine such a vertical plane passing through the line of action of any oblique force Pj, the point A^ being the point of application of it. This force Pp may be resolved into two directions in that plane, one horizontal and the other vertical ; and the same in reference to any other force, P^, P.^, &c. The vertical components are all necessarily parallel forces : the others, though all horizontal, may take various directions. Now, taking any one of these horizontal forces, we may resolve it into two horizontal directions at right angles to each other — any two rectangular directions we please, the two AX, AY, for instance, in the diagram at p. 383. It thus appears that, whatever be the lines of action of a set of forces applied to different points of a body, those forces can always be replaced by three sets of parallel forces, namely, two sets acting horizontally, one of the two in the direction AX (positively or negatively, as to any in- dividual force), the other in the direction AY, at right angles to AX, and a third set acting vertically, or in the direction AZ, as we may call it, perpendicular to the horizon, or to the plane of the paper in the diagram at p. 383. Each of these sets of parallel forces must equilibrate sepa- rately if the body itself be in equilibrium, because, if they had a re- sultant, it would be impossible that it could be counteracted by any forces, at right angles to its line of action. In thus inferring that each group of forces would alone keep the body at rest, even if the others were withdrawn, it must be observed that the cc 2 388 ANALYTICAL CONDITIONS OF EQUILIBRIUM IN GENERAL. inference is justified only as regards jirogressive motion, the resultant of the forces being zero. In order that there may be no tendency to rota- tion about any given axis, the sum of the moments of the forces, in reference to that axis, must be zero, as at p. 384. Now in the case before us, the moments with respect to ^F are due not only to the vertical forces as at p. 384, but also to the horizontal forces whose lines of action are parallel to AX. In like manner, the moments about AX are supplied both by the vertical forces, and by those whose lines of action are parallel to AY. The forces with which we are now dealing are the components P, cos «,, P^ cos ag? <^c., in the direction AX, the components P, cos /?i, P^ cos /Sg, &c., in the direction AY, and the components P, cos yj, Pg cos y^, &c., in the vertical direction AZ ; for the rectangular component of a force is the product of that force by the cosine of its inclination to the line along which that com- ponent is taken : hence the sum of the moments round AY, as furnished by the vertical group, and the sum furnished by that horizontal group which is parallel to AX, are l(P cos y . x), and 5;(P cos a . Z), where z is the distance from ^Fof the point where the direction of the force Pcosa pierces the vertical plane, just as a; is the distance from AY where the force P cos y pierces the horizontal plane. Regarding P cos a, acting downwards upon the horizontal plane, as taking any sign, and P cos y acting on the vertical plane from right to left, as taking the same sign, the above sets of moments produce opposite effects, as to causing rotation about AY ', %o that the rotation resulting from their joint influence is the difference of the two ; that is, it is SP(x cos y—z cos a), which, therefore, in the case of equilibrium, must be zero. Hence that all tendency to motion in the body, whether progressively in either of the three rect- angular directions, or rotatory round either, may be destroyed, the six following conditions must have place, namely, S(P cos a)=0, X(Pcos i3)=0, S{P cos y)=0 [1], '2P{y cos a—x cos /8)=i0, 'SP{x cos y—z cos a)=0, tP{y cos y—z cos j8)=:0 [2]. If the conditions [1] hold, so that progressive motion may be forbidden, and one of the conditions [2] also, so that rotation about a particular axis may likewise be forbidden, then neither can there be motion about any axis parallel to this. For let a parallel to AZ be drawn, meeting the horizontal plane in any point {a, h); and what, in reference to the original vertical, AZ, was the point (x, y), let now be (a/, i/'), in reference to the new vertical ; that is, \etx=x'-\-a, y=y'-^b : then we shall have S(P cos a)=0, S(P cos /3)=0, SP[{2/'+6) cos a—(x'+a) cos A]=0. Now from the first two of these, 52 (P cos «)=0, and a'S(P cos /3)=:0 ; hence the third gives SP(2/' cos a—ccf cos i3)=:0, which forbids rotation about the new vertical. It follows, therefore, when the foregoing six conditions hold, that there can be no tendency to rotation about either of the axes which have the directions of AX, AY, AZ, wherever these may be situated; in other words, no tendency to rotation is impressed upon the body by either of the three groups of parallel forces into which the original forces have been resolved, and .*. the body must be completely at rest, since these forces alone act on it. When the body to which the forces are applied is not free, but is only at liberty to move round a jixed point, then, taking this point for the origin, the equilibrium will be established if the equa- tions [2] alone be satisfied, since these forbid rotation ; and progression ANALYTICAL CONDITIONS OF EQUILIBRIUM IN GENERAL. 889 is prevented by the fixed point. The pressure on this point must be equal in intensity to the resultant of all the applied forces, since the resistance of the point balances them all, and therefore must be equal in intensity and opposite in direction to the resultant.* It is obvious, too, from this, that when the equations [2] are satisfied, that is, when rotation cannot take place, the forces, however applied, must have but a single re- sultant ; and this resultant would be just the same if all the forces were to be transferred parallel to themselves to any point in its direction and to act there: this is plain from equations [1], which imply no restriction as to the points of application of the forces Pj, P^, but take account only of the directions in which they act. Representing the sum of the pressures upon the fixed point, in the directions AX, AY, AZ, by the symbols X, r, Z, we have [1], X=-2(Pcos«), r=-2(P cos /3), Z=-S(P cosy). When instead of the body being at liberty to move round a fixed point, it is restricted to motion round a fixed axis, or axle, then taking this axis for AZ, all tendency to motion round it will be prevented if the single condition ^P{y cos oc—x cos ^)=0 be satisfied; and the three equations just given will express the pressures sustained by the axle in the three rectangular directions. 424. From the foregoing investigation it appears that though forces, which all applied to a single point, will keep that point at rest, provided they satisfy the conditions []], yet that when they act at different points in a rigidly-connected system, the additional equations [2] must be also satisfied to prevent the rotation which otherwise would be communicated to the system. When, therefore, any of the conditions [2] are unfulfilled, and rotation ensues in consequence, it is impossible that any one of the applied forces, taken in opposite direction, can be the only resultant of all the others, as would be the case if there were no rotation. Besides this resultant there must be a pair of forces forming a couple (p. 376), the in- troduction or suppression of which could not affect the conditions [1], since the pair of forces are equal and opposite: but their existence or non- existence, in the system, determines the question as to whether or not there shall be rotation. If [2] are fulfilled, we may be sure that there is no resultant couple furnished by the applied forces, and that then any one of these forces, taken in the opposite direction, is the resultant and the only resultant of all the others. We shall merely observe, in conclusion, that if R be the resultant (of progression) of the forces Pi, F^, &c., and if its inclination to the three axes AX, AY, AZ, be a, h, c, then, since the opposing forces B cos a, R cos b, Rcos c, being combined with the forces i:(P cos a), Z(P cos /3), 2;(P cos y), would produce equilibrium, by representing these latter by X, F, Z, respectively, we must have Jtcoaa=X, R cos b=Y, R cob c=Z, also R^=X^-\-Y^+Z^ [3], because the square of the diagonal of a right parallelepiped — and R is such a diagonal— is the sum of the squares of the three edges about that * It would be well if, in books on Mechanics, wbat is thus always called the resultant^ were, for the sake of precision, called the resultant of progression, to distinguish it from ilte resultant of rotation, which is a couple. A body not kept in equilibrium may have either a resultant of progression only, a resultant couple only, compelling rotation, or both a resultant of progression, and a resultant couple. 390 PROBLEMS ON EQUILIBRIUM. diagonal. The magnitude of the resultant is thus detenninable from [3], and its direction, fixed by the angles it makes with the three axes, is found from the equations X. Y Z cos a=^, cos 6=77, cos c=— . H K K If the original forces all act in one plane, the plane of the axes AX, AY, iQV instance, then the force Z is zero, and we have X. Y Y ^=Z2-f-F2, cosa=— , cos6=sina=— , .-. tan a=— [41 The moments of all the forces about the axis AZ, or rather, as they are in one plane perp. to that axis, about the origin A, are SP(y cos a— a; cos fi)=R(i/ cos a— a/ cos 6), {a;\ y') being the point of application of the resultant R. Now, p. 309, y cosa—x cos/S=the length of the perp. from the originnpon the direction of P, and y'cosa— a:'cos6= ,, „ ,, ,, Ji. Calling the several perpendiculars upon the directions of the forces /*!, /*2' ^^'* Pii Pz^ <^^'» ^^^ ^^^^ ^po'^ t^® direction of R, r, we have 2(Pp)=Rr, that is, the sum of the moments of the components is equal to the moment of the resultant, as was shown for concurring forces at p. 373. And the equations of equilibrium, R being zero, may be written Z=S(P cos a)=0, r=2(P cos A)=0, S(Pi>)=0 [5]. 425. Problems on Equilibrium*— If a perfectly-smooth surface be pressed upon by the extremity of a rigid line or slender rod, in a di- rection inclined to that surface, the force acting along the rod will cause the end of the rod to slide along the surface : for this force may be de- composed into two forces, one acting perpendicularly to the surface, and the other in the direction of that surface. If there be no friction, that is, if the surface be perfectly smooth, there will be nothing to resist the latter force, and it must therefore take full effect, and the end of the rod will slide: this resolvent, therefore, of the force acting along the rod, meeting with no resistance, it is plain that the resistance of the surface, or the pressure upon the surface, is wholly in a direction perpendicular to that surface. In the following problems the surfaces brought into contact, and pressing against each other, will always be regarded as smooth surfaces; so that when the forces acting upon their points of contact are in equilibrium, those resolvents of which the directions are parallel to, or tangential to, the surfaces, necessarily destroy each other : all pressures and their opposing resistances, in the case of perfect smoothness, are perpendicular to the surfaces pressed. Friction will be taken into account hereafter. In the problems about to be given, the forces concerned in each case will all act in one plane, and gravity will always be one of them. These forces will be replaced by their components acting in directions at right angles to each other — usually in the horizontal and vertical directions. The student will remember that although horizontal and vertical axes may be exhibited in the diagram, the sole intention of them is to show the directions in which the component forces, applied at different points PROBLEMS ON EQUILIBRIUM 391 in the plane of those axes, act : they do not act along the axes, but only parallel to them. Yet the conditions of equilibrium are the very same as they would be if these component forces had their lines of action actually coincident with the axes, and were applied at the point which is their origin. The conditions here spoken of forbid all progressive motion of the body or system: there must be another condition to preclude all motion of rotation — the equation of moments. The centre of moments may be chosen at pleasure in the plane of the forces ; for if the moments balance about any one point, they will balance about every other in that plane : but if there be a fixed point in the system, it is usually preferable to take that for the centre of moments. When unknown forces or pres- sures are to be determined, three equations will, in general, be necessary ; the two equations furnished by the components, and which forbid trans- lation, and the equation of moments furnished by the forces themselves : the components, however, may also be employed for this purpose ; when the position merely of the system is required, one only of the former two equations — the equation between the horizontal components — will usually suffice. [See the Note at p. 371.] Prob. I. A heavy homogeneous rod rests, with its lower end on a hori- zontal plane AB, against a smooth vertical wall AC, and is sustained in its position by a string AD, fastened to it at a given point D : required the tension of the string. Let 2a be the length of the uniform bar, and TF its weight, acting vertically at its centre of gravity G, the middle of the rod BC. Let « be the inclination ABC of the rod to the horizon, and the inclination DAB of the string : then, calling the required tension T, and the resistances (opposite to the pressures) at B, C, P, Pp the vertical forces are W and T sin acting downwards, and P act- ing in the opposite direction. The horizontal forces are Pp and Tcos 6 acting in opposite directions. The mo- ments of the forces W, P, P„ to turn the system about A, are W . AE=W . BE=W . a cos a, and P^.AC= Pj . 2a sin a, in one direction, and P . AB—B . 2a cos a, in the opposite direction : hence the equations of equilibrium are P=W-\-Tsmfi, Pi=Tcos^...[l], and PFacosa+2Piasin«=2Pacosa...[2]. Suppressing the common factor a in the last equation, and substituting the values of P, P^, from the preceding equations, we have TFcos a-{-2T&m a, cos ^=2Prcos a+2Tcos a sin 6, .-, T=W -—^^^^-~r, 2 sin {a—ff) the tension required : the pressures P, P^ are given by the first two equations. If the point D to which the string is fastened be so high up that 9= a, no force acting in the direction of the string will be sufficient to keep the bar from sliding ; for T then becomes oo. If D be still higher up, so that 9>a, the force along the string must act in the oppo- site direction for the equilibrium to be maintained, for T is then negative. Prob. II. A heavy uniform rod BC presses with one extremity against an upright wall, and is supported on a fixed point A, at a given perpen- dicular distance c from the wall : required the pressures, and the inclina- tion of the rod to the horizon. PROBLEMS ON EQUILIBRIUM. Let 2a be the length, W the weight of the rod, acting at its middle point G, and a its inclination to the horizon ; and let the pressures at B, A, be P, P^. Resolving the forces into the rectangular directions AC, AP^, we have P COS a=:W &m a, P^=P ain a -^W coa a [1]. Also for the moments about A, we have b P. A B sin a=W{BO-BA) cos a, that is, P.ABsmii=W{a — ^5)cosa; but AB= , .*. Pc sin a=zW(a cos a—c) cos a [21 cos a ' ■- -• From the first of [1], P=W^^^, .-. [2], IFc^^^=T7(a cosS «-c cos «), "■ ■' cos « COS a ^ ' .9,0 , a (sin* a+C0s2 a)c C / C\l .*. C SVn^ a-\-C COS^ ft=a cos** «, .*. cos" a=: =-, /. COS «=( - J , a a \a/ .: P=WUu .=wl(^i-q\ and fro. [1], P,=^=wQK The value of cos a shows that the equilibrium cannot exist if c>a. Otherwise. Taking the horizontal and vertical components, as also the moments about B, we have P, sin«=P, P, cosa=TF...[l], and PTa cos /8=P, c sec a=P, c •••[2], Pi= W Wa cos «= Wc • -o*. as before. The centre of moments may, of course, be taken at pleasure : thus, we might, if we pleased, have combined the equations [1] of the former solution with the equation [2] in this, and we should have arrived at the same result. Prob. III. A heavy bar AB, moveable in a vertical plane about A, is inclined to the horizon at a given angle a : a rope fastened at B, and pulled by a force at C, -4C being •=AB, keeps the bar in that position: required the intensity of the force at C, and the pressure on the hinge A, the weight of the bar being TF, and its length 2a. Let P represent the force in the direction BC, and let the rectangular components of the unknown resistance B, of the hinge A be represented by X, F ; then resolving the other force P acting at C m these directions, and observing that W acting at G, the middle of AB, opposes the vertical components of the other two forces, we have X=P< il«, r=Tf+Psini«, and Wa cos a-=.P . 2a sin From the last of these we have P=W- 2 sin - « •, the force at C. .*. X=- W cos « cot - 2 ^ r=:Tf+-prcos«, PROBLEMS ON EQUILIBRIUM. 393 .'. :P-\-7^=Ii^=-(^W COS « cot ^«^ +^^^1+2 COS ct\ . The square root of this is the intensity of the pressure on the hinge, and Y tan 0=^ is the tangent of the angle at which its direction is inclined to AX. If the vertical through G he prolonged to meet the line of action of the force at in R, AR will be the direction of the force of resistance at A (413), and RAX is the angle here represented by 0. If from any point in RW, G for instance, a parallel terminating in RA be drawn to BC, the triangle thus formed will be the triangle of forces, the sides being proportional to the equilibrating forces acting in the directions of those sides. Prob. IV. A heavy rod of given length, not uniform in thickness, but whose centre of gravity is known, is placed with its ends on two smooth inclined planes : required the position in which it will rest, and its pres- sures upon the two planes. Let AB be the rod resting on the two smooth surfaces CA, CB, in- clined at angles a, a^, to the horizon ; let G be the centre of gravity of the rod, and put AG=a, GB=zb. Then taking G for the origin of the hori- zontal and vertical axes, let the unknown pressures P, P,, or rather the opposing resistances at A, B, be resolved in these directions, then we have, since the pressures are perp. to the planes, P sin «=Pi sin «„ P cos a+Pi cos it{=W [1]. Also if be the unknown angle at which the rod is inclined to the horizon, we have for the moments yP Y about G, Pa cos {a-i)=Pfi COS («,+0> •*• [1]> P sin a, h cos (a, + ^) P^tin » a cos (« —Sf a sin «, cos a, cos ^- sin «, sin 6 cos «,— sin o, tan 6 6 sin at cos a, cos ^+sin a sin 6 cos a-|-siii « tan 6 ' - ... . J cot a, —a cot a , from wnicn we get tan e=. — 7 , the position of the rod. As to the two pressures P, Pi, we have from [1], Pi=P -: sm «i T» / • I • \ TTT • r» TTT sin a, sin o •, P (sm «i cos «-|-cos «! sin «)=TF sin a,, .*. P=zW- — -—^ .'. Py=zW— sin (a+flsi) sin (a+aj' If the centre of gravity be at the middle of the rod, then a=fe, and the tangent of inclination is tan 0=-(cota,— cot a); but the pressures on the /« planes are the same wherever this centre may be, or whatever be the length of the rod. These pressures, too, would be the same if instead of a rod any other heavy body— a sphere for instance — were to rest between the planes, and have a single point of contact with each. Prob. V. A uniform bent bar ACB is suspended at C, about which point it is free to turn in a vertical plane : weights TFj, PF4, are attached to its ends : to find the position in which it will rest. 394 PROBLEMS ON EQUILIBRIUM. Taking the fixed point C for the centre of moments, it will be sufficient for the equilibrium that the moments about € are equal. The applied forces are the weight TFj acting at A, the weight W^, of the bar CA='ila, acting at its middle point, and the weight W.^, of the bar CB=z^a', acting at its middle point, and lastly the weight Wi acting at B. Put «, for the un- known angle ACpi, for the known angle ACB, then tt— (aj + 6) will be the angle «2 or BCp,. The condition of equilibrium ia Wi . Cpi+W, . C!pa=W^3 • Cp^+W^ . Cp^, where Cpi=2a cos a,, CJpa=a cos «„ Qp^a! cos it.^—a! cos (ai+^, CJp4=2a' cos «a=— 2a' cos (a£+^ : hence the equation of equilibrium is (2PFi+ir> cos «i=-{Tf3H-2TF4)a' cos («,+^) [1]. (a[-j-^) sin a, sin ^— cos a| cos 9 Now since— cos «| -=tan a I sin ^— cos 6, tan ai=- [1] is the same as {Vi/7y^-\-W^(i=.{W^-^1W^aH^ «» sin <^-cos 6) (2Tf(+TF',)a+(TF3-f 2TF,)a'cos ^ T* XI. . 1 X X XT- . -I XI . Pr2a+ Wno! cos ^ K there are no weights at their ends, then tan «■= — „, , . ■ — . PFga' sin 6 And these results remain the same whether the two arms CA, CB, are of equal thickness or not : but if they are equally thick, their weights must W a be as their lengths, so that ^=— , and the last expression may then be W\ a written tan a. =-75 cosec 0+ cot 0. Prob. VI. An oblique cylinder stands on a horizontal plane, to which its axis CD is inclined at angle of 60" ; the perp. height of the cylinder is 4 feet, and the diameter of its base 3 feet : required the diameter of the greatest sphere of the same material, that will hang suspended from the upper edge B\ without overturning the cylinder. The centre of gravity G of the cylinder is at the middle of CD, and the forces tending to turn it about B are the weight of the cylinder in the vertical GE, and the weight of the sphere in the vertical B'B : hence, equating the moments, we have ^^XCyHnder=5PX Sphere, where BEr=.BD-DE, and BP=2DE. Now DB=OE ta.n G=2 tan 30°=-^3, 3 3 2 and BE=~—-^Z : 2 3 ^^=3n/3, hence, substituting the volumes of the homogeneous bodies for their weights, we have (~|N/3)x3-2x7854x4=|v'3x|x7854x(diameter)3, PROBLEMS ON EQUILIBRIUM. 395 , (aiam.)'=^^^=^'^=8-074, .-. diam.=2-006 ft. Prob. VII. A. heavy uniform rod AB has one end in a smooth hemi- spherical bowl of radius r, the other end projecting over the rim, which is in a horizontal plane : required the position of equilibrium of the rod. Let be the inclination of the rod to the horizon, the rod being kept at that inclination — first, by the resistance P&t A acting perpendicularly to the curve surface at A, and .-. in a direction passing through C, the centre ; secondly, by the resistance P, act- ing perp. to the rod at D ; and lastly, by the weight of the rod acting vertically down- wards at G, the middle of AB. Equating the horizontal components of P, P,, we have P cos 2 6=P^ sin 0, .'. 77=-^^- Pi cos 20 And the moments about G' axe P . a sin ^=Pi . OD, .*. -=-= — : — -, Pi a sin 6 Now AD=2r cos 0, .*. 0D=AD—a=2r cos ^— a, a sin cos 26' 2r cos 0—a sin sin fi a Bin cos 2^ 2 cos^^— 1 •. 4r cos- 0—a cos fi—2r=z0, a quadratic from which cos may be easily found. If the pressures P, P„ are required, we may readily obtain them from the two equations furnished by the horizontal and vertical components, viz. P cos 20— Pi sin ^=0, and P sin 20-\-PiCos0=W, the wt. of the rod. Thus, to eliminate Pj, multiply the first by cos 0, the second by sin 9, and add ; we then get W sin ^ W Rin tf P(sin 20 sin ^+cos 20 cos 0)= W sin 0, .-. P=_ll— __.= \ = W tan 0, ^ cos {20—0) cos Pcos2^_ cos 20 .". "i — — : — - — —W — . sm cos If the pressures are equal then the inclination must be 30°, since we must have cos 20= sin 0. It may be further noticed that if the length EF represent the weight of the bar, then will FD represent its pressure on A, since P=W tan 0. Examples for Exercise. (1) Two rafters of a roof form an isosceles triangle, each of the base angles being 0, the weight of each rafter is W : what is the horizontal thrust on the side walls ? (2) Six forces P„ P^ Pg, P^, P^, Pg, act at the centre of a regular hexagon, and are directed towards the six vertices : if the magnitudes of the forces are 1, 2, 3, 4, 6, re- spectively, what will be the magnitude and direction of the resultant ] (3) A rope of given length is to be fixed to an upright pillar : what angle must it make with the horizon in order that a given force P, applied at the other end, may be the most effective in overturning the pillar ? (4) A ladder of uniform thickness, of which the weight is 60 lb., is inclined against an upright wall at an angle 60" with the horizon : the foot is secured from slipping by a wedge : a person weighing 12 stone ascends with a burden of 100 lbs : required the pressure against the wall and the thrust at the bottom when the middle of the ladder is arrived at. 896 THE LEVER. (5) Two ladders of lengths 2a, 2b, respectively, and of weights W, W, are raisod against two opposite vertical walls with their lower ends or feet acting against each other: required the distance between the walls when the ladders are at right angles to each other. (6) A door of weight W hangs npon two pivots or hinges in a vertical line : the distance of the centre of gravity of the door from this vertical is a, and the distance between the pivots is b : required the horizontal pressure or strain upon the pivots. (7) Two spheres of weights W, W, rest on two inclined planes, and press against each other : the inclinations of the planes to the horizon are a, a : it is required to find the inclination 6 of the line joining the centres of the spheres to the horizon. (8) A weight W is suspended from the edge of a hemisphere weighing W, resting on its convex surface on a smooth horizontal plane : required the position of equilibrium, that is, the angle 0, by which the axis of the hemisphere deviates from the vertical, the radius being r, and the distance of the centre of gravity from the centre of the sphere being c. (9) Two weights W, W, connected together by a string of given length, rest on the upper surface of a sphere : if a be put for the angle at the centre, subtended by the arc to which the string is applied : required the position of equilibrium, or the angle at the centre formed by the vertical and the radius drawn to one of the weights W. (10) A thin circular plate is supported on a slender vertical rod at its centre : it is required to find at what distances round its circumference three given weights T'Ty W^ W^ must be placed so that the plate may remain horizontaL TV ^ 426. The Mechanical Powers: Machines in Equili- brium. — The mechanical powers, as they are called, are the constituent parts or elements of which every machine is composed. They are six in number : — 1 . the Lever ; 2. the Wheel and Axle ; 3. the Pulley ; 4. the Inclined Plane; 5. the Wedge ; and 6. the Screw. 427. The Lever, — This is a rigid bar or rod : — in its simplest form it is a straight bar, moveable freely, in a plane, about some fixed point (or axis) of support, which fixed point is called the fulcrum. It is divided into three kinds, accordijag to the position of the fulcrum in reference to the force applied and the resistance to be overcome, or balanced : the applied force is usually called the power. The three kinds of lever are represented in the annexed diagrams. Applied as in fig. 1, the lever is of the first kind, the ful- crum being between the power and the resistance. Applied as in fig. 2, the lever is of the second kind, the resistance being between the power and the fulcrum. And applied as in fig. 3, the lever is of the third kind, the power being between the fulcrum and the resistance. In the diagrams, the bar is straight, but, whatever be its shape, if acting as here represented, it is a lever : a perpendicular from the fixed point or fulcrum, to the direction in which the power acts is the arm of the power : the perpendicular from the fulcrum to the direc- tion in which the resistance acts is the arm of the resistance : the portion of the lever between the fulcrum and either end is called an arm of the lever. There will obviously be equilibrium when the power multiplied by its arm is equal to the resistance multiplied by its arm, that is, when the iL. a THE COMMON BALANCE. 397 moments about the fulcrum are equal : the weight of the lever itself being disregarded. Thus, calling the power P, and the weight or resistance to be balanced TT, the ^^ perpendiculars p, Pi, upon the directions in [^ 7'^' ^^^^> ^/ which the forces P, W, act, will be the arms of ]^i ~n^.^/ the power and weight, and the condition of equi- I / w p , . n librium will be Wpi=Pp, .'. -rr=- * that is: ^ F p^ The power and resistance are to each other inversely as their arms, or the perpendiculars on their lines of direction. If the weight w of the lever itself be taken into consideration, it must be regarded as an additional force acting at the centre of gravity of the bar : calling the perp. distance of the direction of w from the fulcrum g, we must then have Wpi±wg=Pp [1] the upper or lower sign being used according as w favours or opposes W : when g=0, that is, when the centre of gravity is at the point taken for fulcrum, the condition of equilibrium is the same as if the bar had no weight. It will be observed that in the first kind of lever, mechanical advantage is gained or not according as the power is farther from or nearer to the fulcrum : in the second kind, mechanical advantage is always gained : in the third, it is never gained. The human arm is a lever of this last kind : — the elbow-joint is the fulcrum, the muscle is the power acting on the lever (the fore arm) near the fulcrum, and the weight supported by the hand is the resistance : the mechanical disadvantage is more than compensated by the extent of range of which the hand is thus made capable. Besides the rough purposes to which the straight lever of the .first kind is applied to ease manual labour, it is of important use in certain delicate contrivances : — especially in those for weighing. 428. The Common Balance.— The common balance, or a " pair of scales," consists of a lever of the first kind, the fulcrum or axis on which it turns being at the middle of the lever, — called, in this case, the scale-beam. When the extremities of the beam are equally loaded, it should rest on the fulcrum in a horizontal position: the loads act by means of scale-pans, which are suspended from the extremities of the beam, and into which the weights, and the commodity to be weighed, are respectively put. A. good balance must satisfy these three conditions. 1. When loaded with equal weights the extremities of the beam should be in a line perfectly horizontal. 2. When there is any — the slightest — difference between the weights, this straight line should cease to be horizontal. 3. When disturbed, or moved out of its horizontal position, it should rapidly return to its horizontal state of equilibrium upon the disturbance ceasing. The first condition is that of horizontality ; the second, that of sensi- bility : — the third, that of stability. Horizontality. — To secure this the line joining the extremities of the beam must be perpendicular to that which joins the fulcrum and centre of gravity when the beam is at rest ; for this latter line is always vertical. The fulcrum must not be at the centre of gravity, because then the beam 898 THE COMMON BALANCE. would remain at rest in every position (418). That the horizontal position may be retained when the extremities are loaded with equal weights, these extremities must be equidistant from the fulcrum, because the moments of the weights must be equal when there is equilibrium. Sensibility. — To secure the greatest amount of sensibility, that is, to cause the beam to pass from a horizontal to an oblique position when there is the least possible inequality between the weights, it is necessary that the friction at the fulcrum or point of support should be reduced to the very smallest amount. With a view to this, in balances for philosophical purposes, the beam rests upon what is called a knife-edge (v), supported upon horizontal plates of polished agate, and is itself usually made of brass, in order that it may not acquire any magnetic properties that might disturb perfect equilibrium and freedom of motion. It is, moreover, necessary that it be as light as is compatible with strength and rigidity, since the greater the pressure on the fulcrum the greater will be the friction. Let C be the point of the knife-edge round which AB, joining the points to which the scale-pans are suspended, turns. If G be the centre of gravity of the beam and pans, of weight W, when unloaded, the line CG will be vertical, and will retain that position when the pans are equally loaded. But if P be a greater weight than Q, CG, perp. to AB, will be thrust out of the vertical, and the centre of gravity of the whole, when AB takes the inclined position in the diagram, _ will be some point vertically under C. Let AM=MB=a, CG=h, CM—k, and the angle which AB makes with the horizontal line pq, 0. Then the moment of P to turn the system about (7, in one direction is P x Cp, and the moments of Q and W, to turn the system in the opposite direction, are ^x Cq, and WxCw re- spectively : the former moment, therefore, must be equal to the sum of these two moments. Now M being the middle of AB, the vertical mM bisects every line drawn through it, and terminated by the verticals from p, q ; hence m is the middle of pq : moreover, the angle CMD being a right angle, the angle CMm = the angle G=B. .'. Cp=mp—mC=a cos ^—k sin fi, Cq=mq-\-Cm=mp-{-Cni:=a cos 6+ J: sin 6, and Cw=h sin ^: hence the equation of moments is P{a cos ^—Jc sin 6)—Q,{a cos 6-\-lc sin ff)-\-Wh sin 6, .: — — -=- an equation which determines G, the angle of deviation from the horizontal line. The greater this angle is, for the same excess of weight, F—Q, the more is the sensibility of the balance increased ; or the less be the differ- ence P—Q necessary to produce a given deviation 0, the greater is the sensibility of the balance : hence ought to be the greatest possible : P — V it may be regarded as the measure of the sensibility: the condition, there- fore, of greatest sensibility is satisfied when -=r — ~- — t— is as great as ^ ; {P-\-Q)k-tWh ^ THE ROMAN STEELYARD. 399 it can be ; the construction, therefore, should be such as to secure as large a value for this expression as is consistent with convenience and with the other necessary conditions ; Qa, for instance, the distance between the points of suspension should be as long as other considerations will allow ; W the weight of the beam and pans should be as much as possible re- duced, and the smaller the distance k of the knife-edge from the line joining the points of suspension, the more is the insensibility increased, as also by reducing the distance h between C and G. There must, how- ever, be a limit to these reductions, for though it is desirable that a large deviation should accompany a small excess of weight, yet it is necessary that the horizontality be restored when that excess is withdrawn, which, however, it would not be if C and G coincided, and the nearer they are to coincidence the more slowly would the beam return to its horizontal posi- tion : stability requires that this return be rapid. Stability. — In speaking of sensibility we have regarded only the effect of a small inequality of weights in producing a sensible deviation of the beam from its horizontal position, but it is necessary, when perfect equality is restored, that the beam returns to horizontality : this requires that the centre of gravity G be below the fulcrum, as we have represented it to be in the diagram. The greater the distance between these two points the longer will be the line Ow for any given disturbance, and there- fore the greater the moment to bring down the arm MB when the dis- turbing cause is removed : in other words, the greater is the stability ; this, therefore, is opposed to sensibility, since for this the smallness of CG is an advantage. But in the balance of commerce a quick return of the beam to horizontality is of importance, inasmuch as in business trans- actions time is. In the balance for purely philosophical purposes time is of less consequence, and sensibility is the main object aimed at ; the best of these balances are so constructed as to detect a difference of weight of a thousandth part of a grain. The only way of increasing sensibility without disturbing stability is to lengthen the equal arms of the balance. 429. The Roman Steelyard.— This is another application of the lever to the purposes of a balance for weighing commodi- ties. It is simply an iron or a steel bar, moveable about a knife- ^„,,,,„„, ,,„,y„„,„„ edge fulcrum O ; the body P to n|'?'"""o""^'"'F'^"" r^-p^ ^ i be weighed is hung at the ex- ^ -W" tremity of the shorter arm A, and a given weight W is moved along the other arm till it balances P, the weight of which is marked by the figure on the arm OB, at which the weight rests : the graduation of this arm is effected thus : — Let G be the centre of gravity of the bar and hook, and Gg the ver- tical from it : let OA=p, 00=p\, Og=g, and put w for the weight of the bar and hook. Then by the last of equations [5], p. 390, W,,+.,=Pp, ... p,+^=^. Make 0.=f , .-. .C=f . Let now the distance p, or OA, be measured from a? to 1, from 1 to 2, from 2 to 3, and so on, and let each of these intervals be subdivided into 400 THE DANISH BALANCE. xC equal parts; then the number marked at C will be — , which call n, and from the above equation, ?iTF=P, so that the weight of P is found by multiplying the known weight W by the number at which it must be placed to produce equilibrium; if TF be 1 lb., that number will show the number of lbs. that P weighs. When the steelyard is so constructed that the centre of gravity G is in the same vertical as O, then ^=0, .-. O:c=0, so that each division 01, &c., is equal to OA. 430. The Danish Balance.— This differs from the steelyard, in having the counterpoise fixed at one end and the fulcrum moveable. To graduate the beam 1, 2, 3, &c,, ounces are placed successively in the scale-pan, and the cor- responding points 1, 2, 3, &c., at which the fulcrum must be placed to keep the beam horizontal are marked on it ; intermediate weights are then put in, and the places of the fulcrum marked in like manner. Or thus : let w be the weight of the lever and scale-pan, and G their centre of gravity, and let be the place for the fulcrum when W keeps the bar horizontal, then W .AO=w . GO=w(^AG-AO\ .-. AO=-:i^AO, so that G, the point at which the instrument would balance when unloaded, and the weight of it vs, being known, the point 0, at which it would balance when loaded with W^ becomes known. If lo, IT, each express so many ounces, the several distances of the fulcrum from A^ for I oz., 2 oz., &c., will be :; , , &c. The reciprocals of 1 2 these numbers are \-\ — , 1 -{--, &c., which are in arith. prog. Hence the WW distances from A are in harmonic prog. (86). This balance is often used for weighing letters. Problem I. A body is weighed successively in the two scales of a false balance : in the one it balances a weight P, in the other a weight Q : required the true weight. Put a and h for the lengths of the two unequal arms of the balance, and X for the true weight of the body ; then by the problem, ax=.hF^ hx=aQ. Multiplying these together, we have ahx^=abPQ, .'. x=^(PQ) ; hence the true weight is a geom. mean between the false weights. Prob. II. A heavy lever AB, equally thick throughout, the weight per inch being given, has a given weight W sus- pended at a given distance CA from the ful- crum A : what length must the lever have so A that the weight P, to keep it in a horizontal la position, may be the least possible? The lever being equally thick, any portion of its length may be taken to represent the weight of that portion : hence, putting x for i n □ p vv COMBINATION OF LEVERS. 401 p^ the required length AB, its weight will be also denoted by a, acting at (?, its middle point: hence, putting ^i for CA, we have [1], p. 397, 1 Wp 1 Wpi-{-x.-x=:Px, .'. P=—^-\--x=u, a minimum^ Wv 1 .*. (p. 148), aP-2ux+2Wpi=0, .*. a;=w, .*. — ^+-«=a;, .-. PFiJ,=ix2, .-. x=^{2Wp,), where for W the length of lever, weighing PT, is to be put. The same expression, namely, P=x= ^/ {)iWpi) gives the weight of P, which is therefore that of the lever. In each kind of lever the perpendicular pressure on the fulcrum is always equal to the algebraic sum of the other perpendicular forces which keep the lever at rest, because it is the resultant of these other forces, inasmuch as the resistance of the fulcrum, which is equal and opposite to the pressure, balances the other forces. In the case above, the pressure on the fulcrum is W+x—a!=W, that is, it is equal to the suspended weight; P therefore merely supports the lever. 431. Combination of Levers. — The mechanical advantage of the single lever will be considerably increased by a combination of several : thus in the system of levers in the annexed diagram, if we call the "S, arms which are on the same side of the power P, p, p^, &c., and the other arms p\ p/, &c., the powers acting at the extremities of these latter being Pj, P2' <^^'» ^® shall have, in the case of equilibrium, Pp=PiP', PtPi=PaPL, PiP^=P3P^', &C., .'. PpPiP2:.=^Pnp'PiP^.„l or, the last power P„, being the weight W, PpPxP2-Pn=Wp%'p^...p'n [1]. Hence the applied power, multiplied by the product of all the arms nearest to it, will be equal to the resistance balanced, multiplied by the product of all the other arms. If the forces act obliquely, the several arms will be the perpendiculars on their directions from the fulcra. In the preceding diagram the levers are all of the first kind, but the reasoning, and the conclusion [1], remain the same whatever levers are combined, and whether they are in actual contact, or are connected to- gether by links, as in the diagram annexed, where P, C, P, are the three fulcra on which the levers of the second kind AB, Afi, A^D, rest. Put- ting, as before, AB=p, A,C=p,, A^D=p.^, and aB=p\ afi=y\, aJD:=p^, we have the same ex- pression [1] for the relation between the power TfF w and the weight, that is, — : _mP2_ 'p'pM An D D 402 THE WHEEL AND AXLE. Examples for Exercise. (1) A uniform lever is 10 feet long, and -weighs 6 lb. : its longer arm is 7 feet, and at the extremity of the shorter a weight of 21 lb. is placed : what weight must be placed at the end of the longer arm to balance the lever 1 (2) A body weighs 10 lb. 9 oz. in one scale of a false balance, and 12|: lb. in the other : what is the true weight ? (3) Two persons, A, B, of the same height, sustain upon their shoulders a weight of 150 lb., suspended on a pole 8^ feet long: the weight hangs at 3^ feet from A : what is the weight borne by each ? (4) The arms of a bent lever are equal, and P : PF : : 1 : s/% the arm from whose extremity P is suspended, rests parallel to the horizon : required the angle between the two arms. (5) At one extremity J. of a uniform straight lever whose length is a, and weight w, a weight W is suspended : where must the fulcrum C be so that the lever, unloaded in the other part except by its owd weight, may rest parallel to the horizon ? (6) The longer arm of a steelyard is 8 feet, and the shorter 6 inches : it is so con- trived that, with its hooks, &c., a weight of 6 lb. on the longer arm, at 1 foot from the fulcrum, balances 121b. at the end of the shorter arm: the moveable weight is 51b., and it cannot be placed nearer to the fulcrum than 6 inches : what must be the gradua- tion, so that by shifting the moveable weight from one mark to the next an additional half-pound may be weighed ?— and what will be the least and greatest weights that can be ascertained by the machine ? (7) In a lever of the third kind the power makes an angle of 48° with the lever, and acts with a force of 2171b. in that direction, and the vertical resistance is 1001b. : re- quired the oblique strain on the hinge which forms the fulcrum. (8) Three levers act in combination, as in the figure at p. 401 ; the arms nearest to the power P, which is 31b., are 9, 11, and 12 ; and the other arms, 1, 2, and 4 : what weight W will be sustained ? 432. The Wheel and Axle. — ^This machine, which is only a modification of the lever, consists of a cylinder called the axle, and a wheel, firmly connected with it ; the whole being moveable round a common axis perpendicular to the face of the wheel. The more immediate use of this machine is to support or raise a weight W, suspended to a rope, which is wound round the ^ axle by means of the power P applied at the circum- ^-n^ ' n->-^ ference of the wheel. Sometimes P acts through a cord wrapped round the wheel ; in other cases, the actual wheel is dispensed with, and spokes only or radii inserted in the axle are used instead, as in the capstan and windlass. Prob. To find the relation between the power and the weight, when the wheel and axle is in equi- librium. Let the radius of the wheel be B, and that of the axle r: then, since the moment of P about the fixed axis to turn the system in one direction must be equal to the moment of W to turn it in the opposite direction, we have p T>—w ^—L Power _ rad of axle ~ ' •'• W~R' '''' Weight~rad of "^l^hiil ^1J» TOOTHED WHEELS. 403 the power and weight being inversely as their distances from the common fixed axis, as in the lever. The greater R is, r remaining the same, the less will the power be which is requisite for the support of a given weight : when it is necessary that a continually-diminishing power should have a uniform effect upon a constant weight, it must act upon a series of wheels continually increasing in radius, the foregoing proportion being always kept up : thus P, F\ being any two values of the powers, we must have for the corresponding values 7?, R', when r and W are constant, the relations In this way the varying power, exerted by the mainspring of a watch while uncoiling, and thus diminishing in elastic force, is made to produce a uniform effect, the continually-decreasing power acting with continually- increasing leverage, by means of the continually-enlarging wheels cut on the surface of the fusee, as in the diagram. Note. — When the thickness of the rope, round either axle or wheel, is considerable, the radius of the rope should be added to the radius of the circumference round which it is wrapped, the power acting along the axis of the rope. 433. Toothed Wheels. —Wheels of this kind are employed alike in the most delicate as well as in the most ponderous pieces of machinery. The teeth or projections round the edges of the wheels acting on one another, as in the annexed figure, are all placed at equal distances, so that when the train is put in motion the teeth of one may enter the spaces between those of the other. When the axle is toothed, or rather when a small-toothed wheel surrounds the axis, the power being applied to the larger circum- ference surrounding it, the small wheel is called a pinion. Let the radii of the pinions be r,, r^, &c., and those of the wheels R^, R^, &c. : then the power P acting on the first wheel equilibrates a weight or power P at its pinion, expresssed ■D by Pi=—^P ; this, therefore, is the power applied to the second wheel, which power equilibrates a weight or power P^ at the second pinion ex- pressed by This increased weight or power P^ is now applied to the third wheel, and thence acts with further increased effect at the third pinion, and so on : lience the power P„ at the nth pinion is p^^w^&Mi-iiJL-P, •[2], so that P,. is the weight that the power P applied to the first wheel will balance on the n\h pinion. And it is plain that in this expression circum- D D 2 404 THE PULLEY. ferences may be put for radii; and then again for these, the number of teeth on each. If, as in the diagram, the first wheel and the last pinion have no teeth, the radii or circumferences of these must be inserted in the expression, or else the number of teeth they are competent to carry. The expression [2] evidently applies, whatever be the magnitudes of the wheels and pinions, or whether the pinions are enlarged to wheels. If the first wheel were toothed, and acted by these teeth on the second, and in like manner the second on the third, and so on, no mechanical advantage would be gained, since then i?,=r,, R.^=r.^, &g., so that the mechanical effect would be the same as if the first wheel acted im- mediately on the last without the intervention of the other wheels. The same expression equally applies, however far the wheels are asunder, each consecutive pair, when in motion, turning on the same axis. Thus, in the annexed diacjram, the wheel F drives the wheel A, called its follower ; the next driving wheel B, in like manner, turns its follower C, which acts upon the driver D, which finally acts on the axle E. And it may be noticed that when the wheels, in any such train, are put in motion, the sym- bols R, r, in the fraction which multiplies P, in the expression [2], may stand for number of revolutions, instead of for radii : it will then denote the number of revolutions of the last wheel, or of the axle E. The foregoing expression [2], being derived from [1], assumes that the power communicated from the tooth of one wheel to that of another on which it acts is in the direction of a tangent to the circle on which the latter tooth is raised : in other words, that the forces act towards points on the circumference of each wheel in directions at right angles to the radii at those points. This requires that the faces of the teeth which come in contact should be of a peculiar curved form : the curve is what is called the involute of the circle. Examples for Exercise. (1) In the figure at p. 403, the radius of each pinion is 1 inch, and the radius of each wheel 10 inches : how much weight will 1 lb. balance ? (2) What must be the radius of the wheel (fig. at p. 402) in order that 12 lb. acting on it may support 120 lb. suspended by a rope wound round an axle of 6 inches radius ? (3) A weight of 500 lb. is sustained by a rope of 1 in. diameter, going round an axle of 8 in. radius : the radius of the wheel is 4 feet, and the power acts close to its circum- ference : required the weight acting on the axle. (4) A train of four toothed wheels act as in the figure above, and can be put in motion by turning a winch of 12 in. radius : the radii of A, (7, are each 4 in., and the radii of B, D, each 15 in. ; also the radius of the axle E is 2| in. : what power must be applied to the handle P to sustain 600 lb. from a rope, of \ inch radius, wound round the axle ? 434. The Pulley. — This consists of a grooved wheel revolving freely about an axis passing through its centre, the ends of which axis are usually fixed in a frame called the block of the pulley, the grooved circle itself, which turns freely between the sides of this block, being called the THE PULLEY. 405 sheaf of the pulley : the sheaf is grooved to prevent the rope passing round it from slipping off when pulled by opposing forces. Single fixed Pulley.— When the pulley is fixed, the forces P, W, which balance, when acting at the ends of a cord passing over the smooth sheaf, as in the diagram, must be equal, what- ever be the directions in which they pull the flexible cord ; and conversely. For the force at P being communicated without impediment to W, can be balanced only by an equal and opposite force, acting on that point of the cord. And conversely, equal forces acting at P and W must keep every point of the cord at rest, since they act upon each point of it in opposite directions. There is .-. no mechanical advantage in the single fixed pulley, but it enables us to give any direction we please to the force P, without interfering with its effect upon the weight W. 435. Single moveable Pulley. — In the moveable pulley, the weight is fastened to the block, the cord is fastened at one end Q, and the power P is transmitted, through the cord passing over a fixed pulley, to the moveable one : the weight W is thus sustained by the two equal forces at P, Q : that these are equal is plain, because the cord is equally tense throughout its whole course PACDQ, and we have to ascertain what relation these equal forces bear to the weight W which they balance. If CA, DQ are parallel, the forces along CA, DQ, balancing TT, must be each equal to half W (p. 376). If they are not parallel let their directions meet in E, at which point the forces P, Q, may be supposed to act, and as they balance W, E must be the line of action of W (p. 363). This force being equal and opposite to the resultant of the equal forces P, Q, the angle E is bisected by its line of action. If .'. a be half the angle of inclination of the cords, we must have F=2Pcos«, .'?-=-^ [11 W^ 2 cos a which is true whether the sheaf be circular or not. If it be circular, and if radius OC=r, then the perp. Cn will be the sine of to that radius, or the cosine of a to that radius, so that the trig, cosine above is [2], that is, the power is to the weight as the radius of the pulley to the chord of the arc which is in contact with the rope. It appears from [1] that the mechanical advantage is greatest when cos « is the greatest, that is, when a=0, or when the directions of the three forces are all parallel; and that when cos a=n' *^^*» ^^ "^^^^ a=120''. there is no mechanical advantage, for then P=W. Cn • — , and .' r chord CD : 2 cos a= , , P r " '• W~choTdCI> ' 406 THE PULLEY. 436. System of moveable Pulleys.— The advantage gained by a single moveable pulley may be increased to any extent by employing a system of pulleys as in the annexed figures : thus putting t^, t.^, &c., for the tensions of the several ropes, the equation [1] gives the series of equations W=2tiC0Sai, ^1=2^2 COS agj ^2=2*3 003 ^3, ..., tn-l=-2tnC0S an=2P COS an [3]; and taking the product of these, expunging common factors, we have PF=2"P cos «! cos asj- • -COS «». . .[4], where n is the number of moveable pulleys. If the angles are all equal, then l^=2»Pcos"«...[5]; and if the cords round the moveable pulleys are all parallel (second figure), T7=2"P,ori>=^ [6]. Suppose the weights uj^ w^, &c., of the pulleys 1, Q, &c., are taken into account, then F will have to sustain these several weights suspended at 1, 2, &c., in addition to the weight W. By the formula just given, the additional power at F to sustain 1, will be -^, since w is merely a new W: the ad- w w ditional power to sustain 2, will be — ^ ; to sustain 3, — -^, and so on, the additional power requisite to sustain the last moveable pulley «;„ alone, w being -^ : hence the power F requisite for balancing W and all the move- able pulleys, when the cords are parallel, is P=Yn^W+w,+2w,+2hv,+ ...-^2^^~^Wn} [7], weight of pulleys included ; or if the weights of the pulleys be equal, it is ^=^{^+(2"— Ijtu}, since the sum of the geom. series 1+2 + 22 + . ..+2«-i is 2"— 1. If instead of fastening the several ropes to immoveable points as in the cases hitherto considered, each is attached to the weight to be sup- ported, as in the third figure, then, the ropes being parallel, the pressure on the first pulley is 2P, that on the second 4P, and so on, that on the nth being 2"P. Subtracting, therefore, P from the whole pressure on the hook, which is the pressure W-\- P, we have TF=(2»-1)P, .-. P=- W .[8], weight of pulleys neglected. But if we take into account the weights of the pulleys w^, w^, &c., we must observe that, contrary to the last case, these weights relieve P instead of making greater demand on it ; the heavier the moveable pulleys are, the THE INCLINED PLANE. 407 less is the force at P that will sustain W: in the former arrangement the moveable pulleys acted on the side of the weight, here they act on the side of the power. Let w', w'\ w"\ &c., be the portions of W balanced by the pulleys themselves, and W the remaining portion supported by P; then from the formula [8], TF'=(2«-1)P, t^'=(2«-1-1)m;„ w"=(2'«-2-1)«?3, &c. .-. TF=PF'+w'+i<;"+...=(2«-1)P+(2»-»-1K4-(2''-2_i)m,j+...+(2-1)w„_j. If the weights of the pulleys are equal, then, since 2»-i4.2'-»+...+2=2"-2, TF=(2'»-l)P+(2"-w-l)i/7i. W If the weight of each moveable pulley were w^^-—^ — - — -^ the n pulleys alone would support TF, since then P=0. Besides the preceding arrangements there are several other systems of pulleys, in which each block embraces only a single sheaf: but a very useful mechanical power of this kind is that in which the system consists of only two blocks, each block containing several sheaves, as in the annexed figures, where only one rope is em- ployed ; and as this is supposed to pass freely over the sheaves without any impediment, it must be equally tense throughout, this tension being expressed by P. The weight W is .*. supported by the equal tensions of the several portions of rope ascending from the moveable block, so that if this block have n sheaves, the tensions amount to 2n, each =P, .-. Tr=2nP, the power multiplied by twice the number of sheaves, where W includes of course the weight of the moveable block of pulleys. And the re- lation between W and P is the same when the moveable sheaves, instead of being inclosed in a single block, are placed as in the last figure, a single rope only being employed. j^»-n— 1' Examples fob Exercise. (1) A weight of 640 lb. is sustained by a power of 6 lb. acting through the system of pulleys in fig. 2, p. 406 : required the number of moveable pulleys. (2) Required the power requisite to sustain 1,020 lb. by aid of the system of pulleys in fig. 3, p. 406. (3) In a system of pulleys six are moveable, and a single rope goes round all the pulleys : what power will be necessary to sustain 1 cwt. ? (See last fig. ) (4) In a moveable block are five sheaves : 1,000 lb. is attached to one end of a rope passing round them : what weight at the other end is necessary for equilibrium ? 437. The Inclined Plane-— This is simply a plane surface in- clined to the horizon. Let i be the inclination of the plane, ? the angle which the direction of the power P makes with the plane, and W the weight supported by P on the plane. There are three forces acting at 408 THE INCLINED PLANE TF, namely, the weight W in the vertical direction, the resistance of the plane in the direction WR perp. to its surface, and the force P in the direction WM, all which direc- tions are given. Taking WC, and WR for axes, and resolving the two forces F, W, along WC, we have Pcos«-PFsini=0, .-. P=TF— * [IJ, COS! which expresses the relation hetween the power and the weight it sustains. To find the pressure on the plane, or its opposite, the resistance R, we have, by resolving the same forces along WR, ^ ^ . »xr . ,v r^n T, TrrCosi cos«— sini sin £ „xos(i4-0 rm *• ■' cost cos« If the power act along the plane, as it usually does, seeing that it is then most advantageously applied, we have £=0, .-. cos £=l, p .-. P=PFsin j, and R=WcQai .'. — =tan i [3]. When the power acts horizontally, e=— i, and .\P=Wtmi, R=W8eoi, .*. ^= sin i [4]. If it act vertically, cos f=sin i, .-. F=W, and 12=0. If I be put for the length, h for the height, and b for the base of the plane, the relations [3], [4], may be conveniently expressed thus : — 1. When the power acts parallel to the plane, T7^=-jt u7=7> p=i;'-[5J> , , . P h R I P h ^„^ 2. „ „ „ tothehomon, -=p -=-, -=-...[6], where we see that either set of expressions is got from the other set, by merely writing b for Z, and I for b. Problem I. The position of the pplley M, in reference to the given inclined plane AC, is given, as also the weights W and P, to determine at what point of the plane W must be placed so that there may be equilibrium. To AC draw the perp. MM\ which is given because M and AC are given. Put a for this perp. ; then a=WM sin ?, and from [1] above, W . . . v^(P2_TF2sin2z) „,,^ a Pa cos 1=^7 sin «, .*. sin«= i .-. WM=- — P ' P ' • sin« sin i >y{F^-W' sin' i)' an equation which determines the distance of the required point from M. Pkob. II. Two weights, W, W\ attached to the ends of a string, which passes over a fixed pulley, mutually support each other on two inclined planes: to determine the relations between W, W\ the tension of the string, and the pressures on the planes. As the string passes freely over the pulley 31 its tension will be the same throughout, and it will exercise the force on the weights which hitherto has been called P. Marking the angles and the resistances concerned, as at p. 407, we K'^ lT:^/^ have THE SCREW. 409 _cos (i+j) P _sin i' R' _ cos {i'-\-%') ^ W _ mii' ~ COS. ' 1F'~cosT" W cost ' " W'~smi P sint R W cos I W which equations express the relations required. If the pulley be at the intersection of the planes, so that the string acts along each plane, then . / « , ^ TF sini' CA I ,=0, . =0, and we have ^.=gi^=-^=i^' that is, the weights must be as the lengths of the planes on which they rest. And the same is evidently true when £=5', that is, when the string makes the same angle with one plane that it does with the other. Examples for Exercise. (1) The inclination of a plane to the horizon is 60° : prove that a force P will sus- tain four times as much weight on the plane when it acts along it, as when it acts horizontally. (2) A person is just able to sustain by his strength a weight of 200 lb. : what weight could he sustain on a plane of 50** inclination, by means of a rope going round it, and fixed to the top of the plane, as in the annexed figure ? (3) Upon a plane of 30° inclination, whose height is 20 feet, a weight of 3 lb. is sustained by a power of 2 lb. acting over a pulley fixed 10 feet above the top of the plane, and in the vertical line through it : at what distance from the top of the plane does the weight rest ? (4) A heavy body is supported between two inclined planes, the angles of inclination being 60° and 30° : prove that the pressure on the former plane is to that on the latter as 1 : ^Z. 438. The Screw. — This consists of a cylinder with a projection called the thread winding spirally round it, and inclined throughout at the same angle to the axis of the cylinder. When the screw is vertical, the projecting thread is nothing but an inclined plane winding round a vertical column : if we could unwind the thread, commencing at the bottom, and keeping the top end of the thread where it is, we should have the ordinary straight inclined plane, the height of which would be that of the screw, the length that of the thread, and the inclination the same as it was before the thread was unwound. Or the course of the thread might be traced on the cylinder thus : Let the base Ah' of the rect- angle ^6 be equal to the circumfer- ence of the cylinder, as also the base fee', &c. : then each of these rect- angles being wrapped round the cy- linder one above another, the united diagonals AB will trace out the spiral thread. Ah forming one turn of it, he another, and so on : and the same power will be required to sustain a weight on the inclined thread in either of the two positions of it. If the power act horizontally, and be applied immediately to the weight, we shall have, as at p. 408, 410 THE WEDGE. TF~^6'"~2err' ""^ 2^)-' where h, is the interval between the threads, and r the radius of the cylinder. The power acts horizontally, as in the annexed figure, which represents the bookbinders' press, iV being the nuty or hollow screw, grooved within, so as to receive the projecting threads of the solid screw. A force applied at the end of the lever P, causes the nut to revolve, and thus to press the solid screw, bearing the press-board, upwards. The books to be pressed are placed between the press- board and the fixed beam at top. The whole pressure on the screw is thrown upon the thread within the nut, so that if B. be the distance of P from the axis of the cylinder of the screw, and r, as above, the rad. of the cylinder itself, the efficacy of the power P will be expressed by P — ; in other n-il: P r -2^r' words, P acting at the extremity of E, is the same as P — acting at the extremity of r: hence, substituting this for P in the preceding equation, we have P h distance between the threads W 2«'i2 circum. described by P The ratio of the power to the weight is .-. independent of the radius of the cylinder: the power is increased by diminishing ^, the distance between the threads, or by increasing the leverage B. Examples fou Exekcise. (1) The distance between the threads of a screw is 1 inch ; a power of 30 lb. acts at the end of a lever, at the distance of 2 feet from the axis : required the pressure produced. (2) The distance between the threads is ^ in-, tke leverage 48 in., and applied power 161b. : required the pressure produced. (3) A power of 6 lb,, with a leverage of 3 feet, produces a pressure of 1 ton on the press-board of a screw : required the distance between the threads. 439. The Wedge. — This is a triangular prism made of some hard material : when used for the common purpose of cleavage or of splitting solid bodies in two, its edge is placed over the place of intended fracture, and the surface of the prism opposite to this edge, called the head of the wedge, is struck by a mallet, and the edge is thus forced forward, splitting or fracturing the body to which it is applied. Annexed is a section of the wedge perpendicular to the edge : when the wedge is used for cleavage, the section is usually an isosceles triangle. The mathematical theory of the equilibrium of this machine, employed in the above manner, is of no practical value, for it proceeds on the assumption that the resisting surfaces, and the faces of the wedge against which the resistances act, are perfectly smooth ; in other words, that the friction THE WEDGE. 411 is nothing, whereas, in practice, the friction is everything : — it maintains the equilibrium when the applied power is withdrawn : in the absence of friction the wedge would be expelled by the pressure on its sides. If, however, we consider a smooth wedge to be kept at rest by a power P applied perpendicularly to its head, and represented in the diagram by the line DE, and by the pressures P^, F,^, perpendicular to its faces, that is, acting in the direction MD^ ND, then completing the parallelogram IK, of which DE is the diagonal ; DE, EI, ID, will then represent the three equilibrating forces ; or, since the triangle DIE is similar to the triangle BJC, the three forces P, P^, F^, will be proportional to AB, AC, BC, that is, P : P^ : P^: : AB . AG : £0, 01 P : P,: P^ :: AB . I : AC. I : BC . I, where I is the length of the edge C, so that the three forces are pro- portional to the areas of the head and faces on which they respectively act. If, as is usual, the wedge be isosceles, and 29 be the angle G of the wedge, then AB-.AO, or Am:\AO:: Bm a :]-, .'. P=2P, sin fi. 440. In the preceding articles the simple machines treated of have all been regarded as motionless under the action of the forces applied to them, the object having been exclusively to ascertain what relations have place among forces which balance through the intervention of one or other of the so-called mechanical powers. We shall now briefly consider the working effect of these elementary machines when they are put into motion ; first, however, recommending to the attention of the student the following judicious remarks from Venturoli. The false opinions which persons unskilled in the nature and the power of machines are apt to conceive often encourages empty errors and mis- chievous deceptions. One of the most common of these conceits is, that of considering machines as available to increase and multiply the force of agents, which is not always true. To form a just notion of the aid which may be expected from machines, looking to the uses to which they are most commonly put, we shall divide them into two classes :— those intended simply to sustain a weight, and those intended to draw it, or to raise it equably. In machines of the first class, both the efifect of the machine, and the immediate efiect of the power, can only be estimated by the weight sus- tained. This being understood, it is evident that the machine increases the effect of the power, so that, for example, a force of 10 lb. will sustain, by means of a lever, 100 lb., provided that the arm of the force be 10 times as long as that of the weight. If it be asked how the force can ever produce an effect so much greater than itself, we shall perceive, if we consider well, that the force 10 does not really sustain the whole weight ] 00, but only the tenth part of it. Let the lever be supposed to be of the second kind ; the force 100 may be resolved into two, the one equal to 90, which acts upon the fulcrum, and the other equal to 10, which acts at the point of application of the power. The first is entirely sustained by the prop ; and the power sustains the second alone. Archimedes required only a fixed point to hold the terraqueous globe in equilibrium. 412 THE MECHANICAL POWERS IN MOTION. If he had found it, says Carnot, it would not in reality have been Archi- medes, but the fixed point which would have sustained the earth. In machines of the second class, neither the effect of the machine, nor that of the power can be estimated simply by the weight raised ; other- wise the measure of the effect would be altogether vague and indeter- minate. In fact, any force, however small, may carry a weight of any assignable magnitude, however great, if it only be granted that the weight admits of being divided and of being carried, one piece at a time. Where- fore it is necessary to take into account the time also in which the power can carry the weight through a given space, or the velocity with which the weight is carried ; and on this account it is that the effect is measured by the product of the weight by the velocity. Now upon this principle the machine does not increase the effect of the force. If a man with a force equivalent to 10, raise, by means of a machine, a weight of 100, he moves with a velocity 10 times as great as that of the weight, and does as much as if, operating without any machine, he carried those 100 at 10 journeys, loading himself with 10 at a time. In a word, what is gained in the quantity of the weight moved, is lost in the velocity ; and the effect remains the same. Between the two classes of machines, above described, there is then this characteristic difference, that the first add to the effect of the power, the second do not add to it. There is another difference, not less remarkable, respecting the resist- ances of friction, and of ropes, and other resistances. In machines of the first class, these resistances are all of them advantageous to the power, and themselves also sustain their portion of the weight ; whence there remains so much less for the power to support. On the contrary, in machines of the second class, the resistances are all of them detrimental to the power, and form part of the weight to be overcome : whence, on this account, a force is required greater than that which would be required, in the imme- diate application of the power. These things being understood, we may now easily bring in review the true scope, and the real utility of machines. Machines of the first class seem to increase the effect of the power ; and they do this by conveniently distributing the weight between the power and the prop. Machines of the second class serve — ^not to augment the effect of the power in quantity, but to modify its quality, as is wanted ; and they do this in the following manner. The effect of the machine being the pro- duct of the weight by its velocity, we can increase, at pleasure, one of the two factors, provided that the other is proportionally diminished. Thus, by means of a machine, we can move a weight enormously great, provided that we are content to move it slowly ; or, vice versa, we can move a weight with very great velocity, provided that it is a small weight ; whereas, by the immediate application of the force, we can hardly go beyond certain limits, either of velocity or of weight (VenturoWs Mechanics, by Creswell, Part II., p. 164). 441 . The Mechanical Powers in Motion.— The principal object of the elementary machines described in the foregoing articles, is not to balance weights or resistances at one part, by the application of LEVER IN MOTION. 413 force to another part, but rather, by aid of this force, to raise or overcome them. With the applied force motion must be combined : both are trans- mitted through, and modified by, the machine ; and the result is work ex- ecuted. Whatever be the machine, there is always an equality between the work applied to it, and the work performed by it, understanding by work the power or weight multiplied by the space through which it moves, or by the velocity, that is, by the rate per minute, or per second, at which it moves. The effect actually aimed at, or what is called the useful effect, does not thus always e'qual the expenditure necessary to produce it, yet none of this expenditure is lost : — what does not contribute to the useful effect, may be said to be spent in wearing out the machine, by the un- avoidable attrition, &c., of its parts. Only a fractional part, therefore, of the force applied is profitably accounted for by the actual result : — this fractional part is called (by Prof. Moseley) the Modulus of the machine : — it is the ratio of the work performed to the work applied. In what follows the fractional part of the work consumed or absorbed, as it were, by the machine, will be disregarded, so that all hindrance to the perfect discharge of its functions being hypothetically removed, the work delivered by it must be exactly equal to the work received by it, understanding by Work the product of the force or power exerted, by the length of path or space through which the exertion is uniformly kept up : there is the same amount of work in carrying 100 lb. ten yards, as in carrying 10 lb. one hundred yards in the same time, and the great advantage of a machine is that it exchanges for us the one work for the other, never, however, giving more than a just equivalent. It is as if we delivered over copper money to the machine — investing it penny by penny — and should receive back at once, so soon as the last instalment had been made, the exact equivalent in the more concentrated form of gold. 442. Lever in Motion. — If a straight lever of either kind be turned uniformly round its fulcrum, or fixed centre of motion, the arc described by the point of application of the power, will be to that de- scribed by the point of application of the weight or resistance, as the arm of the power is to the arm of the resistance ; because arcs measuring equal angles are as their radii : also since both arcs are described in the same time, their lengths must be as the rates or velocities per minute at which the points describing them move, for if one point move over m times the length of path of another in the same time — each moving uniformly— it must necessarily move m times as fast as that other: hence, equating work applied and work produced, Pxarcof P=TFxarcof W, .-. Pxvel. of P=W Xyel of TF, vel. of P arc of P arm of P vel. of \V arc of W arm of W arc of P W By the first of these equations the ratio of the arcs is constant, namely, T^rrr=^-jTi however small they may be ; but when they are each zero, no work is performed, or the 1 • ^ X 1 , , . .,., . ^^ arm of P lever is at rest : hence, when there is equilibrium, -^= 5—=.. ^ P arm of W The fraction vel. P-j-vel. W is called the velocity ratio, and the fraction W-i-P is called the advantage gained by the machine. Required the velocity ratio of F and W in the arrangement of levers figured at p. 401 (fig. 2). 414 WHEEL AND AXLE IN MOTION. vel. W=^ vel. A^ Pi „/ vel. ^2= — vel. A^ Pi v' vel. i4i= — vel. A^ P vel. TF=^^^^vel.P, PP1P2 since vel. P=vel. A^ vel. P PP]Pn vel.* W p'p'iP'2 443. Wheel and Axle in Motion.— When this machine re- volves uniformly, the power moves through a space equal to one circum- ference of the wheel, while the weight moves through a space equal to one circumference of the axle : that is, the vel. of the power will be to that of the weight as the circ.=(7 of the wheel to the circ.=c of the axle, or as the rad.=E of the former to the rad.=r of the latter, so that the . . vel. P C R .... T V J J velocity ratio is — j— ^=-=— , or equating the work applied and per- formed, Wc=PC, .'. Wr=PR : the same is obviously true when similar arcs, however small, of (7, c, are moved through; but when these arcs are each zero, no work is done, and the machine is at rest : hence when there is equilibrium, we must also have W : I* : : R : r. The annexed figure represents the compound wheel and axle, the latter consisting of two cylinders of unequal radii, the cord coiling round one in a direction opposite to that in which it coils round the other, the weight to be raised being suspended to a move- able pulley. Every turn of the handle coils the f'ord once round the larger circumference, while it is at the same time uncoiled one circumference of the smaller cylinder : the part of the rope hanging down is thus shortened by the difference of these circumferences, and .-. the weight is raised through half this difference, so that by the principle of equality of work, Pxc=^x"-i^, .-. Pxie=Trx:^, and !!|:Z=^,=iL. 2 2 velW c—c r—f' where c, c', are the circumferences of the compound axle, r, / their radii, and C, R the circumference and radius of the wheel. Ex. 1. Let the radius of one part of the axle be 3 in., that of the other part 2 in., and the radius of the wheel, or handle, 20 in. : required the advantage of the machine. Here W 2R 40 ^^ ,„„ — = =-, .'. TF=40P; P r—r' 1 hence the weight raised is equivalent to 40 times the power applied, but the latter must move through 40 times as much space. 2. In the annexed arrangement of a pair of wheels and axles, to de- termine the advantage of the machine. Let C be the circumference RACK AND PTNION. 415 of AB, c the circumference of ab, C the circum. of DE, and c' that of de. One revolution oi AB causes one revolution of its axle ah, and therefore BE is turned through an arc equal in length to c; and since de must turn through a similar arc, and that the lengths of similar arcs are as the whole circumferences, the similar arc of de is — , which is .-. the space ascended G by W: hence by the principle of the equality of work, „ ^ ^ cc' W CC RB! C P CC rr which is also the velocity ratio, or that of the spaces uniformly moved through "by P and W, , vel. P ^ CC CC that is, — — — =C/-i-— = — -. vel. W C CC c c The ratios -, — remain the same, however small be the arcs turned through : but when c c W Tin' these are zero there is equilibrium, in which case therefore we must also have — = — j. W Note. — In all these cases we perceive that the ratio — , when the machine vel. JP is at rest, is the same as the ratio ,' „^ when the machine is in uniform vel. W motion. This might have been anticipated, for since the balancing forces are inversely as their arms, and the arms (or radii) directly as the arcs described by their extremities, and these again as the velocities with which they are uniformly moved through, the inference is obvious. But without availing ourselves of the conditions of equilibrium esta- blished in the former articles, we may, as here shown, easily arrive at those conditions from the equation which implies equality of work when vel. P the machine is in uniform motion ; for the ratio — r^-rr^, being invariably the same, however minute the spaces through which P and W move, this unchanging ratio must equally apply to the extreme or limiting case, in which these spaces become each zero ; that is, S and s being the general symbols for the two spaces, and -=a, or which is the same thing, ——a, the s s ratio must still be a, even when s, and conseque7itly S, becomes 0, It is this limiting ratio of the spaces described (whether uniformly or not) by P and W, or of the spaces that would be described by P and IV, if the machine were put in motion, that is called the ratio of the Virtual Veloci- ties of P and W, when these are in equilibrium : it is the ratio of the actual velocities with which P and W would begin to move ; and these initial velocities of P and W, when put in motion, are the quantities called the virtual velocities of P and W when they are at rest. 444. Rack and Pinion. — The circular motion of a wheel may be made to produce rectilinear motion in a bar, by supplying both with teeth, as in the annexed figure : the toothed bar is called a rack, which is made 416 MOTION ON AN INCLINED PLANE. to advance in its own direction by the revolution of the pinion : enclosed in a strong frame, this machine is the Jack, much used for the purpose of raising or moving heavy weights a small height or distance. [By the distance between two consecutive teeth of a wheel, is meant the interval between the two radii through the middle points of those teeth, the interval being measured on the arc of the circle joining the middle points. This interval is called the 'pitch of the teeth, and the circle, whose radius extends from the centre of the wheel to either of the above-mentioned middle points, is called the pitch-circle. It is the radius of this circle that is the radius of the wheel, as the teeth are so shaped that their pressure is in the direction of the common tangent to the pitch- circles of the two wheels which gear or work together.] Let E be the length of the handle, that is of the wheel to which the power is applied, n the number of teeth in the pinion of this wheel, N the number of teeth in the wheel driven by this pinion, nf the num- ber of teeth in the pinion n', and let d be the distance between the teeth of the rack, or of the pinion n\ Then for one revolution of n' or Nf there must be — revolutions of n or P, SO that the rack moves through a space n'd, N while the power moves through 2i27r— , Wn'd= 2PENcr W 2RNt yel. P P' nn'd vel. W Suppose, for ex., that i2=20 in., the number of teeth 7i=6, IV=30, :6, and d=- in., then W 40x30x3-1416 40x5x3-1416 QXQX- 3 =200x1-0472=209-44= vel. P vel. W In all such combinations of wheels and pinions W : P, or vel. P : vel. W : : prod, of circum. of wheels : prod, of circuin. of pinions, or, instead of circumference, we may use the number of teeth, and vice versa. The above is only a particular exemplification of this general truth, ^Rtt being the circumference of the first wheel, and n'd the circumference of the last pinion, so that the pressure produced by the rack is 209*44 times that applied to the handle. 445. Motion on an Inclined Plane.— When a power pulls a weight up an inclined plane it must move through the length I of the plane, in order to raise the weight to the height h of the plane, I 'h' W vel P Heavy bodies raised by the help of this machine would require much ad- ditional power to overcome the friction, which, in practice, however, is THE SCREW IN MOTION. 417 reduced to a very small amount by introdacmg friction rollers between the body and the plane : by which means the mass is made to roll up instead of to slide up. 446. The Screw in Motion. — ^While the power moves through a circumference, of which the radius is its arm (fig. p. 410), the screw moves once round and causes the weight or resistance to move a distance equal to that between two threads of the screw, .-. Pxcirc. of P=TFxdist. of threads. The Compound Screw. — This consists of two screws A, B, the latter turning within the former, so as that when A is turned round once, one interval between its threads descends through the fixed nut n, but the same turn causes the screw B to ascend by one of its intervals through A, the latter screw being free to move in the direction of its length. Hence the weight lifted is raised through a space equal only to the difference of interval between the threads of A and B, .-. TFxdiff. of interval between the tlireads=P X circum. of P. Ex. The distance between the threads of the larger screw is - in., that between the threads of the smaller - in., and the 4 length of the lever is 6 feet : what is the advantage 3 11 gained by the press? Here -—-=-, and circum. of 8 4 o P=60x2,in.,...p^2iL^=3016: 8 hence the weight or pressure is 3016 times the force applied, which force, however, must act through 3016 times the space that the pressure does. The screw, here described, is called, from the contriver, Hunter's Screw. The Endless Screw. — Here the screw and one or more toothed wheels are combined. Each revolution of the handle causes the wheel acted upon by the screw to turn through the space of one tooth only, the distance between the teeth being equal to that between the threads. In the annexed figure let the radius of the handle be 20 in., and the diam. of the axle 5 in. : let the number of teeth in the three wheels be 30, 60, and 40 respectively, the number in the pinions 10 and 8. Then the screw may be regarded as a pinion, which for one revolution of its wheel (the handle) advances the wheel it drives 1 tx)oth: dividing, therefore, the circumferences (or no. of teeth) of the wheels by those of the pinions, we have EE 418 PEINCTPLE OF VIRTUAL VELOCITIES. TT 30X60X40X40^^36000^ . TF=7200P. P 10X8X1X5* 5 Hence the velocity of P will be 7200 times that of TF, and the weight lifted will be 7200 times the force or pressure applied to the handle. 447. Systems of Pulleys. — The principal varieties of these systems have been considered at pp. 406, 407, and since on the principle of W . vel. P the equality of work -^r is always, in uniform motion, = — —r^, we have ■^ r vel. W only to substitute this latter fraction for the former in the deductions there established to get the velocity ratio in each of the arrangements re- ferred to. 448. Principle of Virtual Velocities. — We have already ex- plained at p. 415 what is meant by virtual velocity : if the forces acting at different points of a machine, or of a connected series of points keep the system at rest, and are also of such a nature that when the system is put in motion, the points of application of the forces move uniformly, then these actual velocities, the machine being in motion, are the virtual velocities of the several forces, the machine being at rest. But if, when put into motion, the points of application would not move uniformly, then the virtual velocities are the velocities with which the points would begin to move; the equation, therefore, Px virtual vel. P— IF x virtual vel. W is applicable in either case to the system at rest. Since P and W necessarily tend to move the system in contrary directions, if they be marked with opposite signs, the algebraic sum of the two products here equated will be zero. This principle is general, that is, if any number of forces Pp P.^, r.^, &c., act at any points of a compound machine, and the corresponding virtual velocities be v^, v^, v.^, then if the system be at rest In the preceding articles different simple machines have been com- pounded into one system, and what has been called the resultant work may be regarded as terminating at, or as accomplished by any one of these parts, the power being transmitted through the preceding parts; and it is pretty evident that the principle is true at whatever piece we may regard the compound machine as terminating — as also whatever new forces be introduced to act on one or more of these pieces, provided the equilibrium be still undisturbed.* * Let the spaces s„ s^ ...Sn, be passed througli with any velocities v,, v^, ...Vn, however varying, then by the principle of work, P^s^-\-P2S^-\-...-\-PnSn=0, after any time t; 'ar^'^dt^-^^'' dt' The differential coefficients express the velocities that would have place if the forces were such as to render the actual motion uniform at the time t ; but if the condition have place for uniform motion, it has place for virtual motion, when the system is at rest, in which case the coefficients denote the virtual velocities. (See Differe^jtial Calculus.) FRICTION. 419 Examples for Exercise.* (1) Required the velocity ratio of the power and weight in the system of straight levers in the fig. at p. 401, the arms nearest the power being 54, 6, and 7, and those nearest the weight 2, 3, and 1 respectively. (2) In the compound wheel and axle (p. 414), the length of the handle is 2 ft., the diameter of the larger axle 10 in., and that of the smaller 9^ in. : required the velocity ratio. (3) What must be the difference between the diameters of the axles, in a compound wheel and axle, in order that the weight raised may be 100 times the power applied to a handle 2^ ft. long ? (4) In the wheels and axles at p. 415, the diameter of the wheel to which the power is applied is 20 in., and that of its axle 6 in. : the diameter of the other wheel is 50 in,, and that of its axle 3 in. : how high will the weight have ascended when the power has descended through 4 ft. ? (5) In the wheels and pinions at p. 403, the diam. of the axle to which the weight hangs is 3 in., and the number of teeth in its wheel 30 : the number of teeth in the pinion which this wheel drives is 6, and the number in its wheel 28 : the number in the last pinion is 8, and the diameter of its wheel 27 in. : through what space will the power descend if the weight ascends through 3 ft. ? (6) In the common press (p. 410), if the length of the lever is 3 ft., and the distance between the threads of the screw ^ in., what space must the end of the lever be moved through, in order that the press-board may move through a space of 2 in. ? (7) In the compound screw (p. 417), the distance between the threads of the larger screw is 1 in., that between the threads of the smaller | in., and the length of the lever 4 ft. : how many times the power applied will the weight moved be ? (8) In the mechanism represented at p. 417, let the radius of the handle be 30 in., the number of teeth in the three wheels, 40, 60, and 50 respectively, the number in the two pinions 5 and 10, and the diameter of the axle 6 in,, required the velocity ratio of P and W. (9) Wheel A, 36 in. in diameter, with 72 teeth, and making 120 revolutions per minute, drives wheel B, 12 in. diam. with 24 teeth : on the shaft of B is fixed wheel C, 10 in. diam. with 30 teeth, driving Z>, 6 in. diam. with 18 teeth : required the speed, or the no. of revolutions per min. of D. (10) A driving wheel is to have 10 teeth, and the follower 40 : it is required to find what must be the diameters of the jpitch circles in order that the pitch of the teeth may be 1 in. (see p. 416). 449. Friction. — In all that has preceded, the effects of friction have been left entirely out of consideration : the bodies acting upon one another have been assumed to be perfectly smooth, whereas they are invariably more or less rough, on which account more power must be expended on a machine to produce a given motion, than would be necessary if the resist- ance due to friction did not exist. Sometimes this is a very useful resistance : — useful for the purpose of checking speed or stopping motion • As a first book on the leading principles of mechanism, the learner is recommended to read the neat and instructive little volume of Mr. Tate, formerly of the Battersea Training College ; the above few examples are selected chiefly from that work. The treatise, too, on ** Practical Mechanics, and the Steam- Engine," by John Imray, M.A,, C.E., in the volume on Mechanical Philosophy, in Orr's "Circle of the Sciences," will amply repay the attentive perusal of the student of Machinery — a subject which does not come within the scope of this work. E E 2 420 FRICTION. altogether; familiar instances present themselves in the ropes coiled round windlasses, and capstans, to raise heavy weights or resist great strains ; in the break applied to a wheel of a heavy-laden waggon, or an omnibus, when descending a slippery declivity, as also in the friction- break of a locomotive steam-engine. But to increase speed, the friction must be diminished as much as possible, as it is a hindrance to motion. To estimate the amount of this hindrance, a force must be applied to the machine which will bring it into a state bordering upon motion, in which state it will be strictly in equilibrium. It is found by experiment that for flat surfaces sliding one upon the other, the amount of friction is the same for the same sliding body or pressure, whatever be the dimensions of the side of the body which presses: thus, the friction of a block of material with several unequal but flat faces will be the same whichever face be downwards, if the supporting surface be the same ; also that the materials being the same, the friction varies as the perpendicular pressure on the plane, or that it is equal to a constant fractional part of that pressure or of the opposing resistance. If we call this normal pressure or resistance R, the friction will be denoted by [xR, the fraction fx being called the coefficient of friction. Practically this coefiBcient for given materials is found thus : let the plane AB, supporting a body with a flat surface (each plane being that of a given material), be more and more raised at one end B till the body is in a position bordering upon motion, that is, till the slightest additional elevation would cause it to slide : it is then in that state of equilibrium by the forces acting on it : these forces, resolved along the plane and perp. to it, are, Perp. to the plane, W cos a=:R. Along the plane, W sin a=zfji,R, .*. ya=:tan a ; SO that by measuring the greatest angle a at which the plane can be in- clined to the horizon, without causing the body to slide, the coef. of friction^ for the particular substances experimented upon, becomes known. For instance, the coef. of friction for polished brass and iron is jm.='143, the limiting angle a. being 8° ; so that if a piece of such iron of 1 cwt. rested on a horizontal brass plate, it would require a lateral force of 16 lb. to cause it to slide : for the vertical (or normal) pressure jR being 112 lb., we have ^E=-143 x 112 = 16 lb. If the plate were inclined at an angle of 8°, the iron would be on the point of sliding down it. Again : suppose a polished cylindrical iron shaft, weighing 1 ton, to revolve in a brass bear- ing : the force that must be applied to the circumference to balance the friction would be 2240 x -143=320 lb. : but as the shaft does not slide, but turns on its axis, if the radius of the shaft at the bearing be 3 in., then 3 in. is the leverage at which the friction acts to resist rotation ; Q and since 320 x— =80, it would be balanced by 80 lb. with 1 foot leverage : this force, acting in opposition to the friction, would bring the shaft into a state bordering on rotatory motion. Prob. I. To find the force P, acting at a given inclination e, to a plane which is itself inclined to the horizon at the angle i, necessary to keep a weight W in that state of equilibrium which borders on motion, the friction being taken into account. FRICTION. 491 Let R be the resistance of the plane, then fA,R being the force of friction acting up the plane (since it opposes downward motion), we have, by revolving the four forces perp. to the plane and along it, W COS i—P sin t=R, and W sin i—P cos t=fiR, .'. P=W r— . cos t—fA sm < If P act along the plane, then 8=0, and P=^(sin i— ^ cos i). The equilibrium considered above is that in which the tendency to motion is down the plane; since, however little P be diminished, that downward motion must ensue. But it cannot be said that, however little the force of P be increased, an upward motion will ensue ; because the friction is equally opposed to motion either way: hence, for the other state of equilibrium : — the state bordering upon upward motion, the friction fxR takes a contrary sign ; .1 . , . « , -r^. ,„sm i-\-it COS i so that, changing 4-u, for —u, we have P'=W — : — , cos t-\-fA sin s or if, as before supposed, P' act along the plane, P'=.W{sm i+(A cos i). The weight W will therefore move neither up nor down the plane so long as any force, not without the extreme limits P, P\ be applied to it, and act in the direction WP, making an angle s with the plane. For every state of the system bordering upon motion, the motion being limited to one of two directions, there is thus always what may be called a conjugate state, the friction in both states being the same, but acting in opposite directions : the difference between the two forces P, F\ which, acting in the same line as the friction, apply to these conjugate states, must be twice the force / of that friction, and the coef. of friction might, from this consideration, be found in another way : thus, when P, P\ act along the plane, as above, P'—P=z PF{ (sin i-{-fA cos i) — (sin i—(i. cos i) } =2 Wft cos i, / P'—P .'. /= Wfi. cos iy .*. ^l.=^-^ :, or ft.=.—— .. ^ ' '^ fTcost' '^ 2 IF cos i P'—P If 4=60°, then /» is found from th^ equation ^= . Prob. it. To find at what angle £ the direction of the force P must be inclined to the plane, in order that P may be the least possible to prevent a given weight W from sliding down ; friction being taken into account. In order that P may be the least possible, it is plain that the denomi- nator cos i—yu sin E, in the preceding expression for it, must be the greatest possible. Put sin i=,x ; then we are to have a/(1 — a;^)— ^a;=a maximum. Calling this u, as at p. 148, and rationalizing, we have (l+^2)^2_|_2^Ma;4-t42— 1=0, and since any greater value for u~ renders v^C/ imaginary, this is the greatest value possible, and these values of x and u satisfy the proposed equation. Hence / fA^ U? 1 422 FRICTION. Consequently, the least force P, which will restrain the body W from falling down the plane, must have its direction inclined to the plane at an angle whose tangent is equal to the coef. of friction ; and this angle must lie below the plane. The direction of the least force which will just keep the body from moving up the plane, is got from the above by changing the sign of /t^; that is, the direction i is such that tan s=ix, the angle lying above the plane. Pros. III. To find the equation bordering on motion for a lever free to turn on a cylindrical axle working in a hollow cylinder, or bearing. Let the body be acted on by the vertical pressures P and W: let C be the centre and r the radius of the axle, and P the point on which the axle will begin to turn in the hollow cylinder in contact with it there, on which point the pressure E=P-\-W acts, in the vertical direction. The tangent DE is the di- rection in which the friction acts, and CD is the direction of the normal pressure, or that perp. to the surface pressed, and this multiplied by fx is the force of the friction. If be put for the angle RDT, then p. 420, ^=tan7'=cot^ [1]. Now the normal pressure is R sin 6=R—r- t-t, and .*. the friction / is [1], v'll-l-cot^^ The opposing force /, acting in the direction DE, being thus known, we have, by taking the moments of the three forces P, W, and /, about C, putting r for CD, and observing that / conspires with W to produce rotation in the direction DE, we have P.AC^^W. BC+fr= W . BC+ ..^^ ,. (P+ W)r xyK^+f^ ) for the equation of equilibrium bordering on motion in the direction DE. The equation of equilibrium, bordering on motion in the contrary direction, is got from this by merely changing the sign of fr, or the sign of /t*. It may be remarked, in reference to the foregoing equation for the equilibrium, when nearest to motion, that [x being always a fraction — usually % small one — the second power of it will, in general, have a value so small as to be undeserving of notice : expunging /^^, therefore, the equation is Note. — The friction to be overcome, in passing from a state of rest to a state of motion, is always greater than that which opposes the con- tinuance of the motion when once commenced, although in the case of very hard bodies the excess is very trifling. The experiments of Morin have shown that the friction of motion is wholly independent of the velocity of the motion : the former kind of friction has been called the friction of quiescence. To diminish as much as possible the resistance occasioned by friction is the great object of practical engineers in working machinery, every SCHOLIUM. 423 part of which is therefore kept free of dirt and dust, while the rubbing surfaces are lubricated with oil, or some other unctuous matter. " When a steam-engine is employed as the prime mover of any machine, the power communicated can be readily ascertained by the Indicator. The engine is first worked alone, or with merely the train of wheel-work, in order that the power necessary to overcome friction may be estimated. It is then worked in connection with the machine, and the driving-power required for the machine is ascertained by subtracting the force necessary to overcome friction from the total power, including friction and the re- sistance of the machine. When machinery is driven by some other power, or when the indicator cannot be conveniently applied, the dynamometer (power-measurer) is employed."* [Additional investigations connected with Statics will be hereafter given as applications of the Diffekential and Integbal Calculus.] 450 Scholium. — The foregoing treatise contains as much of the general theory of equilibrium as can be reasonably looked for in an elementary work like the present, in which the higher calculus has not as yet been taught. Within the extent to which our space has compelled us to limit it, as full an amount of detail, and as great a variety of topics, have been introduced as is consistent with the general plan and moderate pretensions of this volume. The student, however, must regard what is here done as little more than a theoretical preparation for the study of those extensive practical subjects that come under the heads of Archi- tecture, Engineering, and Mechanism : on these matters he is referred for ample information to the following works : Imray's " Practical Me- chanics and the Steam-Engine," Moseley's " Architecture and Engineer- ing," Whewell's *' Mechanics of Engineering," Willis's " Treatise on Mechanism," and the neat introductory book on the same subject by Mr. Tate. On the " Strength of Materials," Barlow's is the standard English work ; the practical and professional man will also find a vast amount of available results in Professor Hodgkinson's numerous *' Ex- perimental Researches," as contained in his published contributions to the " Manchester Memoirs," to the British Association, to the Royal Society, &c. The Author of this work has reason to believe that Prof. Hodgkinson is now occupied in collecting together and arranging the chief of these contributions for publication, in a connected form, in one great work on •' The Strength of Materials," on '• Suspension Bridges," &c. End of the Statics. VII. MECHANICS : Part II. Dynamics. 451. The first part of Mechanics considers bodies at rest, or in a state of equilibrium under the operation of applied forces; the second part treats of the motion which ensues when a body is not thus kept at rest. * Imray's "Practical Mechanics and the Steam-Engine:" Orr's ** Circle of the Sciences." 424 UNIFORM MOTION : VELOCITY. We have seen in Statics that in every body there exists one point, and one only, which is distinguished from all others by a remarkable pecu- liarity, which is this : — that, provided only the forces acting upon the body be all applied at that point, the tendency to progressive motion, and the direction and intensity of that tendency, will remain the same : this point is the centre of gravity of the body; it is the point into which the entire mass of the body may be conceived to be compressed and con- densed. If under the operation of forces, thus conceived to be all applied at the centre of gravity, progressive motion actually takes place, the movement of that centre must therefore commence in the direction of the resultant of those forces, and being only a point its path must be a line. If the forces cease to act, the instant this point starts into motion, it must not only commence, but continue to move in the direction of the resultant, that is, its path must be a straight line, so long at least as no external obstacle or influence interferes with its onward progress, and in the absence of all such interference its onward motion must be uniform. These truths must be admitted as self-evident : they are usually enun- ciated in the form of an axiom in the following terms, and constitute what is called — The first Law of Motion. — Mere matter, if at rest and unacted upon by any external force, or only by forces of which the resultant is nothing, must remain at rest. If it be in progressive motion, and unacted upon by any external force, it must continue in motion : the motion must be uniform, and the centre of gravity must describe a straight line. And this is only affirming that inanimate matter is incapable of itself of alter- ing the state into which it is put by any external cause, whether that be a state of rest or a state of motion : this incapability is called the inertia of matter. When in what follows we speak of the path of a moving body it will be understood that we mean the line described by its centre of gravity. 452. Uniform Motion: Velocity-— The motion of a body is said to be uniform when it passes over equal lengths of path in equal times : the term uniform so sufficiently implies this that we have not hesi- tated to use it previously to formal definition. What is commonly called the rate or speed of a body's motion is what is here meant by its velocity ; in uniform motion it is estimated by the length of path described in a given interval of time, as for instance, in one second, one minute, &c. If a body moving uniformly pass over ten feet in every second of time we say that its velocity is ten feet per second. In dynamical inquiries it is the second that is generally chosen for the unit of time, and the foot for the unit of length. For the number of seconds in any portion of time the symbol t is employed, and in like manner the initial letters s and v stand for space {linear space) and velocity ; but this distinction must be carefully observed, namely, that t is always an abstract number: it does not stand for time, but only for the number of seconds in that time, while, on the contrary, s and v each denote the concrete quantity oi feet, the former symbol expressing the whole length of path described in t seconds, and the latter the portion of that length described in one second. It is plain that in uniform motion these three symbols are related to one another as follows, namely, «'— p s=vt, t=- [1], t being measured from the commencement of the motion. VARIABLE MOTION. 425 If, however, t is not reclioned from the instant that the motion com- mences, but only after a certain space s' has been passed over by the body, then the space described in t seconds is only s—s\ instead of s, so that the three equations will then be s—s' , s—s' ^=-p, s=^-\-vt, t=-— [2]. 453. Angular Velocity. — If a body rotate about an axis or point equably, it turns through equal portions of an entire revolution in equal times, and the rate at which it rotates is the angular velocity of the body. This, like linear velocity, is also usually measured in feet, whenever the foot is taken for the linear unit : it is the length of arc, of which the radius is one foot, that is turned through in a second. However different two straight lines may be in length, if each turning about its fixed extremity as upon a pivot complete a rotation in one and the same time, the angular velocities are equal, though the linear velocities of the moving extremities are unequal. If the lengths of the lines or rods be R, r, and the linear velocities of the moving extremities F, v, the angular velocities will be V V -==z- feet. For the uniform linear velocities are as the circumferences rl r described with them, and these are as the radii, so that calling the angular velocity w, that is, the linear velocity at a foot distant from the V V u centre of motion, we have —=-=-. Sometimes, however, the abstract it r 1 number, which this ratio expresses, is regarded as the angular velocity, the linear velocity at the distance r from the centre of rotation being expressed by v=ur. The following are some applications of the foregoing formulae : — (1) A railway train travels at the uniform rate of 40 miles an hour: what is its velocity per second ? s 5280x40 feet 628 ,2, Here v=-= = — =58- ft. t 60x60 9 3 (2) Two bodies a, b, animated by the uniform velocities v, v^ the same instant from the points A, B, and move in the direction of AB prolonged : in what time will a come B a h G up to & ? Suppose that a comes up with h at the point ^-i— ' 1- C, then putting s for AC, and s^ for AB, we have s—vt, s—s^=v^t; Subtracting this from the former, 8x=.0o—V\\tj .*. <=: — ^—, 80 that the number of seconds is found by dividing the space between the bodies at starting by the difference of the spaces denoting their velocities. (3) A wheel whose radius is 3 feet makes 120 rotations in a minute: required its angular velocity per second. Here every point of the circum- ference moves uniformly through a space «=120 X 6«r in 60 seconds, .'. its linear velocity is v=12«r, and .*. the angular velocity is -=4jr=12"566 ft. per second. r 454. Variable Motion. — When the moving body does not pass over equal spaces in equal times its motion is not uniform but variable ; 426 UNIFORMLY-ACCELERATED MOTTON. and its velocity at any instant is the space it would pass over in the follow- ing second, provided it were to move uniformly during that second at the speed with which it is actually moving at the instant. When the variable motion is such that equal accessions of velocity are acquired in equal intervals of time, the motion is said to be uniformly accelerated ; if instead of being equally increased the velocity is equally diminished in equal intervals of time, the motion is said to be uniformly retarded. From the axiom that a body cannot of itself alter its state of rest or of uniform motion, whenever this motion is accelerated or retarded, the change must be due to some external cause : — we call it force ; regarding it as a constant force, if its effects be uniformly the same, that is, if there be the same increase or decrease of velocity in equal times, and a variable force if its effects be variable. 455. Uniformly-Accelerated Motion.— Let the uniform increment of velocity which a body receives from second to second be represented by/; then if v^ be its actual velocity at any instant, at the end of 1 second from that instant the velocity will be v^ +/, at the end of '2 seconds, v^-\-^f, and at the end of t seconds, v— v, -j-f/"; and measuring causes by their effects, the constant increment/ of the velocity is the quantity employed to represent the intensity of deforce, or that constant influence which thus continuously and uniformly accelerates the motion of the body ; the acceleration / of the velocity is hence often spoken of as the accelerating force. The student, however, will bear in mind that/ denotes a length in feet ; the length by which the space described in one second, with the velocity v^-^f, exceeds the space described in one second with the velocity v^, these spaces themselves being respectively v^+f, and v^. The two propositions which follow contain the entire theory of the rectilinear motion of a body uniformly accelerated. 1. If a body be moved from rest, and /be the constant acceleration of its velocity (or the accelerating force continuously influencing it) during t seconds, at the end of which time the velocity is v, then v=ft [1]. This is included in the inference above, v^ being 0, since t is here mea- sured from the commencement of motion. 2. If s be the space through which a body is moved from rest,/ being the constant acceleration of its velocity, or the constant accelerating force, then at the end of t seconds, s==-ft'^. This is proved as follows : — Let the t seconds be divided into n equal intervals, each less than a second, and call each of these smaller intervals r, so that ?iT=t. Then since equal velocities are generated in equal times, the velocities at the end of each of these intervals will be respectively fr, 2fr,2,fT,...,nfr [A], and at the commencement of the same intervals, the velocities will be 0,/r, 2/t,..., (ri-l)/r [L]. Now if the several velocities [A] were uniform duiing the several in- tervals, that is, if the velocity throughout each interval were the same as that at the end, then the whole space S, described in the time t would be S=f-r.T-\-2fT. t+3/t. T-{-...-{-nfr. r UNIFORMLY- ACCELERATED MOTION. 427 And if the several velocities [B] were uniform, the velocity throughout each interval being the same as that at the beginning of it, the whole space aS", described in the time t, would be >S"=0.r+/r.r-f2/r.T+...4-(w-l)/r.r But neither of the spaces S, S\ is that actually described by the moving body in the time t, but some space s intermediate between them, that is, s has a value always between ^^IK^-d'^'^'^K^+i)' however great n may be, which value can be no other than s=- ff^ the value which each of the extremes 8, S\ becomes when n is infinite, since s must be that quantity which coincides with S, S\ when these become equal. We thus have the following relations connecting the terminal or last acquired velocity, and the space described, with the time and the uniform acceleration, or force /, namely, ^=A s=lft^ .-. iP=2fs, s=lut [2]. From these relations the following interesting truths are immediately deducible, namely : — 1. The space described in any time, reckoning from the commencement of the motion, is half the space that would have been described in that time, if the body had moved from rest with a uniform velocity equal to the terminal or last acquired velocity. The last of the preceding equations expresses this truth. 2. The spaces described in equal successive intervals of time from the commencement are to one another as the odd numbers 1, 3, 5, 7, &c. For let t measure the seconds first in one interval, then two, then three, and so on ; then from the equa. s=- jt^^ the corresponding spaces described are Consequently, subtracting each of these from that which immediately follows, the spaces described in the several individual intervals are i/.l,i/.3,i/.5,i/.7,&o., which spaces are as the numbers 1, 3, 5, 7, &c. Suppose for ex., there be an accession of velocity of 4 feet every second ; then / would be repre- sented by 4 feet ; and the spaces described during the first second, the second second, the third second, ...the tenth second, would be 2, 6, 10, ..., 38. The sum of these, or the whole space s, would be s=2(l4-3+5+...+19)=(19-f 1)10=200 feet, which of course is the same as would be given at once by the formula «=:-/{2^ when/=4 ft., and «=10. 428 UNIFOBMLY-ACCELERATED MOTION. If, after sufficient observation of the rectilinear motion of a body, we are con\dnced that that motion is due to a constant force acting upon it, we may readily estimate the intensity of that force by observing the space the body passes through, from rest, in a given time, from the condition /=— : thus, if the time of passing through the space s be one second, then /=2s; that is, the constant accelerating force is measured by twice the space described, through the influence of it, in the first second of the body's motion. When the rectilinear motion of a body is uniform no force can be acting on it during that motion, which can have originated only from an impulse— a force which expires, as it were, in the act : while moving uniformly the body is subjected to no influence whatever. 2s v= ft t =x/(2/^) .=^/. =>' 1 ^ ~2/ -7 _2s ~ V -^7 H 2s ~«2 _i t^ ""2 $ From the equations [1], it is plain that any two of the four quantities /, t, v, s, being given, the remaining two may always be found, and that for each of the four, there are three distinct but equi- valent expressions: — they are exhibited in the mar- gin for convenience of reference. 456. The most important accelerating force with which we are ac- quainted is the force of terrestrial gravity, or the attractive influence which the earth exercises on all bodies, and which all experiments show to be a constant force at the same place, and for all distances above the sur- face that are within our reach. Its intensity is such as to cause a body to fall from rest, in the latitude of London, a distance of 16 1 feet in the first second of time ; consequently in this latitude the expression for the force of gravity, which force is usually represented by ff, from what is shown above, must be g=S'^'^ feet, that is, g is such as to uniformly increase the velocity of a falling body by 32-2 feet at the end of every second during its descent. The ea^act force of gravity, in a few widely-differing latitudes, as determined by very careful observation, is given in the annexed table, the ex- periments from which the different values of g are deduced, having all been made at the level of the sea. It is found that the influence of this force di- minishes the further the body acted upon is re- moved from the centre of the earth, but as there is no sensible diminution within those limits accessible to man, gravity may be regarded, thus far, as a force which in the same latitude is constant. In the foregoing group of equations, the body is supposed to have been put in motion solely by the force /, from a state of rest; but we may con- ceive the actual motion to be due to a two-fold cause : — to an impulse, given to the body (either concurring with or in opposition to the direction, in which / acts), and to the force /; as for instance, when a body is pro- jected vertically downwards or vertically upwards, gravity also acting upon station. Latitude. Gravity. Observer. London . Paris . . . Leith . . . C. G. Hope . New York . Unst . . . 51" 31' 8"N. 48 50 14 N. 55 58 41 N. 33 55 15 S. 40 42 43 N. 60 45 25 N. 32-1908 32-1820 32-2040 32-1412 32-1600 32-2173 Sabine. Borda. Kater. Freycinet. Sabine. Biot. PALLTNa BODIES : GRAVITY. 429 it all the time. Suppose r^ to be the velocity with which a body is pro- jected in the direction in which / acts. If no force modified its motion, it would move uniformly with the velocity v^ that it commenced to move with. Now there is evidently some length of time t^ at the end of which, if the body had moved from rest, by the action of / alone, it would have acquired the velocity v^. Let s^ be the space due to this velocity, then v^=ftj^y and s^=- ft^. Hence t being the time and s the corresponding space reckoned from the termination of i, and Sj, that is, measured from the instant when, and the place where the impulse to the body is given, we have at the end of t seconds that is, v=zv^^ft [1], s=v^t-\--ft'^ [2]. z To eliminate t from these, we have, by squaring the first, r2=t;,2-|-2r,/i5+/2<2=V+2/(v,«+i/<2), that is, [2], v^=v^^-\-2fs [3]. These are the formulae to be used when the body is projected towards the force with a velocity Vy If the projection be in the direction opposed to the force, then/ retards the motion, and its sign must be changed in the preceding formulae, which are then 'VZ=V,-ft, 8=v,t-\fC', v^=v,^-1fs [4]. When gravity is the accelerating force, the symbol g is put for /. 457. Falling Bodies : Gravity. — The following are a few ex- amples on the vertical motion of bodies under the influence of gravity. (1) A body is 4 seconds falling from rest to the earth: required the height from which it fell. Here/=^=32-2, .-. s=^jt'^z=lQ'lXi^=2bVQ feet. (2) If a body be projected vertically upwards with a velocity of 400 feet ; how high will it ascend ? The height will be that from which, if the body were let fall, it would acquire a velocity of 400 feet upon reaching the earth, 1 ^ ,- 1 4002 .*. since s=- — , we have s=- —— =248*4 feet. 2 5r 2 32'2 (3) From an elevated position a body is projected vertically upwards with a velocity of 80 feet : required its position at the end of 6 seconds. Here [4], «='y,<-ip<2=-80x6-16'lX62=6(80-16-lx6)=-99-6, hence the body will be 99 6 feet helow the point of projection. Examples fob Exercise. (1) In what time will a heavy body descend 400 feet ? (2) A body is projected vertically upwards with a velocity of 1500 feet ; how high will it ascend? (3) With what velocity must a body be projected vertically downwards from the top of a tower 150 feet high to arrive at the bottom in two seconds ? 430 MOTION OF BODIES ON INCLINED PLANES. (4) Through what height must a body fall to acquire a velocity equal to the space faUen through ? (5) A body is projected vertically upwards with a velocity of 100 feet : at what time will it be 100 feet from the earth, as well in its descent as in its ascent ? (6) At the same instant that a body begins to fall from a height of 100 feet, another body is projected upwards with a velocity that would carry it 150 feet : at what height would they meet ? (7) With what velocity must a stone be projected down a well 350 feet deep that it may arrive at the bottom in 4 seconds ? (8) When a body has fallen a feet, another, h feet below where the first then is, is let go : how many feet will this latter have fallen when the former overtakes it ? 458. Motion of Bodies on Inclined Planes.— Wheu a body is placed upon a smooth plane, of inclination i to the horizon, the force which urges it down the plane is only that component of gravity which is in the direction of the plane, namely, g %mi; the other com- ponent, being perp. to the plane, produces statical pressure, not motion. Consequently, we have only to substitute g sin i for /, in the formulae at p. 428, to render those formulae as applicable to this oblique motion of bodies as they now are to their vertical motion. Let I represent the length of the plane, and h its height, then sin i=-, L SO that the acceleration down the plane is f=g-, and the velocity acquired I in descending down the whole length is (p. 428) v= ^2fl= s/ ^gh, the same as that acquired in falling down the height h ; and we see that provided the height of the plane be constant, it matters not what be its length: — the velocity acquired in descending down it must be always the same, namely, the velocity acquired in falling through the height. 1. To determine what length down an inclined plane a body will descend while another body falls through the height of the plane. Putting s for the space descended down the plane, in the time t that the body falls vertically through the height, we have h=-gt'', s=-g sm I .t-=h sin i. This expres- sion for the length of plane, descended in the time that would be occupied in falling through the height, suggests an interesting property, namely, 2. If from the extremities A, B, of the vertical diameter of a circle, any number of chords be drawn, a body would fall down either of them in the same time that it would fall down the vertical diameter AB. For prolong ctny one of the chords from A, as AF, till it meets the horizontal line BC; then the part of AC, fallen down in the time that AB would be fallen down, is AB sin C=AB cos BAC=AF : and similarly of any other chord from A. The same is true for the chords from B : take BF for instance : then the chord AD from A, parallel to BF, is equal to BF, since the angles ABF, BAD, and .-. the arcs^i^", BD, are equal : hence the lines ADy FB, being equal in length, and equally inclined to the horizon, THE PARALLELOGRAM OF VELOCITIES. 431 the times down them must be equal, .*. the time down FB is the same as that down the vertical diameter. 3. To find the position of an inclined line through a given point, such that a body falling down it, from that point, may arrive at a given straight line in the shortest time possible. Let A be the given point, and GH the given line. Draw through A the horizontal line AH, meeting the given line in H, and make HE=HA : then will AE be the inclined line required. For let a perp. to AH be drawn from A, and a perp. to GH from E: these will intersect in some point 0; and if OH he drawn, we shall thus have two equal right-angled triangles, so that the two perpen- diculars will be equal: hence a circle from centre 0, wath radius OA, will touch HA, HG, in A, E, and the diameter AB will be vertical. Now AE is the only chord of the circle that reaches the given line ; and the time down any other chord AD from A, and which is too short to reach the given line, is equal to that down AE, which does reach it, .*. the line is reached down AE in less time than down any other straight line from A. Examples for Exercise. (1) If the length of an inclined plane be 60 feet, and its inclination 30°, what velocity would a body acquire in falling down it for 2 seconds ? (2) How long would a body be in falling down the plane Z=100 ft. i=60° ? (3) How far up the plane, h=:.-l^ would a body ascend in 6 sees. Avith a vel. of 50 6 feet? (4) Find the line of quickest descent from a given point to a circle in the same vertical plane. 459. The Parallelograxn of Velocities.— The velocity of a moving body at any instant is the rate or speed at which it is then moving, and this is measured by the space it would pass over in one second, if during that second its speed were to continue the same as at the instant referred to. That a body would continue to move with the same velocity that it has at any instant, it is only necessary that at that instant all in- fluence and obstruction should be withdrawn, and the body be left to itself: its motion then will be the same as if it were originally com- municated by a single momentary impulse. Suppose the body when at A were animated with a velocity (or received an impulse) that would alone carry it to ^ in a certain time, and that it were simultaneously animated with another velocity (or received another impulse), that would alone carry it, in the same time, to C: the joint effect of the two velocities, or of the two impulses, will be to carry it uniformly along the diagonal of the parallelogram to D in that time. For conceive the line AC with the moving body upon it, to be carried parallel to itself, and with the first uniform velocity, up to BB\ then at the instant it arrives at BD, the body will have arrived at the extremity of the moving line, that is, it will be found at D. In consequence of its compound motion, it will have arrived there by the path AD : for in every position d of the moving body, Ab: bd:: AB : BD, since from the uniformity of the motions, the advance Ab of the body in 432 THE MOTTON OF PROJECTILES. the direction AB, to its simultaneous advance hd in the direction AC,\s always in this ratio : the point d is therefore always on the diagonal AD : and thus the single velocity ^Z> is equivalent to the two component velocities AB, AC. Hence, what is proved in Statics, as to the composi- tion and resolution of forces, may in like manner be proved in reference to the composition and resolution of velocities. If 9 be the angle A be- tween the directions of two component velocities v, v', then for the re- sultant velocity F, we shall have 1/ F2=i;24-i;'2_|_2w;'cos^. If ^=90°, then V'^=^-\rv"\ and tan [T, v]=-, V the last expression being the tan of the angle which the resultant velocity makes with v, one of the rectangular component velocities. It has been assumed above that the body arrives at D from A, in con- sequence of its having been subjected, when zt A, to two influences, of which one alone would have carried it to B, with a uniform motion, and of which the other would alone have carried it to C, also with a uniform motion, and in the same time. But the reasoning that shows that the body would arrive at Z), by the joint effect of these two uniform motions, would .equally apply, however irregular the motion that would carry it to B, and however irregular that, that would carry it to C, the time of the arrivals being the same. If the moving line AG just reaches BD, when the body moving along ^(7 just reaches C, it is plain, whether the motions are uniform or not, that the body must, at the end of that time, be found at D ; although its path to that point would be the diagonal of the parallelogram only when the component motions are uniform motions. The principle upon which the foregoing proposition rests is that called The Second Law of Motion : namely, when a force acts upon a body in motion, the effect of this action, in magnitude and direction, is the same as it would be if it acted upon the body at rest. 460. The Motion of Projectiles. — Hitherto the path of a body, under the influence of gravity, has been considered only in those cases in which it is a straight line — the two cases, namely, in which its motion is either vertical or down an inclined plane. But if it were pro- jected obliquely in space, and all resistance and obstruction removed, its path would be a parabola, as will appear from the following investigation. Let AB be the direction in which a body is projected from A with a velocity v^: then, if no other motion were communicated to it, it would move uniformly along the straight line AB with that velocity, and in t seconds would arrive at B, provided AB^=^v^t. But as gravity continues to act on the body from the instant of its motion, this force alone, in the t seconds, would draw the body down along the vertical AC to C, AC being =^f (p. 428). Consequently, complet- ing the parallelogram CB, the body in the t seconds would be found at P, as shown above. Now, since EQUATION OF PATH IN HORIZONTAL AND VERTICAL COORDINATES. 433 CP=AB=v,t, and AC=-gt\ .'. ^=-^, .*. CP^=-^AC ; 2i AC g g that is, regarding AC os the axis of x, and J.B as the axis of y, 7p=z — x, the equation of a parabola (396), the vertical y4C being a diameter, and AB a. tangent to the curve at the vertex A of that diameter. Therefore, abstracting the resistance of the air, the path of a pro- jectile is always a parabola, with its axis vertical. The coefficient of x in its equation is = 4:FA, F being the focus, and FA the radius vector of the origin, p. 343, or it is = 4/ij, h^ being the distance AT) of A from the directrix, which here is horizontal. But the equation — ?-=4/tp gives v{^=^gh^, which expresses the square of the velocity a body would ac- quire in falling through the height h^ : consequently the velocity with which the body is projected is equal to that which it would acquire from gravity by falling down from the directrix of the parabola described to the point of projection. And since any point in the curve may be regarded as the point of projection, it follows that the velocity of the projectile at any point of its path is equal to that which it would acquire by falling to that point from the directrix. Hence, at equal heights, ascending and descending, the velocities must be equal, and the velocity will be least at the highest point, that is, at the vertex of the parabola, /i^ being then the least. 461. Equation of Path in Horizontal and Vertical Co- ordinates. — Taking the point of departure A for the origin of hori- zontal and vertical co-ordinates, the equation of AB, the path the pro- jectile would pursue but for gravity, is y=a! tan a.; but in the time t seconds, gravity diminishes y by -gt'^^ therefore the actual value of y is y=x tan a.—-gf, a. being the angle which the direction of the impulse makes with the horizontal axis, or the angle of elevation of the piece. The velocity of projection being v^ the space AB in t seconds is Vjf, and the a; of j5 is therefore vf cos a, which is the a; of P in that time; hence, «=— ^ — [1], .-. y=a;tan«- ^ / „ x^ [2], i;, cos « '- -• " 2v^ cos2 a "- -■ the equation of the path, which is therefore a parabola (400). If h^ be the height due to the velocity of projection, then v^=^gh^, and the equation becomes x^ y—x tan «-— — [3]. 4A| COS'' tx, Let {a, h) be the vertex of the parabola, then the origin of the axes will be transferred to that point by putting a -{-a for a?, and i/-\-b for y. These substitutions give y=( tan a- "' )x—— --x^+(a tan a—- , ^ „ b), \ 2/t, cos^ a/ 4/i, cos2 « \ 4A, cos^* a /' F F 434 RANGE OF PROJECTILE. and that this may have the form y=:Qx^ we must have tan a=:— -— , 2A, cos-^ a and a tan a—— :;—=b=a\ — — — )=— — . 4/ii cos- a. \2Aj COS'' a 4A.j cos^ a/ 4A, COS"* a From these we get a=27i, sin a cos a=:A, sin 2a, 6=A, siu^ a [4], which are the co-ordinates of the vertex. And the equation of the parabola, the origin being at the vertex, is the curve being wholly below the tangent at the vertex, or the axis of x : the latus rectum of the parabola is therefore 4^^ cos^ a. (388). But the co-ordinates of the vertex may be obtained more readily thus : Since the ordinate of the vertex is greater than that of any other point, the expression [3] for this value of ?/ must be a maximum, ,•. (193), tan a—— ^-=0, .*. x=:2h, cos^ a tan a=h. sin 2a, 2/ii cos-^ a •*• [^]> 2/=2^'i cos2 a tan2 « —A, cos^ a tan^ a=A, sin^ », which values of a: and y are the co-ordinates of the vertex, as before. 462. Range of the Projectile. — The horizontal range is the distance AR from the point of projection, on the horizontal plane through that point, at which the projectile falls : it is of course got at once by putting y=0, in the equation of the path [3] ; we thus get for a;, Range^4^, cos^ a tan a=.2hi sin 2a [5]. This is the greatest possible when sin 2a is the greatest, which it is when 2a=90°, that is, when the elevation of the piece is 45", the length of this range being, ^2 Maxiimim ra7ige^=:.2\=:—, eqnal to double the height due to the initial velocity. As the sine of 2(45"— 6) is the same as the sine of 2(45° + 0), it follows from [1] that the range is the same for angles above 45" as for angles as much below 45", the charge being the same. Calling the maximum range R, we have from [5], for any other ranges r, /, due to elevations a, a.\ the expressions „ . - , T> . ^ / r' sin 2a' , sin 2a' ^„. r=i2sin2a, r'=i2 sin 2a', ,\ -=- , .'.r'= r [61; r sm 2a sin 2a SO that, 1. When the maximum range is known for a given charge, the range for any elevation, with the same charge, is also known ; and, 2. When the range r is found for any elevation a, the range / for any other elevation of with the same charge becomes also known. 463. Time of Flight.— By the equations [I], and [5], the time of flight is 2A, sin2a 4^i sin a , ^_ v? 2v, sin a ., „ „■ ■, ^ t= — ■ = — ■ : but A,=-^, .'. t=— , time of flight. I'l cos a Vi 2g g 464. Greatest Height. — The point of the parabolic path which is highest above the horizontal plane being the vertex of the parabola, and the ordinate of it being the value of h, above [4]; we have n=.hi sin2 a, the greatest height ...[7]. GREATEST HEIGHT. 435 Otherwise : without referring to this expression for the greatest ordi- nate, we have, for the vertical component of the velocity of projection, v^ sin cc : the projectile will .-. continue to rise till this velocity is destroyed by gravity. Now the height or space s, due to any velocity v, is And from similar considerations the time of flight may be found : for as the body reaches its highest point, or describes half its path, in the same time that the vertical component of the velocity of projection would carry it to that height, and since, whatever be the vertical velocity v, this time is i=- (p. 428), .-. iLalf the tune of fliglit= ^' ^^° " , as before. 9 9 Still employing the same principle — the resolution of velocities — we may, in like manner, now deduce the horizontal range : for the horizontal com- ponent of the velocity of projection is v^ cos a, and since the whole time of flight is 2v, sin a 2v, sin a v.^ . ^ ^^ . ^ , ^ -• j , .*. r= — ■ Xv, cos «=— sm 2a=2A| sm 2a, as before found. 9 9 9 And thus from the principle at (459), and the common theory of the vertical descent of bodies, may all the results of the foregoing articles be arrived at, without knowing even the form of the curve. We shall add a problem or two on the foregoing theory. Prob. I. Given the initial direction, to determine the velocity so that the projectile may strike a given point. Let {a/, y') be the given point, then, by the equation of the path [2], , n ^' / 9 y=xtan«— — — : — r—x^, .'. t\= \/ T^m K- 2vi^cos^« cos as V 2(a; tana— 2/) Prob. II. Given the velocity of projection (or the charge), to deter- mine the elevation of the piece so that the projectile may strike a given point. In the equation of the path [3], y=x tan a ; :;-, putting («', yO for the point, 4^1 cos- « and 1-f tan^ a for — r— , we have cos-^* tan^a ;itana=-( — „ +1 ), /. tan«=:. ^ - ■ — ^ ^- -. Hence, whenever the problem is possible, there are in general two dif- ferent elevations, at each of which the mark will be hit by the same charge : but if the initial velocity, or, which is the same thing, if the value of h^, be such that 4Ai2=4^,y+a;'2, or {2h^—y'f=.x'^-\-y'\ there is then only one direction, namely, 2h. tan «=-— . And when r, is such that ih^<4:hy'i/-\-x'\ or {2h^—yy. ;os* cos a I cos 4 4^1 cos a j Under this change of form, the substitution of the above value of x gives for the maximum value of mp, 2 A, sin («—%)( sin («—i) sin (a—i)") Ai sin' (a—i) m»= : < : — >= ■T-. • cos t I cos I 2 cos * J COS" % tan a— tan iz=.- 468. The foregoing articles contain all necessary particulars concerning the motion of a projectile in a non-resisting medium : the several results have been arrived at quite independently of experiment, except only that by which the force of gravity is determined ; special experiments, how- ever, would be altogether superfluous, as the motion of the projectile is governed by influences well ascertained and strictly determinate: — the instantaneous force of propulsion, and the continuous force of gravity, and by these alone. But any disturbing cause, or the action of any ad- ditional force upon the moving body, must of course vitiate the purely mathematical conclusions here reached. In the practice of gunnery, therefore, this theory must be considerably modified, on account of the resistance of the air, a force always as present as the force of gravity itself, and which is of such an effect on projectiles of considerable velocity, that *' some of those which in the air range only between 2 and 3 miles, at the most, would in vacuo range about ten times as far, or between 20 and 30 miles." This conclusion is justified from numerous experiments (by Dr. Hutton) specially undertaken for the purpose of arriving at safe practical rules of calculation for determining the range, time of flight, &c., of Military Projectiles : the principal of these practical rules are the following : — I. To find the velocity of any Shot or Shell. — Eule. Divide three times the weight of the charge of powder, by the weight of the shot, taking both weights in either ounces or pounds, and multiply the square root of the product by 1600. II. Given the range for one elevation, to find it for another. — Eule. As the sine of double the first elevation is to its range, so is the sine of double another elevation to its range. III. Given the range for one charge, to find it for another. — Eule. One range is to its charge as any other to its charge : the elevation being the same. It is to be observed, that of these three rules the first only is entirely the conclusion from experiment; the others are deductions from tlie parabolic theory : the third rule is equivalent to the statement that for 438 EXAMPLES IN GUNNERY. the same elevation the range is as the impetus, or height due to the initial velocity. The following examples, chiefly from Dr. Hutton, are to be solved by combining the first rule above, founded upon actual experiment, with the approximate rules, derivable from the parabolic theory of projectiles, established in the preceding articles. Ex. If a ball of 16 oz. weight be projected with a velocity of 1 600 feet per second when fired with 5^oz. of powder, it is required to find with what velocity each of the several kinds of shells will be discharged by the full charges of powder, namely : — Diameters of the Shells in inches . Their weights in lbs Charge of Powder in lbs. . . . The velocities of discharge . 13 196 9 10 90 4 8 48 2 64 16 1 i 594 584 565 693 693 These are got from Rule I. : thus, for the first velocity we have 2;i=1600Xy/ 27 _ 800 X 3^/3 _ 800 x 5 -1962 196~ 7 — 7 =694 nearly. Examples for Exercise. (1) If a shell range 1000 yards when discharged at an elevation of 45°, how far will it range with the same charge at an elevation of 30° 16' ? (2) The range of a shell at 45° elevation being found to be 3750 feet, at what eleva- tion must the piece be set, to strike an object at the horizontal distance of 2810 feet ? (3) With what impetus (^i), velocity (v^), and charge, must a 13-inch shell be fired, at an elevation of 32° 12', to strike an object at the horizontal distance of 3250 feet ? (4) A shell being found to range 3500 feet when discharged at an elevation of 25° 12' : how far will it range at an elevation of 36° 15', with the same charge ? (5) If with a charge of 9 lb, of powder a shell range 4000 feet : what charge will suffice to throw it 3000 feet, the elevation being 45° in both cases ? (6) How far will a shot range on an ascending plane of inclination 8° 15', and on a descending plane of the same inclination, the impetus being 3000 feet, and the elevation of the piece 32° 30' ? (7) How much powder will throw a 13-inch shell 4244 feet up an inclined plane of 8° 15' inclination, the elevation of the mortar being 32° 30' ? (8) Required the weight of the shot, in order that 6 lb. of powder may fixe it with a velocity of 1200 feet. (9) What must be the elevation of the mortar to throw a 13-inch shell 4000 feet up an inclined plane that ascends 10° 40', the charge being 4| lb. ? Note. — It is found from experiment that the greatest horizontal range, instead of being constantly that at an elevation of 45^^, as in the parabolic theory, will be at all intermediate degrees between 45° and 30°, according to the velocity and weight of the projectile ; the smaller velocities and larger shells ranging farthest when projected at an elevation of almost 45*" ; while the greater velocities, especially with the smaller shells, range farthest with an elevation of about 30°. " But it usually happens, in the operation of natural causes, that near the point where any quantity is greatest or least, its variation is slower than elsewhere : a small difference, therefore, in the angle of elevation, is of little consequence to the extent CIRCULAR MOTION, CENTRIFUGAL FORCE. 439 of the range, provided that it continue between the limits of 45° and 35° ; and, for the same reason, the angular adjustment requires less accuracy in this position than in any other, which, besides the economy of powder, makes it in all respects the best elevation for practice, where the object is to carry a ball or shell to the greatest possible distance."* But for very ample practical information on the subject of Gunnery, the student is referred to the third volume of Dr. Button's Tracts. 469. Circular Motion, Centrifugal Force.— If a body move uniformly in the circumference of a circle, the force which confines it to that curve must reside in, or be directed to, the centre of the circle : for resolving its actual motion at any instant into two — one in the direction of the tangent at the point where it then is, and the other in the direction perpendicular to that tangent — the latter direction will always pass through the centre of the circle : the former will be that direction in which the body tends to move at the instant, and therefore that in which it would move if left to itself. It is thus constrained to deviate from its wonted rectilinear path, solely in virtue of a force soliciting it towards the centre : this is hence called the centripetal force : the equal and opposite of this, that is, that opposing force which, with equal intensity, urges the body from the centre of the circle, is called the centrifugal force: — it is the force with which the body tends to increase its distance from the centre. Of course, it is here supposed that the body is confined to its circular path, or orbit, solely by the unimpeded action of these opposing forces : if a material resistance be interposed between the circumference and the centre, so that the body be thus prevented from yielding to the central force, then this may exceed the centrifugal force to any amount ; on the contrary, if the body's approach to the centre be unobstructed, but a material resistance is opposed to its further departure from it, the centri- fugal may exceed the central force to any amount. Let S be the centre of the circular path, and let AB be the arc de- scribed by the moving body from A in the time t seconds, where t may be a fraction of unlimited smallness. If the attractive force / at S, which when the body is at A acts perpendicularly to its wonted path AM, had continued to act perpen- dicularly to that line during the t seconds, then the body, since it is found at the expiration of that time at jB, would have fallen perpendicularly to AM, the distance DB=AE : so that v being the velocity of the body in the direction AM, we should have AD=vt, and (455) AE=if^, .-. eliminating t, f=2v^ ^,, or f=2iP ^,. Draw the diameter AA', and the chords BA, BA' (the former of which, though not in the diagram, may be supplied); then BA^=AA'XAE, .-. AI!=^„ hence by substitution, f=-—:,( -prA • AA' AA'KEBJ Now the force acting on the body, during its progress from A to B, is * Illusiratious of the Celestial MecLanics of Laplace (By Dr. T. Young), p. 132. 440 CENTEIFUGAL FORCE AT THE SURFACE OF THE EARTH. more nearly perpendicular to the tangent AD, the more the distance AB is diminished ; that is, the expression for / differs less and less from the exact expression, as t becomes less and less, and there is no limit, in the above reasoning, to the smallness of t. But when t decreases down to 0, the vanishing fraction r=r=,=—. — = 1. This is plain, because, calling ° EB sme the diam. of the circle 2r, and AE, x, we have BA"^ 2rx 2r , , -777-:=—— :=- =1, when x=zO. EB^ x{2r-x) 2r-x ' Hence the expression for the constant centrifugal force — equal and oppo- «;- site to the centripetal force at S — is/=— . Or calling the whole time of r describing the circle, that is, the periodic time, T, and remembering that the uniform velocity v is equal to the whole space divided by the whole time of describing it, we have 2irr „ -w' 4flrV ^ 80 that the centrifugal force varies as the radius of the circle directly, and as the square of the periodic time inversely. If the periodic times be the same in two different circles of radii r, Z2, and if /, F, be the respective centrifugal forces, we shall therefore have H t^^ It is worthy of notice that the expression for/ above, namely, /=—, co- incides with the expression [9] for / at p. 427, namely, /=£,wlien.=ir: hence the uniform velocity v, of the revolving body, is the same as that which the body would acquire by falling from rest towards the centre of attraction down half the radius of the circle, observing that /denotes in- differently, as to intensity, either the central or the centrifugal force, since the two forces balance. 470. Centrifugal Force at the Surface of the Earth.— In the preceding article the centripetal and centrifugal forces are accu- rately balanced, so that neither prevails ; but bodies on the surface of the earth preserve their distance from its centre during its diurnal rota- tion, though these forces are unbalanced — the centripetal or attracting force greatly exceeding the centrifugal force : if this were not the case, bodies at the surface of the earth would have no weight: the fact of their having weight proves that the central attraction — gravity — exceeds the opposing centrifugal force ; the superior attraction being further op- posed by the resistance of the solid earth, by which bodies are prevented from approaching nearer to the centre, as they otherwise would do, seeing they have weight. But the foregoing deductions equally apply to cases of this kind : — the revolution of a body in a circle generates a centrifugal force : there is an equivalent and opposite centripetal force which, as above, just balances it, and by so much is the actual force at the centre of the circle counteracted. CENTRIFUGAL FORCE AT THE SURFACE OF THE EARTH. 441 In what follows we have to ascertain the amount of this counteracting force at different places on the earth's surface. By its diurnal rotation, the earth carries round with it, with a uniform velocity, every point on its surface in 86,164 seconds, whether the point be nearer to or farther from the axis of rotation : as the time of rotation of all points on the surface is the same, the centrifugal force, which by [1] increases as the radius or distance from the axis of rotation increases, must be greatest at the equator, of which the radius is E=20,923,596 ft. There the centrifugal force is ^—W- ^86164^ - ^^^^^ *''*• As this force opposes that of gravity, it follows that if the earth had no rotation on its axis, gravity at the equator, instead of being ^=32-088, as experiment determines it to be, would be (t=^ + *11126 foot; so that the weight of a body there would be the part more than it actually is. Since G : F : : 32-199... : '11126, or as 289 : 1 nearly, .'. F=-— ...[3]; 289 or, since g:F\\ 32'088 : -11126, or as 288 : 1 nearly, .•. Fz=:-^ [i']. 288 Representing the centrifugal force in the parallel to the equator, of which the latitude is I and the radius r, by/, we have [2], ^=^, ,.f=F'-=± cos Z, or J^ cos Z, because J=cos Z...[5]. The force of gravity G is not diminished by the whole of the centri- fugal force except at the equator, since elsewhere this force does not act in a direction from the centre of the earth through the body, but from the centre of the rotating circle on which that body is— from the centre of the parallel of latitude. Thus, in the annexed diagram, while gravity, or the centripetal force on P, acts in the direction PO, the centrifugal force acts in the direction Pp. If, therefore, we decompose the force Pp in the perpendicular directions Pp', Pq', the former component being a force directly opposed to that of gravity, and the latter a force acting tan- gentially, and therefore urging the body P towards the equator EQ, we shall have Force opposing gravity, Pp'=Pp co^p'Pp=fco^ ;=-— cos^ I. Hence that part of the centrifugal force at any point of the earth's sur- face which is exercised in directly opposing gravity, varies as the square of the cosine of the latitude of that point : the fraction — - has been already shown to be -11126 foot. The other component Pq' of the centrifugal force at P being in the direction of a tangent at that point, is exerted in driving the particles 442 MOVINa FORCES. from the region of the pole N, to that of the equator, and to cause the oblate figure which the earth is known to assume on account of the accumulation of matter about the equatorial regions. The expression for this tangential force is Tangential force, Pq' =:/ sm l=z—— sin I cos 1=—^ sin 2?= '05563 sin 2L 289 57o This value is the greatest possible when sin 0,1=1, that is, in the latitude of 45", where the tangential force is equal to half the centrifugal force at the equator : on each side of lat. 45® the tangential force diminishes, and it vanishes altogether at the equator and at the poles. From the above principles we may readily determine the time in which the eartli must perform its daily rotation, in order that the centrifugal force at the equator may be exactly equal to G, the force of gravity there ; that is, in order that a body at the equator may lose all its weight. Let T' be the required time of rotation, the time at present being T ; then, the circle of rotation being the same, the centrifugal forces are inversely as the squares of the periodic times (p. 440) ; hence, the centrifugal force corresponding to the time T^ being Gj and that corresponding to the time T being — -, we have Q T T Q • ___ . . T^ . T'^ . T'—————— ' 289"* • ' •• ""V289"~17' Consequently, if the diurnal rotation of the earth were performed in the 17th part of the time that it really occupies, that is, if it were to rotate 17 times as rapidly, all bodies at the equator would be without weight, and, if placed at a small distance above the surface of the earth, would remain there unsupported. If the rotation were still more rapid, bodies would be repelled from the equatorial parts of the earth's surface. Examples foe Exercise. (1) What is the effect of centrifugal force in diminishing that of gravity in the latitude of London, 51° 31' N. ? (2) Express the value of that component of the centrifugal force in the latitude of London which is exerted in transporting the materials on the surface towards the equator. (3) Calculate both forces for the latitude of Dublin, 53° 21' N. (4) If the length of a sling be 2 feet, with what velocity must a stone placed in it revolve in a vertical cii'cle in order that it may be just prevented from falling out ? (5) If the central be n times the centrifugal force, prove that the velocity of revolu- tion is the same as that which the body would acquire in falling through a distance - nr A towards the centre, r being the radius of the circle. 471. Moving Forces. — In the foregoing articles on motion, no account has been taken of the bulk or weight of the moving body, except, indeed, in the few remarks concerning military projectiles : we have been chiefly occupied with the path described, the time of describing it, and the velocity of the motion, particulars which, in reference to an indi- vidual body, moving under the influence of gravity, or of any similar accelerating force, remain the same whether the body be great or small — MOVING FORCES. 443 whether it he a single particle or the aggregation of any numher of particles, seeing that gravity affects all alike. But if instead of a single isolated body, left free to obey this force, two or more bodies are so connected together that the motion communi- cated to either modifies that which, without such connection, would be communicated to the others, then the investigation of this motion renders the consideration of the quantity of matter moved indispensable. 472. Mass. — This term is used instead of the words " quantity of matter." If a body be compressed into one-half of its original bulk or volume, the quantity of matter would be the same in each of its states ; but a cubic inch or foot of the compressed material would contain twice the quantity of matter in a cubic inch or foot of it in its original state, and we therefore say that the mass of the former inch or foot is double that of the latter inch or foot. But, as in other cases of measurement, it is necessary that some definite unit of measure should be fixed upon by general consent, and accordingly it has been agreed upon that, taking the cubic foot as the unit of volume, the quantity of matter in that bulk of pure distilled water should be regarded as the unit of mass. 473. Density, — This has reference to the closeness, more or less, with which matter is packed — to the volume in which it is comprised. In the illustration above, though the quantity of matter is the same whether it occupy a cubic foot of space, or be compressed into only half a foot, yet in the latter case, since it is packed into half the space or bulk it originally filled, it is said to be twice as dense. Hence, if there be two bodies A, B, of equal volumes, and A be found to be m times the weight of B^ we infer that there is m times as much matter in A as in B ; and, therefore, that the matter in A has m times the density of the matter in B. But although, by thus weighing two bodies of the same volume, at the same place, we could discover the ratio of their masses, and also the ratios of their densities, yet it must be observed that weight, mass (or quantity of matter), and density, are three very different things : weight is the statical measure of the external influence exercised upon mass by gravity ; an influence which varies according as we transport the mass nearer to, or farther from, the centre of the earth : the same mass, at the pole, has a different weight from what it has at the equator. Mass and Density are, therefore, terms irrespective of, and having no necessary reference to, weight : — if bodies had no weight, these terms would be equally significant. In like manner, as distilled water, at a certain temperature (60** Fahrenheit), supplies, by universal convention, the unit of mass, so the same fluid equally supplies the unit of density ; that is, we regard the density of water as I ; so that when the density of any material is said to be D, we understand that it has D times the density of distilled water; any given volume of it will therefore weigh, at the same place, D times an equal volume of distilled water. Mass, density, and volume, are therefore thus related, namely, Mass=DeiisityX Volume, or, in symbols, M=DV : mass is therefore the numerical expression for units of magnitude or volume : — not of the volume V, but of the volume that V would become if the body originally of volume V were expanded or compressed till it had the density of distilled water. Suppose, for example, the volume of a body to be 1 cubic feet, and 444 MOVING FORCES. that in each cubic foot there is three times the quantity of matter that there is in a cubic foot of distilled water, that is, that the body has three times the density of distilled water, then, for the mass M of the body, we have the value M"=-DF=3 x 10=30 cubic feet of matter equal in density to distilled water. And it is easy to see that, for the purposes of calcula- tion, there is great convenience in thus reducing different denominations to a single one : — replacing all matter — of which the varieties are so great — by its equivalent of a specific kind — pure water. 474. Momentum. — The momentum of a moving body is the- product of its mass by its velocity : it properly represents the force of the blow with which the moving mass would strike an obstacle. Thus : taking the mass M in the example above, suppose it to move uniformly with a velocity v of 12 feet per second, then we should have Mv= the momentum of 30 cubic feet of a substance of the same density as water, moving 12 feet per second. Calling the momentum of one cubic foot of water (or of any matter of the same density as water) moving one foot per second, the unit of momentum, we have in the case before us Mi;=360 of these units. 475. Moving Force. — In uniform velocity the momentum of the mass moved is constiant : in uniformly accelerated velocity, additional momentum is generated every second, the increments of momentum being constant like the increments of velocity. This constant increment of the momentum is called the moving force of the mass. Moving force is thus related to momentum, as acceleration is to velocity; that is, putting, as heretofore, V for velocity, and / for acceleration (or accelerating force), we have V : f : : Mv : Mf, the moving force, F. The statical effect of a constant force on a mass M is continued constant pressure or weight : the dynamical effect is constant moving force : — un- resisted pressure. It is important that the student acquire correct notions of the terms here defined. If a body in motion be acted upon by no force, its velocity is constant : this constant velocity affects equally every particle of the mass : the sum of all the particles, that is, the whole mass, multiplied by this constant velocity, is the constant momentum Mv. If a constant force / act on the mass, it acts on every particle, and constant acceleration of the velocity takes place : this multiplied by all the particles, or the whole mass, is, in like manner, the constant increment of the momentum, and is therefore represented by MJ, and called the moving force. The symbol/ here, as elsewhere, though called the constant /orce, stands solely for the effect of that force — acceleration of velocity : in the case of gravity it is S^-^feet. When we speak of multiplying this by M, we mean the midtiplying it by the number of cubic feet of distilled water to which M, as a mass, is equivalent. The only concrete quantity entering these expressions is therefore feet, but we may, if we please, eliminate even this, and employ abstract numbers only, thus : Let the quantity of matter in a cubic foot of pure water be taken for the unit of mass, and the density of pure water for the unit of density. Let the momentum of the unit of mass (one cubic foot of pure water, or of matter equally dense) moving with a uniform velocity of one foot per second, be taken for the unit of momentum. Let the constant increment of momentum of the unit of mass, generated in each second, when the constant acceleration is one foot per second, be taken for the unit of moving force. Then V representing the number of cubic feet in the volume of a body B, M the number of units of mass, v EXAMPLES OF MOVING FORCES. 445 the number of feet per second of velocity, / the number of feet per second of acceleration, and s the number of feet passed through in t seconds, we shall have M=.DV= the number of units of mass in the body B of density D and volume V. Mv^ the number of units of momentum of B moving with velocity v. F=:Mf=the number of units of moving force of B, when its acceleration is /. «=-/){'=: the number of units of space passed through in t seconds when the acceleration is /. F From the equation moving force F=:Mf, we get acceleration /=— [I.], that is, it is the moving force divided.by the mass moved. Note 1 . In solving particular dynamical problems, the student will in general find it to add to the clearness and intelligibility of his symbolical results if one concrete quantity — the concrete quantity linear space — be preserved; that is, to regard s, v, g,f, as representing /<3<3i. If the problem be statical, s and v can neither of them enter the inquiry, but g and/ may, not however as generating motion, but weight, which is the only concrete quantity in Statics (see the problem which follows). 2. It is well too for the student to notice that, as the velocity acquired in t seconds, when the constant acceleration is /, is v=ft, the expression Mv, for the momentum of a moving mass, is equally expressed by Mft. m 476. Examples of Moving Forces. — Pbob. I. Two heavy bodies W, W\ whose masses are M, M\ are connected by a string passing over a fixed pulley : required the circumstances of the motion. Let M be the greater of the two masses. If M were supported, the pressure on the support would be the difference of the weights: this pressure, .-. when unresisted — that is, the moving force — j@ is F=(M-M')^. The mass moved is M-j-M'; for not only is M moved downwards by this force, but M' is equally moved M—M' ^ upwards. Hence [1],/=-— — —g. Also for the velocity of M downwards, or of M' upwards, and for the space passed 'wi through equally by each, from the commencement of motion, we have [2], page 427, ^ M-M' 1 M-M' ^ in which, since masses are proportional to their weights, W, W\ may bo W—W substituted for M, M' : we may therefore write /=t77 — ^tp, (J- Tension of the String. — If the string T were fastened to the pulley so as to prevent motion, the whole force of gravity acting then upon M units of mass, it would stretch the string with the whole weight W : but as it is, only a part of gravity acts upon M to stretch the string, for the part /= ^ — 77?, of gravity, is expended in producing motion. Hence, calling the tension T, we have W: T : : g : g—f, that is, W—W W—W 2WW WT- '1-1-- — ' T—W—- ~W= 446 EXAMPLES OF MOVING FORCES. The same expression for T would of course be given by considering the force acting upon the other mass W ; this force is not only that of gravity W—W acting downwards, but also the force/, or the — — —part of gravity acting upwards, both conspiring to stretch the string : we thus have W :T ::g : g+f, that is, W~ W W— W 2 WW w-\'W" ' w-\-w' ir+ir 477. The foregoing investigation involves considerations partly dynami- cal and partly statical, tension being of the nature of weight, or statical pressure. This effect of gravity canuot, with any propriety or consistency, be represented by g, which has always been employed hitherto to stand for 32-2 /(?^f, at least in the latitude of London, nor yet by the moving force Mg, unless due warning be given to the reader that the symbol is to take a different meaning. Without such change of meaning the state- ment W=Mgj given in most books on the present subject, is simply absurd. It is plain that the weight of any mass must be W=Mx the weight of the unit of mass, M being the number of such units in the mass. Now, by general consent, the unit of mass is 1 cubic foot of distilled water, which is found to weigh exactly 1000 ounces ; this, therefore, is the value of g in the expression W=Mg, which, with this statical interpre- tation of g, correctly expresses the weight in ounces of any body when its mass is given. It is evident that it must do so, for (by def ) M is the number (whole or fractional) of cubic feet of pure water to which the body is equivalent, as to weight ; but 1 cubic foot of this weighs 1000 oz. ; hence the weight of the body is W=M xlOOO oz.=Mg=DVg (p. 443). And in like manner, if any other unit were taken for that of mass, g would stand for the weight impressed by gravity on that unit. With this interpretation of g and of / (which above is a fractional part of g), let us return to the statical part of the present problem. „, . . , / M-M' \ 2MM' The tension, or weight on the string, is T=M{g-f)—M{^g-j^--^ J=^-^.g. W 2WW' But M= — , .*. T= , as before. [Here g and / both represent weight.] Pressure on the axis of the Pulley. — Since the tension or weight acting on the string is the same on each side of the pulley, therefore the pressure or weight on the axis is 4MM' 4WW M+M"" W-\-W'' Suppose the weights TF, W\ were 7 lb. and 5 lb., then the tension of the string would be 5| lb.; that is, a weight of 5| lb., if suspended to a fixed string, would produce the same tension as is actually produced in the moving string. And the axis of the pulley would in like manner sustain a pressure of I If lb. Also the acceleration / would be 7—5 /=- — -32-2 ft.=5'36 ft. per second, T-f-o being J of the accelerating force of gravity. [Here / represents linear siiace.l atwood's machine. 447 478. Atwood's ]M[achine. — This was a machine based upon the principle of the preceding problem, and contrived by Atwood to render the acceleration produced by gravity, or the dynamical value of g, very easily deducible from observation. Omitting here the vertical scale to measure the space descended, the friction wheels to diminish the friction on the axis of the pulley, &c., it is plain that the weights PT, W\ may be W—W so chosen as to render the fraction = — =, of any value we please, that is, so that the acceleration/, of the descending weight, may be any fractional part we please of the acceleration g due to gravity. Hence the motion of the descending weight may be rendered sufficiently slow for the space passed over in one second, or in any number of seconds, to be accurately observed ; and since we know this space s to be \ W-\-W s=--ft\ we thus readily find /, and thence gr= _ /. Prob. II. Two weights connected by a string passing over a small pulley Care placed upon two smooth inclined planes CA, CB: it is re- quired to determine the circumstances of their motion. As before, putting M, M\ for the masses of the weights W, W\ the expression for the moving force of W, in the direction CA, is Mg sin i ; and the moving force of W\ in the direction CB, is Mg sin i' : these forces oppose each other, so that the moving force F, to which the motion is due, is their difference, namely, F M sin i—M' sin i' i5'=(JJf smi-lf'sm i") g, .'. [I.], /= 7= rj— T77 ff- \ /i/> L jw mass moved M-}-M This, therefore, is the expression for the acceleration of W down the plane CA, and consequently for the accelera- tion of W up the plane BC. And since the velocity generated in t seconds from the com- mencement of motion is ft, and the space passed over -ft^, the velocity acquired by W downwards (supposing this weight to prevail), or that acquired by W^ up- wards, in t seconds, is M sin i—M' sin i' , M sin i—M' sin i' „ where ^=32-Q feet: the only concrete quantity in these expressions for the velocity and space. If the two planes were vertical the problem would be identical with the former one, sin i and sin i' being each=l. But if one only of the planes were vertical, then the problem would be to determine the motion when one body W, hanging vertically, draws another W up an inclined plane. In this case we have only to make sin i=l in the foregoing results. In like manner, if one of the planes, AG, were vertical, and the other hori- zontal, then in the same expressions, M sm z=l, and sin i'=Q, .-. f=j^^^. And in all these expressions W, W\ may be substituted for M, M\ since 448 MOVING FORCES — WHEEL AND AXLE. the masses are as their weights, and since the foregoing fractions remain the same, whatever quantities proportional to M, M\ be substituted for them. Tension of the String. — Under the original conditions, W being the descending weight, we have, for the tension of the string, the whole pres- sure of W in the direction of CA, diminished by so much of it as is expended in producing the motion, that is, it is m n^/ . . ^ sin i—M' sin i \ - (sm *+sm tjg^rf—-—— (sin ^-|-sm t% which becomes the same expression as that in last problem, when i and i' are each 90°, that is, when the weights hang vertically. Prob. III. The height and length of the plane AG (last fig.) are given : to find the length of the plane BC, so that a given weight W may draw another given weight W up BC in the least time possible. Let h and I represent the height and length of the given plane, and put X for the length of the required plane ; then since . . h . . ., h ^ Whx-W'hl , . sm 1=^ , and sin i = -, .'. /= ^ ^ gr, and since s=a7, ^ 2s 2a; 2(W-\-W')lx'^ . . a? .". ^=— r=-r=777;^ r— — — -= a mimmttm, .*. --7 =—=a minimum, / / {Wx-W'l)hg Wx—Wl * .'. (p. 148), x^-Wux-\-W'lu=0, .-. (p. 148), x=i-Wuy and 4tW'lu=Whi^, 2 AW'l 2W'l .*. tt= , .*. x=. , corresponding to the minimum required. If the weights are equal, then x=2l ; hence, that a weight W may draw an equal weight (TF') up the plane BC, in the least time possible, the length of BC must be double the length of AC. Pros. IV. A weight W is attached to the axle of a wheel, and another W^ to the wheel itself: to determine the motion. Put r, R, for the radii of the axle and wheel, then, since the weights act at these distances from the axis of motion, their efficacy in producing motion round that axis will be expressed by ^.r and W\ R, so that, putting M, M\ for the masses moved, the moving forces will be Mg . r, and M/g . R, and since these oppose each other, the actual motion is due to only the moving force {M'R—Mr)g, a portion of which is expended in moving the mass M' downwards, and the remaining portion in moving the mass M upwards. Let the accelerations of M, M\ be/,/, respectively, then the moving forces, or moving pressures, on these masses — acting as they do at the dis- tances r, R, from the axis of motion — are Mf.r^ and Mf.R. But the whole moving force is [MR—Mr)g, hence {M'R-Mr)g=M'f. R-\-Mf. r [1]. Now the accelerations /',/, are to each other as the radii R, r, the veloci- ties themselves being always in this constant ratio, since the wheel and axle both rotate in the same time. .'.f=-f, .:{M'R-Mr)g=(^M'~+Mry, .-. /=^ \9- M'R-^-^Mt^' This is the expression for the acceleration of the ascending weight W; and since PRINCIPLE OF d'aLEMBEBT. 449 /=-/, ••• / = M'mj^Mr^ ^^ '' *^' acceleration of W . Also, since the masses are as the weights, TF, W\ may be put for M, M', that is, we may write {W'R- Wr)r (W'R-Wr) R For the velocity of the ascending weight, at the end of t seconds from the commencement of the motion, we have ^ ( If '72- Wr)r , , , 1 ^ o 1 ( 1^'^- ^»')»* ^ In like manner, for the descending weight W, we have ^_ {W'R-Wr )R _\ {W'R-Wr )R If Iiz=r, the problem will be the same as Prob. I., and / and /' above will become each converted into W— W -r— — — , W being here the weight moving downwards. W -\-W If we divide the linear velocity v (acquired in any time t"), of the circum- ference of the axle, by the radius of it, r, we shall get the angular velocity of the axle (453) ; or if we divide the linear velocity v\ of the wheel, by its radius R, we shall get the angular velocity of the wheel : but as each performs a revolution in the same time, the angular velocities must be equal (453), and accordingly we find, whether we take the axle or the wheel, that W'R-Wr ^ , W'R-Wr angular velocity=-^p;^^-^5'<, and ang. accel= ^,^^_^^ , gr. It is plain, from the property of the lever, that the numerator expresses the efficiency of the moving force to produce rotation : — the denominator expresses the resistance to angular motion, and is analogous to the mass moved. 479. Principle of D'Alembert.— When masses are perfectly free, whatever forces may be impressed upon them must have full effect ; but when they are connected together, as in the foregoing problems, so that one mass is, as it were, a drag upon another, the effect of the force impressed upon each must fall short of what it would be if no such con- nection between the masses existed. Let the forces which would actuate the masses if they were perfectly free be called the impressed forces ; and let those which would suffice to produce the actual effects, be called the effective forces : these latter applied to the several masses, simultaneously with the former — if applied in a contrary direction — would obviously prevent all motion in the system ; that is, there would be equilibrium. This axiomatic truth constitutes the Principle of D'Alembert : it may be thus formally enunciated : — There must always he equilibrium between the impressed and the effective forces, provided the latter be made to act opposite to their real directions. The foregoing problems may all be solved by aid of this general prin^ ciple, which we see reduces every dynamical inquiry, involving the con- sideration of moving forces, to a statical condition. Let us take the problem just solved : — The impressed forces, in obedience to which the masses would move if 450 MOMENT OF INERTIA. free, are Mg, acting with the leverage r, and M'g, acting with the leverage R, the motions being downwards. The effective forces, in obedience to which they actually move, are for M, upwards, —Mf, and for M\ down- wards, M' f\ the former acting with a leverage r, and the latter with a leverage U. Taking these with opposite signs, we have for the equation of equilibrium {Mg+Mf)r={M'g-Mr)R. which is substantially the same as the equation [1] in the former solution. The student is recommended to solve Prob. II. by this principle. Examples for Exercise. (1) A weigM of 4 lb. is attached to a string passing over a pulley, and dra-ws up a ■weiglit of 1 lb. fixed to the other end of the string : how far will the heavier weight descend in 3 sees, ? (2) Two weights of 1 lb. each are suspended at the ends of a cord passing over a pulley : an ounce weight is added to one of them : how long will it be in descending 12 feet, and what velocity will it then have acquired ? (3) What weight must be added to one of the equal weights W in last example, in order that in 6 sees, it may acquire a velocity of 48 feet ? (4) The weight W (Prob. I., p. 445), instead of descending from rest, has a velocity of V, feet per second communicated to it : what velocity will it have acquired at the end of t seconds ? (5) A weight hanging vertically from a string passing over a small pulley fixed at the top of an inclined plane draws an equal weight up the plane. If the length of the plane be 12 feet and its height 10 feet, in what time will the ascending weight reach the top of the plane ? (6) Griven the height h of an inclined plane, to find its length I, so that a given weight W^ descending vertically, shall draw another given weight W up the plane in the least time possible. (7) The common height of two inclined planes is 20 feet ; the length of one is 30 feet, and that of the other 40 feet : equal weights connected by a string passing over a pulley at the top (as in Prob. II., p. 447), are placed on these planes : how long will the descending weight be in falliag from the top of the plane along which it moves to the bottom ? (8) A weight of 20 lb. raises a body of twice the volume and of three times the density by means of a wheel and axle : the radius of the wheel is 3 feet, and that of the axle 4 inches : through what space will the descending weight fall in 10 sees. ? (9) A given weight P raises Q by means of a wheel and axle, of which the radii are R, r : what must the weight of Q be, so that in t seconds the momentum of it may be the greatest possible ? [See Note 2, page 445.] (10) The radius r of the axle only is given : find the radius R of the wheel, so that a given weight P may raise another given weight Q through any assigned space in the least time possible. 480. Moment of Inertia. — It has been seen in the solution to Prob. IV. that when masses are so connected together that the result of the forces acting upon them is rotation, the denominator of the fraction for the acceleration / is no longer the sum of the masses moved, but the sum of the products of each mass into the square of its distance from the fixed axis. In the problem referred to, the forces act, through the inter- vention of the strings, as if they were immediately applied at the ex- tremities of the radii of the wheel and axle, so that these radii may be IMPACT. 451 regarded as the respective distances from the axis, of the masses acted upon, the masses being viewed as condensod into mere particles. If by the distance of a mass from the axis of rotation we always understand the distance, from the axis, of the point into which if that mass be condensed, the rotation will be the same, then the product of the mass by the square of its distance is that which is called the moment of inertia of the mass, with respect to that axis. A point having the property here alluded to may always be found for every macss of mathematical form, and in refer- ence to any axis: — it is called the centre of gyration of the body; and the distance of this centre from the axis — which distance is usually re- presented by k — is called the radius of gyration : hence the moment of inertia of any mass, in reference to a given axis, is the product of that mass by the square of the radius of gyration, that is, in symbols, the moment of inertia is Mk\ The determination of k, for different forms of bodies, and for any given axes, requires the Integral Calculus, under which head the necessary formulae will be investigated. Since not only the masses themselves, but the rotating parts of the machine with which they are connected, are put in motion by the im- pressed forces, the moment of inertia of the latter should, in strictness, be taken into account. In the problem above, on the wheel and axle, this has been omitted; so that the denominator of the fraction/ contains only a part of the whole moment of inertia. Supplying therefore the part neglected, namely, mk'% m representing the mass, put in motion, of the machine, and k its radius of gyration, we have for the correct values of/,/'. {M'R-Mr)r _ {M'R-Mr)R If r=E, the problem becomes that of the weights and pulley (Prob. I.), and the correct expression for /, or /, these being then equal, is (M'-M)R^ ^"{M' + M)R^+mlc^' M' being here the descending mass. As mass a weight, W, W\ w, may be put for M, M\ m, in these expressions, where it is to be observed that IV is not the weight of the machine, but of that part of it only which is made to rotate by the action of the impressed forces : the denominator of the expression for / denotes the sum of all the resistances to rotation, that is, to angular motion. If m' be the mass, which, placed at the extremity of Z?, would offer the same resistance to angular motion as the mass m of the rotating part of the machine, m' would be such that 'p) ' (k \^ ^ j , is usually so small, comparatively, that, in the mechanical powers, it may in general be disregarded. 481. Impact. — When a moving body strikes another, whether at rest or in motion, the shock which takes place is called impact or collision. A body may so strike an immoveable obstacle as to adhere to it, a a ^ 452 DIRECT IMPACT. if sufficiently soft, or else to rebound and return along the line of its original motion ; or in the case of a second moving body it may follow in its path, or meet it in that path, so that whatever motion after collision takes place, the common path of that motion may not be departed from. All these are cases of direct impact, or direct collision : these we shall consider first, previously, however, making a few remarks in reference to the elasticity of bodies of different material. It is found by experiment that when impact takes place between two bodies the shock occasions some amount of compression of the parts brought into contact, so that an in- stantaneous change of figure is produced. If, after the collision, the original form of the bodies is as instantaneously restored by a force of restitution, equal and opposite to the force of compression, the bodies are said to be perfectly elastic. On the contrary, if no force of restitution be brought into action, the bodies are said to be perfectly inelastic. There is no reason to suppose that any such bodies actually exist in nature : whatever be the substances experimented on, it is always found either that the force of restitution is less than that of compression, or that the effect of this latter force is less instantaneous, so that if the original figure be restored, the restoration occupies more time than the compression, the bodies being driven asunder after the impact before all the force of restitution is expended, and therefore with less violence than that with which they struck. And although the material may be such that the impinging body may adhere permanently to that struck, and unite with it in one mass — as in the case of a lump of clay or putty — ^yet experiment finds that some force of restitution is always exercised, and that even clay and putty are not perfectly inelastic. The ratio which the force of restitu- tion bears to the force of compression, as respects any substance, is called the modulus of elasticity of that substance. Newtons contrivance to ascertain this modulus for hard substances was this : he suspended two balls of the same material from two slender threads, so that when they hung vertically they might just rest in contact : upon being drawn asunder each would describe a circular arc. These arcs were accurately graduated, and each body being drawn from the vertical, through the same extent of arc, was let go at the same instant : the force of their rebound after impact was always found insufficient to carry them back through the same extent of arc. It was inferred from the experiments that the modulus of elasticity for ivory was - of perfect elasticity, and for glass — . From experiments of a different kind, Prof. Hodgkinson determined the ratios to be for ivory '89, and for glass, "94, results that differ very little from those of Newton. If e be put for the modulus of elasticity, V for the velocity of direct impact upon an immoveable plane, or the velocity of approach, and v for the velocity with which the body is driven back, that is, the velocity of recoil, we shall have v=eV [1]. 489. Direct Impact.. — Let two bodies, of which the masses are M, M\ moving with velocities V, V\ strike with direct impact ; and first suppose that there is no elasticity. The momentum of M is MV, that of M\ M'V\ .', the momentum of the compound mass, when they both unite, will be MV-\-M^V\ where the symbols F, V\ involve their pr(/;;^er bigns, these signs being like, if the motions of both bodies are in the same DIRECT IMPACT. 453 direction, and unlike if the motions are in opposite directions : — motions from left to right we shall as usual regard as +• If the common velocity after impact be F^ then, since the two masses are united, we must have MV+M'V {M+M')V,=MV^M'V\ .'. V^=-^^^-^r- [2]- If the bodies move in opposite directions, and the momenta before collision be equal, then F, F' taking opposite signs, F^=0, that is, each body will be brought to a stand-still by the shock. If one of the masses 3f ' be at rest before impact, MV then since F'=0, F,=— -rr^. The principle assumed above, that the momentum after impact is the algebraic sum of the momenta before impact, is implied ia what is called The Third Law of Motion, which, as stated by Newton, is this : — " Action and reaction are equal and contrary," that is, A cannot act mechanically on B, without A itself being reacted upon equally, but in an opposite direction. In the case before us the momentum lost by one of the bodies is the momentum gained by the other. Let M be the mass which loses momentum, then vel. of M before impact = F ) , , , ^^ „ tt „\ .: vel. lost by if= V— Fi- „ after „ =F,J Butr2i V v-v ^^+^'^' {v-r)M' rg. But [2], V-V,-.V ^:;p^7— j^_^j^, -l^}- The difference F—F' of the velocities of approach (always taking these with their proper signs) is the relative velocity of approach, so that we may put the foregoing conclusion thus : — Vel. lost by M M' Eelative vel. before impact M-\-M.' Vel. gained by M' M And, in like manner, Relative vel. before impact M-\-M'' And from these equations it follows that The relative vel. before impact = vel. lost by iff -f vel. gained by M'...[i\ for they give Vel. lost by M : Vel. gained by M' :: M' : M^ . Vel. lost by M M' Vel. lost by M-{-\el. gained hYM'~M-\-M'' Ex. A body of 51b. weight moves from left to right with a vel. of 12 ft., and comes into direct collision with another of 9 lb. weight moving also from left to right, with a vel. of 8 ft. : required the velocities lost and gained. 9 4 The masses being as their weights, vel. lost by M [3], =ft^(12-8)=2^. A O And the relative vel. being 4, .-. [4], 4—2 -=1 == vel. gained by M' The common vel. after impact is [2], 60+72 3 ^'-"14—^7' agreeing, as of course it ought to do, with the above results ; 454 OBLIQUE IMPACT. for 12-2^=9? and 8+1^=9^. If the bodies moved in opposite directions, the second mass M' moving from right to left, then the sign of 8 would be minus, and the relative »felocitj would be 12 + 8=20 : we should then have rel. lost by il!f=-^(12+8)=12^, and vel. gained by if' =20-12 -=7=. 6 6 Hence M is driven back with a vel. 12—12 -=— ^, 7 7 1 fi and M' proceeds with a vel. —8+7 ^=— ^> that is, both proceed, in union, from right to left, with the common velocity—-. Let now the bodies have elasticity, the modulus of elasticity being e. Let the velocity of M after impact be F^ and that of M\ F/, then the momentum lost by M (that is, communicated to M') is M (V—Vj), and that gained by M' is M' (V^'-W), . •. 3I{ F- Fi) =ir ( Vi'-V) [6]. Also V—V is the velocity of approach, and V^^—Vj^ the velocity of recoil, .:V,'-V,=e{V-V') [6]. Solving [5] and [6] for F^ and F/, we have _ 3IV-\-M' r-eM'( V- V) _ MV-\-M'V'-eM{V'-y) '~ M-^M' ' ^~" M+M' ' each of which coincides with [2] when e—0, or the bodies are non-elastic. If e=l, or if the bodies be perfectly elastic, then the velocities after impact are {M-M')Y-\-1M'V _ {M'-M)V+2MV '~ M+W' ' ^"" W^M' ' SO that if in this case M=M\ we must have V^=V\ and V/=V; that is to say, if two perfectly elastic and equal bodies strike with direct impact, each, after the stroke, has the same velocity that the other had before the stroke. In the case of perfect elasticity, [6] is F/- Fi= F- F, .-. F+ F,= F/+ V. Multiplying [5] by this, we have Hence, when the bodies are perfectly elastic, the sum of the products of each into the square of its velocity is the same after as before impact. The product of a mass by the square of its velocity is called the vis viva (or living force), so that in the collision of perfectly elastic bodies, no loss of vis viva is occasioned by the impact. 483. Oblique Impact. — 1. Let a ball whose elasticity is e strike a perfectly hard and immoveable plane : to find the motion of the ball after impact. Let AB be the path of the centre of the ball before impact, and BC^ its path after impact: then if DBD' is perpendicular to the plane CD, the angle ABD'=^ is called the angle of incidence, and OBLIQUE IMPACT. 455 D'BC'=^\ the angle of reflexion. Let 7, the velocity along ABC, be represented by EG : the components of this — one perpendicular to the plane, and the other parallel to it — are BD, BE. The component BE, being parallel to DC, is un- affected by the action of the plane ; but BB, by the force of restitution, is converted into BD', .'. com- pounding the velocities BE, BD', the path and velocity Fp after impact, will be represented by BC\ vel. ^^ = F sin ^ \ '' ^1 =(^^ ^^^ <»)2+( Fsm ^,2= 72(^2 ^^^2 ^^^,^. ^). „ . ^ DC Fsin^ 1^ ^ tan ^ Also tan ^=-Erf^=-^ y=- tan 6, .'. 6=- — r,. -Bi/ e F cos ^ c tan ^ Thus knowing the modulus e, and the direction and velocity of inci- dence, the direction and velocity of reflexion may be easily computed, or, observing the angle of reflexion consequent upon a given angle of inci- dence, the modulus e may be ascertained. If e=\., that is, if the elasticity be perfect, then Fi=F, and 6'=0, showing that then the velocity is the same after impact as before, and that the angle of reflexion is equal to that of incidence. But if e=0, then Vi=V sin 9, and 6'=90° ; that is, if the body have no elasticity, and the plane be perfectly hard, the former, after oblique impact, will slide or roll along the plane with the diminished velocity F sin 0. [In the figure, BD^ should be less than BD,] It may be further noticed that in the case of imperfect elasticity the vel. after impact will always be less than that before impact, but the angle of reflexion will always be greater than that of incidence. 2. The annexed figure represents two bodies A, B, impinging upon each other at the point : CD represents the magnitude and direction of the velocity of A, and (7'Z>' the magnitude and direction of the velocity of B. Let mn be the line perpendicular to the surfaces at 0. Then, by the resolution of the velocities, A may be regarded as animated by two velocities, one (7^ parallel to nm, and the other CF perpendicular to nm. In like manner, B may be regarded as animated by the velocity C^E\ parallel to nm, and the velocity C'F', perp. to nm. If A, B, at the instant of contact, were animated solely by the velocities CF, C'F\ they would merely slide past each other, and would experience no shock ; the shock they do experience is therefore due solely to the velocities CE, C'E' ; and even then, that any shock may take place, the former velocity must exceed the latter. The intensity of the impact is the same, therefore, as if the two velocities CE, C'E', alone existed ; and the bodies will move as in the case of direct impact, only that these motions will be combined with the unchanged velocities CFy C'F\ Hence we shall merely have to com- pound the velocity of A, in the direction nm, after the direct impact spoken of, with the CF, perp. to nm, in order to obtain the magnitude and direction of ^'s velocity after the shock. And in a similar way are the magnitude and direction of ^'s velocity to be found. Examples for Exercise. (1) An inelastic body A weighing 10 lb., and moving with a velocity of 3 ft. per second, impinges directly upon another B weighing 7 lb., moving in the same direction with a 456 TRANSMISSION OF PRESSURE. velocity of 2 ft. : required tlie common velocity after impact, the velocity lost by A, and the velocity gained by B. (2) A weighing 2 lb. and moving with a velocity of 6 ft., comes into collision with By weighing 3 lb., and moving in the opposite direction with a velocity of 5 feet : the coef. of elasticity is "6 : required the velocities after impact. (3) If a ball falling from the height of 100 ft., on a perfectly hard plane, rebound to the height of 70 ft., what will be the coef. of elasticity ? 15 (4) An ivory ball — ^the modulus of elasticity of which is ——falls from the height of 100 feet upon a perfectly hard plane : including the several rebounds, what will be the whole space described before the motion ceases ? (5) A ball is to be projected from a given point, so as that after reflexion from a given plane it may pass through another given point : it is required to find where the ball must strike the plane : the elasticity between the ball and plane being c. 484. Scholium. — The foregoing short treatise on Dynamics compre- heuds as much of this extensive science as can be conveniently introduced without the aid of the Differential and Integral Calculus : — a department of pure mathematics which is in especial demand in the higher class of inquiries respecting motion. Exemplifications of this will be furnished in the subsequent treatise on that subject. End of the Dynamics. VIII. HYDROSTATICS. 485. Hydrostatics treats of the equilibrium of fluids, of which there are two distinct kinds, namely, compressible fluids and incompressible fluids: — the latter being generally called liquids, of which the most important is water. Of the former kind the most important is the air we breathe. Air, the gases, and vapours, are called compressible fluids, because by the application of pressure they may be forced to contract in volume, and thus to occupy less space ; and since on the removal of the pressure they resume their former bulk, they are also called elastic fluids. Water, on the contrary, and liquids in general, are called incompressible fluids, because they will sustain a large amount of pressure without any sensible diminution of volume. By the application of enormous force, water has indeed been made to contract its bulk in a slight degree ; but the pressure must be so great, and even then the contraction is so trifling, that no sensible error can arise from regarding it as absolutely incom- pressible : it is therefore said to be an inelastic fluid. 486. Transmission of Pressure. — The fundamental property of all fluids is this, namely, that pressure applied to any part of the fluid is transmitted equally in all directions throughout its entire volume. This remarkable truth is proved experimentally as follows : — Let a covered vessel, of any shape, be tilled with any fluid, as water ; and let any two perforations, fitted to receive pistons, whose faces A, B, have equal areas, be made anywhere in the vessel. If these pistons be introduced, a certain amount THE HYDROSTATIC PAKADOX. 457 H C 1) E i» TB of pressure must be applied to each to keep the fluid in its place ; but if any additional pressure be applied to one, it will always be found that an equal additional pressure must be applied to the other to preserve the equilibrium : — any greater pressure on one than on the other would drive the latter out. Suppose, for instance, one side of a closed vessel be flat, and that its superficial area be 10 square inches : then if a single piston, with a plane surface of 1 square inch, be introduced into a perforation, which it fits, at any other part of the vessel, and the vessel be filled with water, then, if besides the pressure on the piston necessary to keep the fluid in its place, an additional pressure of 10 lb. be applied to the piston, the flat side of the vessel will sustain a pressure of 100 lb., because each square inch of it sustains a pressure of 10 lb. Instead of a piston, we may conceive a tube of one-inch section inserted in the top of the vessel, and 10 lb. of water poured in : the pressure will be the same. Let AB be such a tube inserted in the top of the vessel CE, the area of its base £ bein^ 1 sq. inch, and that of the side FE, or of the base DE, 1 sq. inches : then the 10 lb. of water in the tube AB will cause an additional pressure of 100 lb. on EF, and also upon DE, if the area of the bottom be equal to that of the side FE. If the tube be enlarged to 2 sq. inches section, and 20 lb. of water be poured in, these pressures will remain unaltered, be- cause there will still be only 10 lb. of additional pres- sure on each sq. inch. And, BA being the height of the tube, if it be so enlarged as to fill up the entire space CG, — in other words, if the vessel DF be enlarged to DG, and then filled with the fluid, the bottom DE, and the portions EF, DC, of the sides, will each sustain no more pressure than when, by means of the 1-inch tube, the shape of the vessel was ABCDEFB. It thus appears that the pressure on the bottom of a vessel does not at all depend on the quantity of fluid introduced into it, but only upon the height of the upper surface of the fluid : if the bottom of a vessel support the pressure of a ton of water, then, without interfering with the bottom, we may so alter the shape of the sides that a pound of water shall exercise the very same pressure. In the figure above we see that the pressure on DE, when the vessel of shape DG is filled, is no greater than the pres- sure on the same base when the shape of the vessel is altered to ABEC, and that filled with the liquid. 487. The Hydrostatic Paradox.— This is the name given to the remarkable property just mentioned, namely, that any quantity of liquid, however small, may be made to balance or overcome a pressure, however great. In the diagram, ^ is a vessel, the top and bottom of which are stout boards, connected together by loose leathern sides, like the boards of a pair of bellows : — the contrivance is, in fact, called the Hydrostatic Bellows. Into this vessel water is poured through the bent tube communicating with the interior : the up- ward pressure of the water thus introduced raises the board above A, and upon stopping the supply, the water will stand at the same level hE both in the tube and in the receptacle A. Suppose now that a heavy weight W is placed upon the board, a weight 458 THE HYDROSTATIC PRESS. equal to that of the mass of water that would fill up the entire space CE above the board E, supposing this space to be closed in ; and at the same time let the tube be filled to the same height c : the small additional quantity of water, thus poured into the tube, will balance the additional weight W; and if more be poured in, the weight will gradually rise. And all this will take place, however narrow the tube may be ; for the downward pressure of the slender column of water cb is transmitted as an upward pressure upon every portion of the under surface of E, which is equal in area to the area of a section of the tube ; so that if the area of the under surface of the board E he m times the area of a section of the tube, then the weight W, balanced by the column of water ch, will be ni times the weight of that column. Instead of pouring additional water (be) into the tube, we may apply a pressure to the surface b, equivalent to the weight of that water, and similar effects will follow. It is upon this principle that Bramah's Hydro- static Press is constructed : the bellows A is replaced by a strong hollow cylinder CF of metal, to which a piston E is fitted ; the tube be, which is very much smaller in section than the cylinder, and which communicates with it as in the figure, has also a piston fitted to it. A quantity of water being introduced under the two pistons, the rod of the small piston in bo is connected with a lever, to the arm of which a moderate force being ap- plied, a considerable pressure is communicated downwards on the surface of the water b : this pressure acting upwards upon the piston E raises it and whatever heavy weight may be connected with it, in the manner above described. Suppose for ex. the diameter of the smaller cylinder be to be - inch, and that of the larger cylinder CF be 1 foot, and that a pressure of 10 lb. only be applied to the smaller piston : then since the areas of circles are as the squares of their diameters, we have (1\2 -\ : 122 : : 10 n,^ . g^gQ lb.=the upwaxd pressure on W. In this machine, as in the mechanical powers, the power applied multi- plied by the space it passes through is equal to the weight raised multi- plied by the space it passes through. For F being the pressure applied to the area b, and B being the area of a section of the cylinder, we have for the weight raised, TF=— P. Now the column of water of height H which is forced out of the tube and into the cylinder, and which therefore increases the cylindrical column by a height h, has for its volume bH=Bh, .-. |=f , .-. PF=f />,.-. Wh=Pff. on a 488. The upper surface of a fluid at rest is always horizontal. — Obser- vation sufficiently shows this to be the case : it is a necessary consequence of the equal transmission of pressure in all directions. If we could con- ceive any portion of the still surface of a fluid to be raised up higher than the surrounding parts, the pressure of this protuberant part could not be transmitted laterally or horizontally, for as it is unsupported horizontally, such lateral pressure would prevail, and the heap of fluid would spread itself horizontally : the fact that all fluids press laterally necessitates the UPPER SURFACE OF FLUID. 459 consequence that their surfaces when at rest must present a horizontal level. The pressures here spoken of are of course, entirely due to gravity, which, although acting upon all masses in a strictly vertical direction causing weight, yet in the case of fluids this weight or pressure is trans- mitted equally on all sides. The top particles, in the heap of fluid above supposed, press with their weight the particles beneath them in a down- ward direction, but these latter are equally pressed in a horizontal direc- tion, and meeting with no resistance, they necessarily yield to this lateral pressure, and continue to spread till the horizontal pressures are balanced, or, which is the same thing, till the vertical pressures on the particles con- stituting any horizontal layer of particles are all equal, that is, till the vertical distances of these several particles from the upper surface are all equal, which cannot happen till this upper surface becomes also horizontal. 489. Not only is the upper surface horizontal when that surface is con- tinuous, but the two separate surfaces of a fluid, in a bent receptacle con- taining it, are both in the same horizontal plane. In the annexed figure let A, B, be the points at the extremities of the horizontal line AB, lying wholly in the channel of communication between the two ascending portions of the bent tube, these portions being of any relative bulk whatever. Then since evei7 particle in the line AB is at rest, the horizontal pressures at A, B, must be equal : the pressure on A is the weight of the ver- ^psjM^ I " tical column of particles CA : the pressure >t^ on B is equal to the pressure on D (or I< xR^ the reaction of this pressure), augmented 1^4^!^,^^-^ J by the weight of the column of particles ^^g^|^5__5-^|: DB. But the pressure on D is the* same g^^ulSMs; as that on E, the line of particles DE being at rest. The pressure on E, however, is equal to that on F, aug- mented by the weight of the column of particles FE, /. the pressure on B is equal to the pressure on F, augmented by the weight of the two columns DB, FE. Proceeding in this way, and observing that there is no pressure on the point M at the surface, we see that the pressure on B is equal to the weight of the five vertical columns of particles DB, FE, HO, KI, ML. And since the pressures on A, B, are equal, it follows that these five vertical lines are together equal to the single vertical line CA. Consequently every point in the two surfaces at C, M, is at the same vertical distance from the horizontal plane passing through the two points A, B : these two surfaces are therefore themselves in one horizontal plane {De Launay: " Cours Elementaire de Mecanique.") In the figure, the arm AC oi the tube has been considered to be up- right, so that C is at the surface of the liquid : but if this arm were inclined like the arm BM, it might be proved, as in the case of BM, that the pressure on A is still that due to a vertical column of particles whose height is AC= the height of M. It will be noticed that what is called a horizontal plane is a flat surface perpendicular to the vertical line drawn from any point in it : on account of the remoteness of the centre of the earth, all the verticals from the surface of a fluid, enclosed in a vessel, being undistinguishable from 400 SYMBOLS USED IN HYDROSTATICS. parallel lines, that surface is undistinguishable from a strictly plane surface. But in extensive sheets of fluid — as, for instance, the surface of the ocean — the condition of equilibrium would be that that surface be spherical, every point in it being equidistant from the centre of the earth. If any portion of the water were farther distant, there would be lateral pressure without lateral resistance, which would necessarily result in motion. In the actual case, however, the sphericity of the surface of the ocean is modified by a force opposed to that of gravity : — the opposing component, namely, of the centrifugal force, brought into action by the earth's rotation (p. 441). 490. Levelling Instrument.— From the property that the two surfaces of a fluid contained in a bent receptacle are always in the same horizontal plane, the levelling in- a a strument derives its value. The common level consists M M of a bent tube, as in the figure, open at the two ends. |I V M This tube is nearly filled with a fluid — generally \IMI|||||illi|i||^ mercury — which supports two floats bearing sights, with a slender wire across each, the wires being equidistant from the surface of the fluid. When the instrument is held in the hand, the two surfaces bearing the floats are necessarily horizontal, however the tube itself may be inclined, and .-. the wires are always in the same horizontal plane ; and whatever objects which, seen through the sights, are on the same level as the wires, must be in the same horizontal plane as those wires. Suppose, in the second figure at p. 179, the levelling instrument is at C, and that A is a. distant object seen to be in the plane of Cs horizon : the chief use of the instrument is to enable us to find the situation of the point JB, beneath A, which is on the same spherical level as C, that is, to enable us to find the depression AB. Calling the distance CA, d, which is, of course, very small in comparison with the rad. B of the earth, and putting h for the depression, we know (p. 179) that /i=^- It is tlius found that the depression for - of a mile is - of an inch, taking the radius o o of the earth to be about 4000 miles. And in general, (p 5280 2 since 775 wi«^es=— — - cP/ee«=-66cP, .*. h= oP very nearly, AM oUUU o where h is the number oi feet in the depression, and d the number of miles in the distance CA. Hence, if rf be 1 mile, the depression is 8 inches ; if it be 2 miles, the depression is 32 inches ; and so on. 491. Symbols used in Hydrostatics. — The symbols em- ployed in this subject, and the meanings attached to them, are the same as those already explained at p. 443, of the Dynamics, with a single ad- ditional one, standing for a term not hitherto introduced: — the term specific gravity. If a body weigh S times as much as a mass of distilled water of equal volume, the number S is said to denote the specific gravity of the substance of which that body consists. The term, therefore, PROPOSITIONS IN HYDROSTATICS. 461 merely signifies the ratio of the weight of any volume of the substance, -whether solid or fluid, to the weight of an equal volume of distilled water. This ratio [S) is the same abstract number as the ratio D, for the density of the substance : — the former ratio, however, is that of weiyhts, the latter is that of quantities of matter : — of masses. In treating of liquids, the symbol g will always stand for the weight impressed by gravity upon a mass of distilled water, of which the volume is I cubic foot, that is, it will denote 1000 ounces. And just as, at p. 446, the weight W of any substance of volume V was represented by W=DVg, so here it will be equally expressed by W=SVg, D and S being the same abstract number: the only concrete quantity concerned in the following investigations is weight or pressure. 492. Proposition I. — If a body be immersed in a fluid, the pressure on any portion A of its surface, is equal to the weight of a column of the fluid whose horizontal base is equal to the area A, and whose altitude is the depth of the centre of gravity of A below the surface. Let a, be the length of any linear column of fluid particles, F^ being the point of the surface pressed. Regarding this point as a small area, the weight of the column will be SP^a^g, where S is the specific gravity of the fluid. Similarly, for the weight of another column of the fluid of length a^ pressing upon another point F^, we have SF.ji.jf ; and so on, .*. Pressure on surface A^S{P^a^-\-P^^■\-...•\-PnOt'J)g=:Sg^{Pa)^ where Y.[Fa) stands for the sum of P^a^, F.ji.^, &c. Let now G denote the depth of the centre of gravity of the assemblage of points P,, P.„ &c., that is, of the proposed area A, then by [1], p. 384, Y.{Pa)=l.{F)G^AG, .'. Pressure on surface A:=SAGg=SVg:=vreight of F, AG being the volume F of a column of fluid of horizontal base A, and of altitude G. If the fluid be water, then *S'=1, and Pressure=F/7=TF. If any side of a vessel incline inwards, the pressure on that side is of course an upward pressure, the amount of which is calculated as above. This pressure, reacting downwards, adds to the pressure on the bottom arising from the weight of the fluid : these two pressures together amount to the weight which the bottom would sustain, if the inclined side were replaced by a vertical side. On the other hand, if the sides of a vessel incline outwards, as in the figure at p. 464, the pressures on those sides are all downwards, and the bottom sustains a pressure less than that due to the weight of the fluid ; inasmuch as, in all cases, the pressure on the bottom is the weight of a vertical column of fluid having that bottom for its base, and which reaches upwards to the surface (p. 457). 493. It follows from the preceding proposition that if a given surface immersed in a fluid rotate round its centre of gravity, the pressure on that surface will be the same in every position of it. Also, if the surface be a rectangle, the pressure upon it, when it forms the bottom of a vessel, will be double the pressure upon it when it forms one of the vertical sides of the same vessel, supposing the fluid to fill the vessel, because the centre of gravity in the former case is twice the depth of the centro of gravity in the latter : hence the pressure sustained by the four upright 46Q PEOPOSITIONS IN HYDROSTATICS. sides of a cubical vessel, filled with any liquid, is equal to twice the pres- sure on the base, that is, to twice the weight of the liquid. This proposition enables us readily to calculate the pressure against a i-ectangular dam, or against a pair of flood-gates, for we have only to multiply the area of the dam, or flood-gate, by half the depth of water to get the volume of water, the weight of which will be the pressure : thus, suppose the water is 8 feet deep, and the breadth of each flood-gate 5 feet ; then the area of the surface pressed is 40 feet, and the depth of the centre of gravity of it is 4 ft., ,-. Pressure=40x 4=160 cubic ft. of water=160000 oz. =10000 lbs. Pressures on straight lines. — The centre of gravity of a straight line being at its middle point, if two straight lines «,, a^, be placed vertically in a fluid, the upper extremities of each being at the surface, we have a^-^-, orai so that the pressures are as the squares of the lengths of the lines, if the lines are inclined to the surface of the fluid at the angles a^ respectively, then the depths of the centre of gravity are But fli sin a, 2 I , a, sin ao , , , ■, and — - ; so that the pressures then are as a, sm «, ttj si^ tti :: '. Cbf TT- or as a^ sin a^ : a^ sin a.^ Prop. TI. A circle being just covered by vertical immersion in to draw from its lowest point that chord on which the pressure shall be the greatest. Let BC be the chord, and G its centre of gravity, or middle point : Put BF=x, then BD=2x, and BO^=BA . BD=irx, also AF=2r—x=:. the depth of G below the surface : hence the pressure on BC h {2r—x)j^irx=:ai.maodmum, or squaring and dividing by ir, x^—4:rx'^-\-4r^x= a maximum, .-. (p. 148), Zx^-8rx-{-4r^=0, .-. x=^ r, o Prop. III. To divide the perpendicular face of a rectangular embank- ment, or of a flood-gate, into component rectangles, so that the pressure upon each may be the same. Let the division be into n rectangles, of which AF is the lowest. Then since the centre of gravity of a rectangle is at its middle point, the depth of the centre of gravity of ACis - DA, and the depth of the centre of gravity of EC is - DE : the pressures on these rectangles are therefore as \dA.DA : i DE. DE, or as DA"" : DE^. 2i 2 Now since the pressure on AC is to be divided into n equal pressures, and PROPOSITIONS IN HYDROSTATICS. 463 that on EG into n—1 of these, the pressures on the two rectangles must be to each other &s n to n—1. .'. ea=da-de=da(i-^—\ Suppose the rectangle ^5 is to be divided into two rectangles, such that the pressure on the upper may be equal to that on the lower : , ^„ DA DAJ2 then »=2, and DE=-j-=: — ^. Hence, the proper degree of strength at the bottom of the embankment being secured, it would be waste of material to make it equally strong throughout: the strength upwards may diminish as the square of the depth diminishes. Prop. IV. To determine the pressure upon the interior surface of a hollow sphere filled with a fluid of given specific gravity S. The centre of the sphere is the centre of gravity of the surface : the dis- tance of this from the surface of the fluid is the radius r, and the area of the spheric surface being 47rr- (326), the pressure is equal to that of a vertical column of the fluid of base 47rr^ and altitude r, the weight of which column is 4:7rr-Sg. This, therefore, expresses the amount of pressure sustained by the interior surface of the sphere. But since the volume 4 of the sphere or of the contained fluid is -7rr\ the weight of it is only o 4 -■n^r^Sg : hence the internal pressure of the fluid is three times the weight o of it. Prop. V. A hollow cone rests with its base on a smooth horizontal plane, and water is poured in at an orifice at the top ; to what height will the water rise before it lifts the cone off its support and escapes ? Here we have to find the height of water, the upward pressure of which against the interior surface of the cone is just equal to the weight or downward pressure of the cone itself. Let E be the point in the axis of the cone to which the water must rise : then the pressure on the base is made up of the weight of the water above it, and of the reaction of the upward pressure against the in- terior curve surface. But the pressure on the base is the same as it would be if a cylindrical vessel of water DB stood on that base (p. 457) : consequently the pressure on the interior curve surface must be equal to the weight of that portion of the water in the circumscribing cylinder DB, which is exterior to the cone. This weight therefore must be equal to that of the conical shell. Put 00=a, OAz=.h, CE—x : then the vol. of the cone CAB is - a'&'a, o and since similar cones are as the cubes of their altitudes, the vol. of the cone whose alt. is CE is 404 PIIOPOSITIONS IN HYDROSTATICS. 3 a^ The difference of these volumes is the vol. of the water in the cone, that is, vol. of water in conQ=z-A+ • • • +P-Sn)g. Hence, if the area of the base be multiplied by the sum of the products arising from multiplying the thickness of each layer by its specific gravity, the numerical result will be the number of thousands of ounces of pres- sure on the base. If the thickness be the same for each layer, then the pressure will be found by multiplying the area of the base by the common thickness, and the product by the sum of the specific gravities, and by 1000 oz. For ex. A cylinder is filled with mercury to half its height, and the other half is tilled with water : taking 14 for the specific gravity of mercury, required the pressure on the base, and on the concave surface of the cylinder. Let h be the height of the cylinder, and r the radius of its base, then the area of the base will be 7!r\ and since 5j=14, and ;S^2=1, andjp,=:jp2=/ /i, '^"^e have 1 15 Pressure on base =- *j-2A(14 + l) 1000 oz.=— arr-ZtXlOOO oz. Z 2 Now disregarding the water, and measuring downwards from the surface of the mercury, the depth of the middle section, and therefore the depth of the centre of gravity of the surface pressed by the mercury, is -of - h, that is, - h, .: since, generally, Pressure =S grains, .-. ^= \X 132^^43^= 'l^^l^^linclies. m 500. The Hydrostatic Balance.—Specific gravities of sub- stances are ascertained by means of the hydro- static balance, to the under part of one of the c= ^ scale-pans of which a hook is fixed. The body \ to be experimented upon is suspended by a cjj fine thread to this hook, and then immersed in a fluid of known specific gravity, as distilled water at a temperature of 60° of Fahrenheit's thermometer. The weight of the body in the fluid being thus found, and its weight in air, or in a vacuum, being already known, its specific gravity is determined as in Prob. I. ; or if the body be so light as to require a sinker, it is determined as in Prob. II. If the substance consist of loose materials, as a powder, its specific gravity is found by imbedding a known weight of it in wax, or some such substance, employed as a sinker, and then computing S by Prob. II. For determining the specific gravities of liquids, an instrument called the Hydrometer is employed. 501. The Common Hydrometer.— In its simplest form the Nicholson's hydrometer. 473 hydrometer consists of two hollow spheres, usually of glass, a larger and a smaller one below it, as in the annexed diagram. Into the lower sphere mercury or small shot is introduced, in order that the spheres may sink in the liquid. The axis of the stem Z) is a straight line through the centres of the spheres, so that when the instrument rests in the fluid the stem may he vertical. Suppose that when placed in distilled water, the stem sinks to some point A, and that when placed in another liquid L, it sinks to the point B : then by the property (499), namely, ^HP' S'_V^ Specific gravity of L_ yolume of AC _^o\. of (DO— DA) g-'y,, we ave - — Volume of BQ~\o\. of {DO-DB)' Let the whole volume of the instrument be considered to consist of 4000 equal parts, and the stem be divided so that each portion is one of those parts. Suppose the stem to contain 50 of these parts, numbered from I) downwards, and that it sinks to 30 in one liquid L, and to 10 in another liquid U, then Specific g ravity of i;_4000— 10_3990 Specific gravity of i/'~4000— 30~3970* And in this way the specific gravities of different liquids may be com- pared. When used for the purpose of testing the strength of alcoholic liquors, as by officers of the Excise, the point on the stem to which the hydro- meter sinks in distilled water is marked 1-000, and the point to which it sinks in a lighter fluid of known specific gravity, say '900, is marked •900. The interval between these points being then divided into 100 equal parts, the instrument becomes adapted to show by inspection the specific gravities of all liquids whose sp. gravities are between these limits. And the scale may in like manner be extended in either direction, and thus be available for other purposes. Proof spirits consists of half alcohol and half water: its sp. gravity is '920, which point is marked *^ Proof on the Exciseman's scale: if the liquor be below proof the instrument will not sink so far, and if it be above proof, it will sink farther. 50*2. The Specific Gravity Bottle. — But there is another way of de- termining the specific gravity of a liquid. A bottle, made to contain 1000 grains or 500 grains of distilled water, is accompanied with a weight which is the exact counterpoise of the empty bottle and its stopper : this is filled with the liquid, and the weight which, in addition to the counter- poise, balances the full bottle, is of course the weight of the liquid. This weight, divided by 1000 grs., or by 600 grs., according to the size of the bottle, is the sp. gr. of the liquid : it is therefore expressed by the number of grains in the weight of the liquid, converted into a decimal, or into a whole number and a decimal, by advancing the decimal point in that number three places forward for a 1000 grain-bottle, and by doing the same with double that number for a 500 grain-bottle. There must of course be liquid sufficient to fill the bottle. 503. Nicholson's Hydrometer. — By means of this instrument ^ 474 Nicholson's hydrometer. the specific gravities may be compared, not only of two fluids but of two solids, or of a solid and a fluid. The stem of this instru- ^ ment is a slender wire of hardened steel, having a dish at each end, and a hollow globe or cylinder between the two, as in the figure. The adjustment of the instrument is such that when a given weight is placed in the upper dish A, it will sink, in dis- tilled water, to the point P, in the middle of the stem AB : it is used as follows : — 1. To compare the specific gravities of two liquids. — Let W be the known weight of the instrument, w the weight which must be placed in the dish A to sink the hydrometer to P in the liquid L, w^ the weight required to sink it to P in the liquid U : then Weight of the vol. of L displaced = W-{-w : Weight of vol. of L' displaced= TF+w„ C Specific gravity of L W-\-w Specific gravity of L' W-^-Wy 2. To compare the specific gravities of a solid and a liquid. — Let w be the weight which must be placed in the dish A to sink the instrument to P. Replace w by the solid, and let w^ be the additional weight necessary to sink the instrument again to P. Remove the solid to the lower dish C, and let w^ be the additional weight to be placed in the upper dish to sink the instrument to P : then we shall have Weight of the solid in air =w— w, ) , ^, - . , [ .'. wt. of fluid displaced =:w^—w,, „ „ m the fluid =w—W2J ■< " Specific gravity of solid w — w, Specific gravity of fluid w^—wi Example 1. A body whose sp. gr. is iS'=*6 floats with - of its volume o below the surface : required the sp. gr. fi'j of the fluid. By the principle at (499), the whole body (=1) X sp. gr. of the body = the part (2\ 2 =q )x sp- g^' 0^ *1^6 fl^d, that is, '6=- -S^„ .*. /S',= '9. 2. A piece of cork weighing 6 oz. is attached to 24 oz. of brass, of which the sp. gr. is 8-7 : the weight of the whole in water is 2*24 oz. : required the sp. gr. of the cork. Here the weight lost by the compound is 30 oz. —2*24 oz.z=Wi—W\ (p. 471). 24 The weight lost by the brass sinker is — - oz. =zw—uf. 8"7 The diff'erence of these is the weight lost by the cork alone ; and since the sp. gr. of the fluid is S-^=\, we have for the sp. gr. of the cork (p. 471) 5= ^ = ? 1-24 27-76-— 27 •76-2-76-25'" ^** 8-7 8. A mass of gold being immersed in a cylinder of water caused the surface to rise a inches : a mass of silver of the same weight W, caused it to rise b inches ; and a mass, still of the same weight, but composed of Nicholson's hydrometer. 475 gold and silver, caused it to rise c inches : required the proportion of gold and silver in the composition. Let x be the volume of the gold, and y that of the silver; then x-\-yi^ the volume of the compound, and since the elevations of the surface of the water are proportional to the volumes immersed, we have c : a : : x-\-y : - (a;4-y)=the volume of the gold mass of weight W ; .b ::x-{-y :-{x+y)= „ „ sUver mass „ c - (x-\-v) :x : : W :-. W= weight of gold in the compound. c "^ a x-\-y ^-icc+y):y:'.W:l.^W= „ silver ' a{x-\-y) b{x+y) .: {ic—ba)x={ah—ac)y, ,: x :y : '.a(b—c) : b{c—a). This gives the ratio of the volumes of gold and silver. To find the ratio of their weights let 8 be the sp. gr. of the gold, and /S" that of the silver ; then the weights of a and y will be to one another as Sx to S'y, .-. wt. of the gold : wt. of the sHver : : a(b-c)S : blc-a)S' : : ^~l ^, : 1. o{e — a) o It was probably in some such way as this that Archimedes discovered the proportion of gold and silver in the crown fabricated for Hiero, king of Syracuse, who had ordered the workmen to make it of a given weight of pure gold. Examples for Exercise. 2 (1) A body floats in a fluid of 13*5 sp. gr., and - of its volume is above the surface : o required the sp. gr. of the body. (2) If a piece of gold weigh 15 lb. in air, and 441b. in mercury, and the sp. gr. of the gold be 19, required the sp. gr. of the mercury. (3) A piece of cork, whose sp. gr. is '25, floats in water : how much of it will be above the surface ? (4) A block of wood weighs 3 lb. in air, a piece of lead weighing 6 lb. in water, is at- tached to it as a sinker : the compound weighs 5 lb. in water : required the sp. gr, of the wood. 2 (5) A body whose sp. gr. is - floats in water : required the ratio of the parts of it in o and above the water. 12 (6) A cubic foot of wood, of sp. gr. — , is put into water : with how much more wood 13 must it be loaded, in order that its upper surface may sink to the level of the water ? (7) The specific gravities of two bodies weighing W, and W, are S and S' : show that the specific gravdty of the united mass is - ,..^. , .../ ■ „ SS'. vro -f- rv o (8) A piece of cast iron, of sp. gr. 7 '425, weighed 40 oz. in air, and 35*61 oz. in a fluid : required the sp. gr. of the fluid. (9) The sp. gr. of a composition of 1 cwt. of tin of sp. gr. 7*32, and of copper of sp. gr. 9, has for sp. gr. 8*784 : what are the weights of tin and copper in the mixture ? 476 WEIGHT OF THE ATMOSPHERE. (10) 120 grains of brass, of sp. gr, 8, are counterpoised in air by a copper weigbt of sp. gr. 9 ; both, still hanging to the balance, are then immersed in water - what weight must be applied to the ascending arm of the balance to restore the equilibrium, the ad- ditional weight not being immersed ? (11) 37 lb. of tin loses 5 lb. in water, and 23 lb. of lead loses 2 lb. in water : a mixture of tin and lead, weighing 120 lb., is found to lose 14 lb. in water : how much of each metal is there in the mixture ? (12) A man, whose weight is 168 lb., can just float in water when a certain quantity of cork is attached to him. Assuming the sp. gr. of the man to be 1 '12, and that of the cork "24, what quantity of cork is necessary ? 504. Elastic Fluids. — The properties of fluids established in the foregoing articles belong to fluids in general : elastic fluids, however, have additional properties peculiar to themselves, and these, therefore, demand a distinct examination. Of such fluids, the atmosphere which surrounds us is of the most importance; and it is in reference to this, that the science, which treats of the mechanical properties of elastic fluids, is called Pneumatics : — from a Greek term, signifying the air we breathe. In the following articles atmospheric air will be the only elastic fluid con- sidered : but the investigations would have been the same, and the conclusions similar, if any other elastic fluid had been the subject of examination. 505. Weight of a Cubic Foot of Common Air.— The air which surrounds the earth, like every fluid, has weight, as may be proved by many conclusive experiments. We may, in fact, take any definite volume of it, as we would take so much water, and actually weigh it in a balance, as follows : — Let a vessel, the neck of which is provided with a stop-cock, have the capacity of a cubic foot, that is, let it be capable of holding 62^ lb., or 1000 ounces of distilled water. This vessel may be exhausted of its air by means of an air-pump, a machine which will be presently described. Thus emptied, and the stop-cock closed so as to prevent the re-admission of air, let the vessel be accurately weighed in a very sensible balance. The exact weight of the empty vessel being thus ascertained, let the stop- cock be opened and the air admitted, and when thus filled with a cubic foot of air, let the vessel be once more weighed : it will be found necessary to put about 536 grains of additional weight into the other scale-pan to restore the balance, thus showing that the weight of a cubic foot of air at the surface of the earth is about 536 grains. It is thus found that on the average the weight of any portion of air at the earth's surface is about — — of the weight of an equal volume of dis- tilled water. Taking the cubic foot for the unit of volume, that bulk of air, when the thermometer stands at 60% and the barometer at 30 inches, weighs 536 grains : hence the pressure communicated by gravity to a cubic foot of air in this state is 536 grains, which is therefore the value of g in reference to elastic fluids, just as 1000 oz. was the value of g in reference to liquids. 506. Weight of the Atmosphere. — if a tube about three feet LAW OF MARIOTTE AND BOYLE. 477 in length, and closed at one end, be filled with mercury, and the open end be then closed with the finger, and the tube inverted in a basin of mercury, the finger being removed while the mouth of the tube is below the surface, it will be found that the column of the fluid, originally 3 feet in height, will fall only six or seven inches, and that 29 or 30 inches above the surface will remain suspended in the tube. Hence the surface of the mercury in the basin which surrounds the inserted tube must, on every portion of it equal in area to a horizontal section of the tube, sustain a pressure from the atmosphere which it requires a column of 29 or 30 inches of mercury, on the same area, to balance (486). Mercury is about 13i times as heavy as distilled water, so that the pressure of the atmosphere on any given area of the earth's surface is the same as the pressure of a column of water on that area, and of height about 134X30=405 inches =33| feet. The truth of this conclusion was actually put to an experimental test by Pascal in 1647. He procured glass tubes, closed at one end, of the great length of 40 feet, and found that when filled with common water in a deep river, and then raised vertically with the open end downwards, the fluid ceased to fall when the water in the tube stood at about ^2\ feet above the surface of the river. Had the water been distilled water, which is somewhat lighter, a still higher column would have been supported. Since 30 cubic inches of mercury weigh about 15 lb., it follows that every square inch of the surface of the earth at the level of the sea sustains, on the average, an atmospheric pressure of 15 lb. If the atmosphere were to be removed, and to be replaced by its weight of mercury, the depth of the mercurial coating would be about 30 inches. If M denote the radius of the earth in feet, and r the proper depth of mercury, S being its specific gravity, we shall have for the weight W of the enveloping coat of mercury H'=(i^-l^)sj=l«(iP+iJr+i,-)sxlOOOoz. which weight the celebrated Cotes calculated to be equal to the weight of a globe of lead of 60 miles in diameter, or upwards of 77,670,000,000,000,000 tons. Still this coating of air is comparatively thin : the thickness of it is estimated at not more than 50 miles, so that, taking the average radius of the earth at 4000 miles, the thickness of the surrounding fluid is only — th of the radius : on a twelve-inch globe this thickness would be only about — .th of an inch, lo 507. Law of Mariotte and Boyle. —It was proved experi- mentally by both these philosophers at about the same time (1662), and independently of each other, that — The elastic force of air (at a given temperature) is directly proportional to its density, or, which is the same thing, inversely proportional to its volume. pB ^A. 478 LAW OF MARIOTTE AND BOYLE. Let DAG be a bent tube of uniform bore, open at both ends C, D ; and let mercury be poured in so as to fill the bend of the tube, AB. Let the end C be now closed, and such a quantity of mercury be poured into the tube at D as will cause the surface originally at B, to rise to B' halfway between B and C. It will be found upon measurement that the column of mercury AE thus poured in will always be such that the portion of it A'D above the level of B' will be exactly equal to the mercurial column that balances the atmospheric pres- sure at the time. We thus see that when the surface A sustained the pressure of only the column of air above it, the air in the shorter leg occupied the space BC, the upward pressure upon it at B being merely the weight of the air which enters the tube at D ; but that when twice this pres- sure acts down the tube, and is communicated upwards to the air in the shorter leg, that air is then compressed into half its former volume, BC. Let now another column of mercury (29 or 30 inches high, according to the state of the atmosphere) be poured into the tube at D : the air in B'C will be found to become compressed by this additional pressure into a volume B"C exactly equal to one-third of its original volume BC, and 80 on. Similar results are found to follow, whatever fractional part of a column of mercury equal to A'E — which represents the whole atmo- spheric pressure — be poured into the tube, the general conclusion being that at whatever points B\ B'\ the columns CB\ CB'\ of compressed air may terminate, Pressure on 5 " (7 : Pressure on B'C:: Vol. BV : Vol. B " C. And since these pressures are balanced by the downward reaction, or force of elasticity of the condensed air, we have Elastic force oi B"C: Elastic force of B'C : : Vol. B'C: Vol. B"C, and because the density increases as the volume diminishes, the last two terms of either proportion may be replaced by Density of B'^C: Density of B^C, so that the pressure divided by the density is constant ; and the same proportions hold for other elastic fluids. [The foregoing Law of Mariotte is that for the same elastic fluid VP=V^P^ (representing the pressure and corresponding volume by symbols), provided the temperature remain unchanged. The experiments of Gay-Lussac and Dalton further show that if the temperatures of the two volumes V, V\ are T, T\ then the equation is yp v'p' , _— — m=T~, — ^/' where a is constant, at least for the same gas, and =-00366 for air when the degrees of tem- perature are measured on the Centigrade Thermometer. The constant is called the coefficient of dilatation. Suppose a volume V of air, under a pressure P, has the temperature T, and it be required to find to what temperature T' it must be raised in order that it may occupy a volume V, The preceding equation gives \uJ J VP a ALTITUDES BY THE BAROMETER. 479 Let, for instance, the volume 7' be double F, and the pressure P' two- thirds of P, the first temperature being jr=10° (Centigrade) : then since a =-00366, we find the required temperature to be r'= 104-4°]. 508. As in inelastic fluids the standard of density is pure water, so in elastic fluids it is atmospheric air taken at the ordinary temperature of 60°, and when the mercurial column measuring its weight or pressure is 30 inches : — the density of such air is 1. When by compression a portion of this air is reduced to half its bulk, the density of it is 2, and the cor- responding pressure on it = two atmospheres, and when compressed to - of its bulk, its density is w, and the pressure upon it = ti atmospheres. Prop. I. If altitudes increase in arithmetical progression, the corre- sponding pressures of the atmosphere decrease in geometrical progres- sion. Conceive a vertical column of the atmosphere to be divided into equal strata, so thin that each stratum may be regarded as of uniform density throughout. Let -p^ be the pressure on the base of the column, that is, on the bottom of the first stratum, and l^ the density of that stratum. In like manner, let;?2 be the pressure on the bottom of the second stratum, and ^2 the density of that stratum, and so on : then putting A for the area of the base of the column, and t for the thickness of each stratum, the dif- ference between the pressures on the bottom and top of the nth stratum, that is, between the pressure on the bottom of the nth stratum, and that on the bottom of the (n -|- l)th, must be the weight AylnQ of that wth stratum (446), that is, i^n— jPn+i=-4rS„gr. Now^ is constant (p. 478) : Call it ^, 9(1 T Ti •pn-\—'Pn . rg In like manner, =J. ^-, Pn-\ k . Pn—pn+\ Pn-\—pn Height in Mfles. Times Barer. 1 3| 2 7 4 14 16 21 64 28 256 35 1024 42 4096 Pn Pn-l Hence the pressures p^ jOg' Pa^ •••» ^^^ consequently the densities, are in geometrical progression ; the heights 0, t, 2t, 3t, ... being in arithmetical progres- sion. From observation it is found that at 3^ miles above the surface of the sea the air is only about half as dense, that is, it is twice as rare : hence the construction of the small table annexed, which, however, can be regarded only as an approximation to the truth, since the difference of temperature of the various strata, and the diminution of the force of gravity, have been neglected. Prop. II. To investigate a formula for finding the difference of the heights of two stations above the surface of the sea by means of the barometer. Let z be the height in feet of the upper station above the surface of the sea, and z' the height of the lower station. Then dividing the intervening air into strata of 1 foot thick, the pressure may be taken as uniform throughout the same stratum ; taking then t=I foot, and also ^=1, that 480 . ALTITUDES BY THE BAROMETER. is, regarding the area of the base of the atmospheric column as 1 sq. foot, we have [1] l_^!i±J=.^, ...theratio^"±i=l-f...[2]. Pn h Pn k Now let the mercury in the tube for measuring the atmospheric pres- sure, that is, the height of the barometer, be h at the upper station, and h^ at the lower ; then these heights being as the pressures on the zth and zWi strata, they must be as the corresponding powers of the ratio [2] by Prob. I., that is, .-(-i)-(-fX=(-ir''. From the small value which observations assign to the fraction '-, powers of it may be neglected ; and since, by the logarithmic theorem (11 2)^ •-(^-f)=-{f+Kl)+-}' we have, by disregarding all the small terms after the first, k The constant fraction - is determined thus : Let JE?' be the height of 9 the barometer at a foot above the sea-level, and H its height at a known altitude of a feet above that station, then by the formula, since z—^-^a^ we have Tc, IT h , E'. a=-log.-, .-.-=«-%, ^ft. And thus the constant fraction in the general formula may be determined by experiment : it has been in this manner found to be 64,0*20, when the Napierian are converted into common logs, so that dividing by 6, the practical formula for the computation of any altitude Ay above the level of the sea, would be ^=10670 log Height of tarom. at alt. ^ ^ Height of barom. at the sea-level provided the temperature of the air were uniformly 55** Fahr., for which temperature the constant was determined. But experiment has shown that the altitude of a place, as deduced from this formula, will vary by about j^th of its whole value for every degree, by which the mean of the temperatures at the two stations differs from 55° : this variation to be added when the mean exceeds 55", and to be subtracted in the contrary case. In order, therefore, to convert the constant multi- plier into the more convenient number 10,000, we proceed thus. The difference between 10,670 and 10,000 is 670, and the difference between 10,670 and the value it would have for 1" less of temperature, is THE BAROMETER. 481 10670 =24-5 : hence to find for how many degrees less of temperature the difference is 670, we have the proportion 24-5 :670 : :1° :27°, .*. 55°-27°=28°. At this temperature .-.the altitude A in fathoms is ^««^«, Height of barom. at alt. A ^=10000 log , Height of barom. at the sea-level * and this value must he increased or diminished by the jo^th part of itself for every degree which half the sum of the temperatures at the two stations exceeds or falls short of 28". We are thus furnished with a very simple practical rule, but it must be regarded only as a tolerably close approximation : when great accuracy is required corrections must be made for the variations of gravity depending upon the height of the station and the latitude of the place, for the dila- tation of the mercury in the barometer which lengthens to the extent of about th part of the whole for every additional degree of tempera- ture, and for the state of the atmosphere at the time of observation in reference to the moisture suspended in it. These corrections are made by the help of special Tables, constructed conformably to the results of observation : those which are at present regarded as the most accurate are the tables of M. Delcros, published in the " Annuaire Meteorologique " for 1849, and those of M. Plantamour, in the same periodical for 1852. By the former set of tables the height of Mont Blanc, calculated from very careful barometric and hygrometric observations, made on the 29th of August, 1844, by MM. Bravais, and Martitis, was concluded to be 4814-5 metres. By the latter tables, the height determined was 4811*7 metres. [The metre is 39-37 English inches.] 509. The Barometer. — This instrument is so well known that in the preceding articles we have presumed the reader to have at least a general acquaintance with its object. It may be regarded as a balance to determine the weight of a column of air on a given base, and reaching ver- tically upwards to the extreme limit of the atmosphere, the counterpoise being a column of mercury upon an equal base. The tube containing the mercury is either straight or bent, as in the annexed figures : when straight, its lower extremity, which is unclosed, is inserted in a c] small cistern, or box, or bag, of mercury ; the pressure *^'^ of a column of air downwards on an area =J. of the surface of the mercury in the cistern, is communicated upwards at A balancing the weight of the column AB in the tube previously exhausted of air. Every other area equal to ^, of the surface surrounding the tube, is of course equally pressed, and equally communicates that pressure up- wards, but the area covered by the tube BA is the only one of these areas upon which an equal downward column of air does not press. The mer- 1 1 482 THE SYPHON. curial column ^^, therefore, is equal in weight to the weight of the atmo- spheric column on an equal area A : the height of the mercury in the tube above the surface of that in the cistern always ranges between 28 and 31 inches, and scales are attached for reading off the height at any time. The bent tube is the form used for the Wheel Barometer, or Weather Glass, as it is sometimes called : the end A is open to the air, the other end B is closed, and usually enlarged into a globe, the upper part of which, not filled by the mercury,* being a vacuum. Upon the surface of the mercury at J. , is an iron ball, connected with another rather lighter, by a string passing over a pulley P. As the ball at A rises and falls with the mercury, the string turns the pulley by its friction, and an index, turning with it, points to the different degrees marked on the surrounding circle. A very slight variation in the height of the surface at A will make a considerable difference in the position of the index J, if this sur- rounding circle be tolerably large : but the friction of the pulley is an impediment. 510. The Thermometer. — The lower end of the tube in this instrument is expanded into a bulb, and no part of the fluid within is in contact with the air. As in the barometric tube the space above the fluid is a vacuum, and the fluid itself is either alcohol or mercury. The bore of the tube is so small that a very slight expansion of the fluid which occupies it makes a sensible difference in the length of the fluid column, and thus renders it a fit measurer of the heat of any fluid in which it may be plunged (the surrounding air for instance), or of any body with which it may be brought in contact, since bodies in general expand by the appli- cation of heat, and contract by cold : — water, however, being an exception, within certain limits. In the thermometer of Fahrenheit, the point at which the mercury stands when the instrument is plunged into melting snow, and allowed to remain there till the mercury will fall no lower, is marked 32° : — this is the freezing point. The point at which the mercury stands when the in- strument is immersed in the vapour of boiling water, is marked 212° : — this is the boiling point ; and the intermediate degrees are marked on the attached scale. In the Centigrade Thermometer the freezing point is marked 0°, and the boiling point 100° : — whence its name. In Eeaumur's, these points are 0° and 80" respectively. Distinguishing these three kinds of degrees by F°, C°, and R"" — the initials of the names they bear — we may convert the number of degrees which marks any temperature on one scale into the number which marks the same temperature on either of the other scales, from the relations 0° 7}° P° ^2° 36(7°=45i2°=20(ii'-32°), or ^=^= , . 5 4 9 511. The Syphon. — This is a bent tube ABC, with unequal legs AB, BC, used for the purpose of transferring liquids from one vessel to another. The end of the shorter leg is inserted in the liquid, and the air in the tube withdrawn by the application of the mouth to a small suction- pipe, communicating with the tube near the end C of the longer leg. THE COMMON PUMP. 483 There is a stop-cock between the insertion of the suction-pipe and the open end C, to cut off all communication of the atmosphere with the in- terior of the tube, preparatory to the exhaustion of air within it. This exhaustion having been effected, the pressure of the air on the surface DE of the liquid forces the liquid up the leg AB, the highest point B of which is much below the height to which the liquid would ascend but for the bend : it is .'. driven downwards by the reaction of B with a pressure equal to the weight of the atmosphere pressing on Ay dimi- nished by the weight of the sustained column aB, a being at the surface. If the leg BC terminated at E, on a level with a, the two columns aB, EB would balance, since the upward pressure of the atmosphere at E, diminished by the weight of the vertical column of liquid of the height of B, would be equal to the upward pressure at a, diminished by the weight of a like vertical column of liquid, in which case no liquid would flow out ; but if the end C be below the level DE, the equilibrium is destroyed : — the atmospheric pressures upward at a, C, are still the same, but there is more weight of liquid pressing downward at C, than there is sustained at a ; and since the passage is free, the liquid must be forced out by this excess of pressure, and must continue to flow till the surface of it in the vessel falls to a level with C when it will stop. If the syphon have no stop-cock, it must be inverted and filled with liquid, and both ends being closed, and the instrument turned to its proper position, the shorter end must be inserted in the liquid, and then both ends unclosed. If the liquid be water, B must not be higher above its surface than 33 or 34 feet (p. 477). If it be mercury, it must not be higher than about 29 or 30 inches (p. 477). 512. The Common Pump.— The annexed figure represents a section of the common suction pump. AB is the pump-barrel, communicating by the pipe below it (called the suction pipe), with the water in the well. C is the air-tight piston (the sucker), in which there is a valve opening upwards : there is another valve F, also opening upwards, which, when shut, closes the communication between the barrel and the pipe. The piston C is worked in the barrel by aid of a lever H : — the pump- handle. Suppose no water to be as yet in the pipe : the pump-handle is raised, and the piston connected with the pump-rod de- scends, and as V is closed, the air in the barrel below C is thus condensed, and by its upward pressure opens the valve in C, and escapes : the handle is ttien pulled down, and C raised, the valve in it being closed by its own weight, and by the down- 11 2 484 THE FOKCING PUMP. ward pressure of the atmosphere above, while it leaves a vacuum or a partial vacuum in the barrel below it : this vacuum is immediately filled up by the expansion of the air in the pipe, the upward pressure of which forces open the valve V, the downward pressure upon V having been wholly, or at least partially, removed. Instead, therefore, of the barrel and pipe being filled, as at first, with atmospheric air of the same density as that which presses upon the surface of the well, the pump contains only rarefied air; so that the pressure down the pipe is less than the pressure up it from the weight of the external air, and therefore water is forced up the pipe by this excess of external pressure. By another stroke of the pump-handle another barrel-full of air is pumped out, and the water in the pipe, being thus relieved still more of downward pressure upon it, ascends still higher ; and after a few strokes enters the barrel, and it is then lifted up by the piston and pumped out ; more water following till the barrel is full, when the water can be made to fliow out in a continuous stream. It will be observed that whenever the upward pressure against either of the valves does not exceed the downward pressure the valve closes by its own weight. Since the pressure of the atmosphere on any area is equal to that of a column of water on that area of only 33 or 34 feet (p. 477), it is plain that the water cannot be made to pass the valve V, into the barrel, if V be not less than at this height from the surface of the well : nor can the barrel be filled to the spout if the spout exceed this height. Supposing the barrel to be constantly full, the quantity discharged at the spout, at each stroke of the handle, is a column of water, the base of which is the horizontal section of the piston, and the altitude the height to which the piston is raised — called the length of the stroke. Let the radius of the section be r feet, and the length of the stroke I feet : then — Quantity discharged = ^'liierH cubic feet =S-U16rHxQk gallons, Since a cubic foot of water or about 1000 oz.=62^ lbs. avoirdupois. 513. The Forcing Pump. — In order that water may be puftiped out of a well, in a continuous stream, the spout must not be higher than 33 or 34 feet from the surface of the water in the well, if the common pump be employed : but the forcing pump will deliver water con- tinuously at any height. The annexed diagram is a section of this pump. As in the common pump, F is a valve opening upwards at the junction of the suction-pipe and barrel, or cylinder AV. The piston C is air-tight as before, but has no value : — it is solid. Conceive the piston C to be raised to its highest point, the valves V and v, both opening upwards, being closed. As C de- scends the air below it presses V closer down, opens v, and rushing up the pipe vF, escapes : the valve v then closes by its own weight. As C ascends, a vacuum beneath it would be formed ; but the upward pressure of the air in P forces V open, and by its expansion fills VC, V THE DIVING BELL. 485 then closing by its own weight ; v also remains closed from the superior downward pressure of the atmosphere : water, however, ascends in P, on account of the diminished pressure down the tube. In this way every descent of C compresses the air between it and V till the elasticity of it becomes sufficient to force open v, when another portion of the air, originally in CP, escapes. And every ascent of C is followed by the air, still in P, forcing open the valve V, to fill up the vacuum VC, and by a further ascent of water in P, in consequence of the further diminution of pressure down P. At length the water, by thus rising higher and higher, passes through V, when the next descent of C will force it through v, which instantly closes upon the withdrawal of the upward pressure, that is, upon the ascent of C ; and the return of the water in vF is thus prevented. The next downward stroke of C forces more water through v, till at length the column rises to F; and it may then be forced out in a continuous stream. The Fire Engine is a combination of two forcing pumps, the pistons of which are worked by a lever, having its fulcrum at its middle ; and the power which raises and depresses the piston-rods is applied at each end alternately. The water from the fire-plug is pumped into a central receptacle (with which the two cylinders containing the pistons communi- cate), called the air-vessel, the air in which being forced to contract the more as more water is pumped into it by the descending pistons, exercises the greater pressure upon the water, and forces it with considerable velocity out of the delivery pipe. 514. The Diving Bell is so called, because in its original con- struction a bell-shaped figure was given to it. At present, it is a heavy iron chest, open at the bottom, with a seat in the interior sufficiently high to enable persons sitting on it to keep their heads above the water, which gradually rises in the bell as it descends, but which can never completely fill it on account of the inclosed air, which as gradually becomes more and more condensed. A flexible pipe, communicating through the top (or bottom) of the bell, and furnished with stop-cocks, allows of the escape of the air unfit for respiration, and of fresh air being pumped in. We may find the space into which the air, originally in the bell, will be compressed, when at any depth below -=,_^ _ , --=^-^_::-^ the surface, thus : Let AB be the depth of the fe=3^^^^^^ft^& surface of the water in the bell below the sur- face of the water above. Then taking 33 feet for the height of a column of water, the pressure of which would be equal to that of the atmosphere, the condensed air in the bell sustains a pressure equal to 33 feet of water (the original pressure) -+- the pressure of another column of water of height AB. Calling this height h, and remembering that the spaces into wh^ch air is condensed are inversely as the condensing pressures (507), we have ^ Vol. of conde psed air_i)5_ 33 Vol. of original air ~Uc~ZZ-^h* /. putting i)C=a, the depth AI)=zd, and DB=x, -= — — — , 'a m-\-d+J i€ -^/ ■ D H8 — ' m^ 486 THE AIR PUMP. .-. a;2+(33+c?)ic=33a, a quadratic which has only one positive root ; and this gives the thickness x—DB of the layer of air above B. 515. The Air Pump. — This is a machine constructed for the pur- pose of exhausting a vessel of as much of the air con- tained in it as possible. There are several forms given to it, but that best known is the double-barrelled pump of Hawksbee, or the common air pump. The vessel to be emptied of its air, and which is called the receiver, com- municates by means of a pipe inserted at Z), with two smooth barrels, in which two closely-fitting pistons with valves are moved by rack-work. The four valves a, b, e,f, all open upwards ; and the results produced by working the handle H are as follows : — As one of the pistons descends, its valve is forced open, and the air in the barrel passes through : as this goes on, the other piston ascends, with its valve closed, emptying the barrel up which it moves of its air ; the vacuum being instantly filled from the air in the receiver, through the pipe at D. Eeversing the motion of the handle, the first piston now ascends, emptying of its air the barrel up which it moves ; and, as before, fresh air from the receiver rushes in. These alternate ascents and descents of the pistons so rarefy the air in the receiver, that at length its elastic force is too feeble to raise the valves, and thus to pass into the barrels : when, therefore, this stage is arrived at, the exhaustion cannot be carried on any further. Prob. I. To find the density of the air in the receiver after n strokes, or turns of the wheel. Let R be the capacity of the receiver, and B that of each barrel : then regarding the density of the atmospheric air as 1, let ^^, ^^, ..., ^„, be the densities of the air in the receiver after 1, 2, ..., n, strokes, or turns of the wheel. Then since at every turn a barrel full of air is withdrawn, the volume R of air is expanded into R-^B; hence (507), Consequently the density decreases in geometrical progression. The rarefied air of volume B, which remains in the receiver after n strokes, is equal only to a volume ^,Ji of the original air ; and since, as just (7? \7l i^"+' ) E=— , it follows that this last fraction denotes the part of the original volume of air which still remains. R R \" rTb) The quantity of original air diminishes, therefore, in geometrical pro- gression ; and since a decreasing geometrical progression may be continued to infinity, it follows that the receiver could never be completely emptied of air by any number of strokes of the air pump, even should the air, however rarefied, be always sufficiently elastic to open the valves. Prob. II. To find what number of strokes are requisite to reduce the air in the receiver to a given density. THE CONDENSER. 487 Let S be the given density, that of atmospheric air being 1, and put n for the required number of strokes : then by last problem, logS=«{logi2-log(i2+^)}, log 5 'logR-\og{R-^By single Since both numerator and denominator of this fraction are negative, ^ being alwaj's less than 1, the fraction itself is positive. As inferred above, so here, if ^=0, that is, if the receiver is to be completely exhausted, n must be infinite, since log = — oo . 516. Smeaton's Air Pump. — This consists of but barrel AB, communicating with the receiver through the pipe C. There are three valves : — one at B, one in the piston D, and one in the top plate A of the barrel: — all opening up- wards. As the piston ascends from the bottom of the barrel, D is the only valve that closes ; for A is forced open by the upward pressure against it, and D closed by the same pres- sure downwards; and the under portion DB of the barrel being exhausted of air by the stroke, the valve B is forced open by the air rushing in from the receiver. If, as before, R, B, represent the capacities of the receiver and barrel, we shall have the same expression for the density ^„ after n — — - J , the original density being 1. Although the density after n strokes is thus the same in L^j i^ both machines, yet a greater degree of rarefaction may be ob- \Jc tained by Smeaton's than by Hawksbee's machine ; for in Smeaton's the valve of the descending piston is relieved from the down- ward pressure of the atmosphere, and therefore will open, and allow air to escape, for a much less upward pressure than is necessary to open the valves a, b, in the older air pump of Hawksbee. 517. The Condenser. — The object of the condenser is the reverse of that of the air pump : it consists of a receiver R, a portion of which is represented in the annexed diagram. Into this vessel, containing at first only atmospheric air in its ordinary state, additional air is forced by the descent of a solid piston moving in the cylinder B, which communicates with the receiver through the valve C opening with a spring downwards, immediately above which is a stop-cock. Near the top of the cylinder is an aperture a, which, when the piston is at its greatest elevation, is immediately below it. Sup- pose that from this elevation the piston is forced down : — the barrel full of air beneath, becomes more and more condensed, till its pressure at length opens the valve, and the air in the barrel is thus driven into the receiver. The piston having reached its greatest depression, the air immediately under it, and that in the receiver, must have the same density, so that the valve C closes by the action of its spring. Again, raising the piston above the aperture a, through which a second barrel full of air rushes, this further quantity is 488 SCHOLEUM. forced into the receiver as before : and so on, till the upward pressure of the condensed air is so great against the valve, that the downward pressure of the condensed barrel full of common air is insufficient to overcome it and force the valve open. Since a volume B of common air is forced into the receiver at every stroke, the quantity of common air condensed into the volume H, after n strokes, must be R-\-nB. Putting 5 for its density, that of common air being 1, we have (507), Hence the density increases in arithmetical progression, the increase being — times the density of the external air at every stroke. .lI Examples for Exercise. (1) How much of the original air will be left in the receiver of an air pump after four strokes of the piston, the capacity of the receiver being ten times that of the barrel ? (2) Required the pressure per square inch, on the interior of the receiver in the last example, after two strokes of the piston ; the pressure of the atmosphere being 143 lb. per sq. inch. (3) Required the pressure per sq. inch on the interior of the receiver of a condenser, after two strokes of the piston, the atmospheric pressure being 14? lb. per sq. inch. (4) When the mercurial barometer stands at 30 inches, at what height will a water barometer stand, the density of mercury being 13*5 ? (5) At what height will a water barometer stand when the atmospheric pressure is 15 lb, per sq. inch? (6) A diving-bell is sunk till its top is 45 feet below the surface of the water : what height of air is there then in the beU, the barometer outside standing at 30 in., and the mercury being 13*5 times the density of the water, the height of the bell being 5 ft.? (7) What number of degrees of Fahrenheit correspond to —3°, and to 49° Centigrade ? And what degree of Reaumur corresponds to 39° Fahr. ? (8) A barometer which exposed to the atmosphere would stand at 30 in. , is placed under the receiver of Smeaton's air pump : the barrel and receiver have equal capa- cities : what will the height of the barometer be after three strokes of the piston ? (9) A cylinder 36 inches high, and 24 inches in diameter, is filled with atmospheric air when its pressure is 12 lb. on a square inch : to what depth will an air-tight piston of 3 tons weight sink in the cylinder ? (10) A spherical shell or balloon is to be formed of material of which the thickness is h, and its density, relatively to the surrounding air, d ; and it is to be filled with gas of density S : prove that, in order that it may ascend in the air, its exterior radius must exceed Jc-r-< 1~"V T^If \' 518. Scholium. — On the subject of the foregoing short treatise, the student may consult with advantage the neat little " Manual " of Professors Galbraith and Haughton, and the larger works of Bland and Webster. But equally as in the other branches of the mixed sciences, its complete development requires the aid of the Differential and Integral Calculus, a department of pure mathematics upon which we shall now enter. End of the Hydrostatics. DIFFERENTIATION 489 IX. THE DIFFERENTIAL AND INTEGRAL CALCULUS. 619. The symbols employed in the Higher Calculus — the principles of which we are now about to explain — are the same as those always used in the operations of common algebra, with the addition of merely one or two, peculiar to the present subject. But it is more especially necessary here to class the quantities which those symbols represent under two distinct heads : — constant quantities, and variable quantities ; for no analytical in- vestigation into which variable quantities do not enter, or in which a change of value may not be allowably assumed, can ever involve the pro- cesses of the differential calculus, which is essentially the science of variable quantities. The distinction between constants and variables has been sufficiently exemplified in the Analytical Geometry : the former have values absolutely fixed and determinate, like the figures of common arithmetic : — the latter may take any values we please to give them, consistently, of course, with the condition or equation which fixes — not their values — but their relation of dependency on each other. In ordinary algebra the earlier letters of the alphabet, a, b, c, &c., are used to represent known quantities, and the final letters, z, y, x, &c., to denote unknown (though constant) quantities. Here, however, the leading letters will be employed for constant quantities, whether known or unknown, and the final letters for variables. 520. Differentiation* — Before we can explain the meaning of this term, it will be necessary to examine into the effect produced upon a function of a variable x, by changing it from one value to another. (1) Let the function be ax^ + b: we shall represent it by y, writing y=:ax--{-b, and shall observe what change takes place in the function y by changing x into x-^-h. Putting y' for the changed function, we have i/=a{x-\-h)^-\-b={ax^-^b)+2axh-\-h% from which, if we subtract the original function y=ax'-\-b, we shall have y'—yz=2axh-\-h' for remainder ; that is, using the customary form of ex- pression, if h be the increment of the variable, 'iiaxh-\-h~ will be the incre- ment of the proposed function of it, whether h be positive or negative. In the latter case, h would more properly be a decrement: but it is usual to call it a positive increment in the one case, and a negative increment in the other. The ratio of the increment of the function to the increment of the variable in the case before us is ^^=2ax+h [1]. (2) As a second example let the function be y=ax^+bx—c. Giving to the variable x the increment h, the increment y'—y of the function will be y'-y={a{x-\-h)^-]-h(x-^h)-c}-(ax^+bx-c)={2ax+b)h-^ah\ and the ratio of the increment of the function to that of the variable will be y-^=2ax-\-b-\-ah [2]. fi (3) Suppose, lastly, that the function is y=ax^—bx-^c, then we have y'-y={a{x-\-hy-b{x-{-h)-c}-{ax^-hx-c)={dax^-b)h-^Zaxfi^-^ah\ 499 DIFFEEEKTIATION. /. ^^=2aar-b+Saxh-\-ahP [3]. h The increment, or difference h between the original value a of the variable and the new value x-\-h, is frequently represented by the notation ax, where A stands not for a quantity, or factor, but for the words " difference of," or "increment of;" and, in like manner, for y'—y, the increment of the function, the analogous symbol Ay is employed ; so that the three ex- pressions marked above may be written thus : — (1) ^=(2ax)+Aa:. (2) ^={2ax-\-h)+aAx. (3) ^=(3aic2_6)+3aaAa;+a(Acc)2...[l]. Each of these is of course an identical equation : neither implies any condition : the first member merely indicates what in the second is actually performed, that is, the second member merely interprets the meaning of the first. It is of importance to notice, in reference to the three preceding results for — , that the quantity within the vinculum in each case — and which we AOS may regard as the first term of each result — is independent of Ax or h, so that this first term will remain unchanged, however we alter the value of Ti : suppose, then, that assuming any particular value k for h, we con- ceive that value to continuously diminish from h=^k down to ^=0, the aforesaid first term, which all along will remain undisturbed, will then Aw f/ y express the complete value of — , or ; for all that follows it will vanish. In all these instances, therefore, this first term accurately denotes Ay the limit of the ratio -^, or that final or ultimate value of it to which it Ax tends as £^x or Ji continuously diminishes, and at which it actually arrives when these continual diminutions bring h down to its ultimate value A=0. Ay It is this limiting value of the general ratio —^, in any particular case, that AX the calculus selects for its operations. Aw In the foregoing examples we have arrived at this limiting value of — Ax by first getting the general value, namely, that value which is independent of all restriction as to the value of the increment Ax, and then introducing the hypothesis of Aii?=0. The calculus will furnish rules for enabling ua Ay to obtain the limiting value of the ratio — , whatever function of x y may Ax be, without first developing its general value ; that is to say, if for distinc- tion, we use the notation -~ for the limiting value of the general ratio dx Aw dv --, the calculus will enable us at once to find -^, without encumbering the A.r dx ° DIFFEKENTIATION. 491 expression with the terms involving Ao?, (Aa;)'^, &c. (as above), which are afterwards to be rejected. Employing the notation here proposed, the limits of the ratios [1] are severally <1) '^=.2ax. (2) ^=2ax+6. (3) ^=3aa^-&...[2]. Cu2/ dX (J/jC It is plain that these are only so many values of distinct vanishing fractions : — the values, in fact, of the fractions 2axh-\-h^ (2ax-\-b)h-^ah^ (SaxP-h)h+SaxhP-\-ah^ , , ^ , ^^ , , when hz=0, each of which fractions, in the proposed hypothesis, before the division by h is executed, assumes the form -, and it is this form that the more signi- ficant notation -^ replaces. We say this is a more significant notation, because it clearly implies that the general fraction or ratio, of which this Av denotes a single specific case, was — , the numerator of which stands for Ax a certain fixed and definite algebraic expression of the form F{x + Aoo)— F(x), where Aa? is of arbitrary value : the notation indicates, in fact, what we may call the pedigree of the particular - we are thus symbolizing. 521. From what has now been said it is presumed that the student will clearly understand what is meant by the expression the limit of the ratio of the increment of a function to the increment of the variable. If, as above, x be the variable, and y be put for the function, or the proposed algebraic expression involving that variable, the limit here spoken of is represented by -~, which is called the Diffeeential Coefficient, derived from the function : thus in [2] above, ^ax is the differential coefficient derived from the function aa;'^ ; ^ax-\-b is that derived from aar + bx+c ; and 2>ax^—b is the differential coefficient derived from ax^—bx-\-c. The relations [Q] necessitate the relations (1) dy=2ax.dx. (2) dy={2ax-^l)dx. (3) )+{^x^+^)d{3^-\-a). And by that for powers, diSx'^-\-h)=Qxdx, d{a^-\-a)=ZxHx, .-. dy={x^+a)6xdx-\-{ZaP-^h)Ba^dXy .:^^=15x*-[-Zhx'-\-6ax. ' dx (4) y=(a+6a;»0". The differential of the root, or expression raised to the power n, is mbx'^-^dXf hence, by the rule for powers, dy=n{a-{-hx^)^-^mhx^-^dx, .'. -^=6mw(a+6a5'»)"-Ja;»»->. dx (5) y=y/{a-\-hx'')={a+hxh^. . The differential of the expression under the radical is 2bxdx, hence, by the rule for powers, dy=- {a+hx^) ~ 4 2lxdx^ hx . dy_ *' dx'^^{a-\-bx^' (6)y= tions, dy=. {x^-2Y By the rule for frac- {x^-2r-d{x-]-Zf -{x-irZ?d{x^-2)^ (a;2-2)* By powers, rf(a;+3)3=3(a;+3)2c?.r, d(:c2_2)2=2(x2-2) X 2xdx, ^ dy_ Z{x'^-2Y{x-^2>Y-ix{x+2>f{:>?-2) '''dx~ {x^-2y {S{x^-2)-^x(x+S)}Cx-{-S y ~ (a;2-2)3 _ (■r''+12a:+6)(a;+3)'' ■~~ {x^-2)^ (7) y=a-\-s/(b-\-^^. By the rule for fractions, X^XO i^-^y- 2cx , 2c , — idx= ^dx. dy dx -^v(*+^.) (8) y2=aar'+&, ••• 2ydy=2axdx, dy ax ax dx'~ y ~^J (cu;*+6)' Examples for Exercise. (1) 2/=2ar^-4xH3a;-2. (2) y=a-Zx^. (3) y=(2+3a:)x3. (4) y=a+^. (8) 2/= >/(l-^') (5)y= (6)2/= a;2 a2+a;2 x'' \a+x^)' (7) y=(a+6a:)'. (9) y=a^x+-. (10) 2/=a+ 3+x2* (11) y={a:+V(x»-l)}'. (12) y=-3{a:'+arN/(aHa?«)}. APPLICATIONS TO GEOMETRY. 497 528. Applications to Geometry. — The general equation of a secant passing through two points {x\ y'), {x'\ /'), in any plane curve, is (352), (367), &c. v' — y" Aw' y—y'=.- — ^ (a?— a/), or y—y'=-^, {x—x'), whatever be the value of Ax . When Aa/, hy continuously diminishing becomes =0, and consequently ^2/^=0, the secant becomes a tangent at the point {x\ y') : the equation of the tangent is therefore y-y-%i^-^) \M 80 that if from the equation of the curve we find the general value of dy -J-, and then in the result we put the particular values of x, y, belonging to the proposed point, [1] will become the algebraic equation of the tangent at that point. For ex. Let the curve be the ellipse aY+lPaP=a^l^, or aY=a^^-^^' DiflFerentiating, we have 2a^y^=—2b^Xf .*.—=— ^. dy 6^ X dx ' " dx o?' y If, therefore, {afyy') denote the point of contact with the tangent, [1] gives the equation of the tangent where a, 6, are any semi-conjugates (see p. 320). Again : let it be required to find the general equation of the tangent to a line of the second order, the equation of the line being af-^-hxy+cx^+ey-^-lx-irp^La [2]. Differentiating, 2ay-^-{-l(x-^-\-y\-\-2cx-\-e /+^=0, dx \ Q/X / dx " dx 1ay-\-hx^-e " y ^ 2ay-\-bx-\-e^^ ^' is the equation of the tangent at the point [x\ y') of the curve [2]. We thus see that in the application of the calculus to curves, the dv differential coefficient -p always denotes the coefficient of inclination of a tangent to the curve at tlfe point (a?, y). In the straight line too, namely, y^ax-\-h. we also have -i-=a. dx When the axes to which the curve is referred are rectangular, — is the trig, tangent of the angle which the linear tangent makes with the axis of X : in reference, therefore, to such axes, the general equation of the normal, at any point, (a/, y') of a curve, is KK 408 LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 3,™,'=-^(.-.0 [2]. da! And from these equations [1], [2], it is easy to deduce, as in the Analytical Geometry, expressions for the lengths T of the tangent, iV of the normal, 7\ of the subtangent, and 1^^ of the subuormal, whatever be the curve : thus Let the curve be the ellipse : then, as above, <^'/ 6^ a;' -^,= -. — : and the subtangent is dx c^ y and similarly of the other expressions. These few illustrations of the geometrical meaning of the differential coefficient — will serve to give the student an insight jnto the use of this important symbol in the higher geometry : further examples of its application will be given hereafter. 529. Logarithmic and Exponential Functions.— i To differentiate log x, of base a. Put y= log a;, then Ay= log (ar+Ax)— log ar, JF+Aa; / Aa;\ (^+T> • y_ ^ _ * * Aa; Ax Ax - (no). .oy(l— as*) dy -1 (2) To differentiate y=cos-*a:, .*. coay=x, .: —sin ydy=dx, .: j-= //-■_, 2\ ' (3) To differentiate y=versin-» x, .*. versin y=x, .*. sin ydy=dx, dy_ 1 (4) To differentiate y=:tan~* 05, .*. tan y=a5, .*. sec^ ydy=dx, .'. — =- da; l-\-x^' (5) To differentiate y=cot-» 05, .*. cot y=a;, .*. —coaec^ ydy=dx, .'. ;;/=fT~2' (6) To differentiate y=sec~i 05, .'. sec y=x, .*. tan y sec ydy=.dx, .*. 3^= dx Xs/{x'^—V) 1 (7) To differentiate v=cosec- ^x, .'. cosec y=x, .'. —cot w cosec ydy=dx, .'. -r-=- / / ., , x - '' "^ ' dx Xy/{a?—l) (8) To differentiate y=log x, .'. log 2/=a5, .'. —=.dxj .*. — =log-' 05. y dx This last example is the same as y=e', since e* is the number whose Nap. log is X, that is, it is log~'aj. The preceding forms, like those in the former article, being in frequent request, should be kept in the memory. But -^ here is always the re- dx All ciprocal of -^ ^^^ ^^^ direct functions, when y is put for x. (1) y=Bm^x, .'. dy=navD^*~^ d Bin 05, =n sin"-^ cos xdx, dy .'. -—:=n sm" - 1 cos x. dx (2) y=co8 jc+cos 2a;+cos 3a;+..., dv .*. — -=— (sino5+2sin2a5-|-3 sin 3a5+...)' dx (3) u=y tan x^, y a function of 05, du=yd tan x^-j-tan x^dy. But c? tan 3i?=^aeQ^ji?d. q^=.1x sec^x^cZar, dw „ dy .'. -— =:2va5sec2ic'4-tana;2--i. dx c?x (4) 2/=xe*''**, .-. dy=xdec<'**+e<^'«^dx, .-. -^=e^''*'(l-a;sinx). (5) y=x sin-^ x^, .•. dyz=xd sin-^ x^^gjn-i jpS^j^jp But d sin-i x2_ 2x Va-x*) dx, dy 2x2 dx Va-x*)^ (6) y=tan-i|, " dx -i,/^^ 4+x2* (7) y=cot-» (a+6x)2, .•. dyz -d{a+hx)\ l + {a+hxY But d(a+&x)2=2(a+&x)&dx, dy_ &(a+&x) **d^~~l + (a+6x)** (8) y=(sin-ix)2, .-. d2/=2 sin-^ aid sin-^ x, . cZy 2 . dx /^(l— x^) 602 IKVEKSE FUNCTIONS. (1) y=cos ax. (2) y=log{xe<='>'''). (3) y=a^««*. (4) u=cotxy. (5) y=sm~'^ax. (6) y=:seG~^x^. (7) y=versin-* e*. (8) y=cosec-i wx^. Examples for Exercise. 1— JC* (9) y=sm-l ^-j-^. (10) y=tan-» j-^^, (11) y=cos" ^ n/(1+^)' /^«v , /1 4- sin a; 532. In the preceding articles rules have been given for differentiating any elementary function, and by the combination of these rules, functions however complicated may be differentiated, as several of the examples worked out above sufficiently show. Whatever be the function F(a;) therefore, we can always find the limit of the ratio — • — . If F{x-\-h) be developable in a series of terms, proceeding according to the positive integer powers of h, that is, if F{x-\-h)=A-\-Bh+Ch^-\-..., the limiting ratio spoken of will always be the coefficient of the first power of h in that development, that is, it will be B. For the first term A must be =F{x), since in the above identity, if ^=0, we must have F{x)=A, so that F{x-\-h)-F{x)=Bh-\-Ch?-\-,.,, dF{x) and therefore the limiting ratio must be ' =B. Hence, whenever dx F{x-^h) is developable in the above form, we may be sure that that development is always Fix+Iij^Fix)-^"^^ h-hCh^+ ax If page 116 be referred to, it will be seen that what was there called the limiting equation, derived from the algebraic equation F(^)=0, is no other than what is now shown to be the difi'erential coefficient derived dFix) from F{x) equated to 0, that is, the limiting equation is — p- =0. In like dx manner, in Newton's method of approximation (p. 135), the correction fir) a/=—jY^, is no other than the original polynomial, divided by its dif- Jiv) ferential coefficient, when in each r is put for x. And similarly in the theory of vanishing fractions (p. 144) F,(x) FJx) , , . 1 XI- . , , !> -^(«) -r^-;, -7=^, &c., merely imply that ntim. and den. of -rH fii^yfii^) /{«) are each to be differentiated^ then the num. and den. of the resulting . fraction, and so on. INTEGRATION. 503 533. The student must not fall into the mistake of supposing that he may apply to an algebraic or numerical equation the operation called dif- ferentiation, as he would apply the ordinary operations of algebra, and thus get a new equation which holds simultaneously with the original : the equation jP(x)=0, does not imply that jP,(x)=0 f where F^{x) is put for — -^ \ unless F{a:) fulfils a special condition : — unless, in fact, it involves equal factors, that is, unless F(a;) = has equal roots. It must be always re- membered that the differential calculus is exclusively occupied with quan- tities which, without any violation of the conditions to which they may otherwise be subjected, may be regarded as varying continuously. If the conditions be such as to forbid this continuous variation, the calculus be- comes inapplicable. In the algebraic equation F(^) = (), x has certain determinate values : — the roots of the equation, to the exclusion of all other values ; that is to say, x has, not variable^ but constant, values. If, however, we remove the restriction that no values of x are admissible, except those which make i^(^)=0, and take the expression F[x) inde- pendently of all conditions, then, of course, we may treat ;p as a continuous variable and freely apply the calculus. Thus, putting x-^h for X, we may write, as above, and are at full liberty to impose the conditions jP(a;)=0, Fpi;):=-0 ; the consequence of which would be that for every value a of a? which satisfied them both, we should have F(a+h)=Ch^-{- Now, by giving a sufficiently small value to h, the series on the right, for that small value, and for all values still smaller, would, as a whole, have the same sign as C, whether that small value of h be positive or negative (p. 150). But we know that if F{a-^h) and F(a—h) have the same sign, then between the numbers a-\-h and a—h there must be an even number of roots of the equation F{x) = (), or else no roots at all. Now here the condition is that one root, namely, a, necessarily lies between a-{-h and a—h : hence two roots, at least, must so lie, and that however small be the interval a-{-h, a — h, .-. the equation must have two roots at least equal to a, provided the condition F^(x)z=0 have place simultaneously with F{x)—0. This accords with what has been already established in the theory of equations, and we advert to that subject here only for the purpose of showing thus early, as at art. (528), that differential coefficients have interesting and important bearings upon geometrical and algebraic inquiries. Having thus established the fundamental rules and principles of the differential calculus, we shall postpone for a while the further develop- ment of those principles; and shall, in like manner, now unfold the leading elementary operations of the Integral Calculus. 634. Integration.^ — This is the name applied to the operation which is the inverse of differentiation. When a function is given, its differential may be found by the preceding rules : when, on the contrary, the dif- ferential is given, the corresponding function — called the integral of that 504 INTEGRATION OF THE FORM {FxYd{Fx). differential — must be found by the rules of the integral calculus. The symbol used to indicate that the integral of any proposed differential is to be taken, is f, placed before the expression ; so that the symbols d and J' are symbols of operation the reverse of each other : — both pre- fixed to a function, leave that function in its original state : for instance, J'dF{x) is F{x), and dfF^[x)dx is {F^x)dx. The following three prin- ciples are at once suggested by the direct process : — 1. Since daF{x) is the same as adF{x), namely, aF^{x)dx, it follows that in the reverse operation, J'aF^{x)dx is the same as aJ'F^(x)dx', so that, whether in differentiating or integrating, any constant factor or divisor may be taken from under the sign of operation and placed before it : thus, daX is the same as adX, and faXdx the same as afXdx. 2. Since the differential of the sum of any number of functions is the same as the sum of their several differentials, it follows that, when we have to integrate the sum of any number of differentials, as for instance, Xdx-\-Xidx-\-X^x-\-hc., the integral will be equally indicated, whether the sign / be prefixed to the sum as a whole, or to each individual term ; that is, f{Xdx-\-X^dx-\-X^x-\-kQ.)=fXdx-\-fX,dx-\-fX^x->tkQ. 3. Since a constant connected with a function, by addition or subtrac- tion, disappears in the differential, it follows that in the result of the reverse operation of returning from the differential to the primitive func- tion, the constant should reappear. But as the differential remains just the same, whatever the constant in the primitive or integral may have been, we cannot possibly know, from the mere differential being given, what particular constant has thus been discarded. All that we can do, therefore, is to integrate as if no additive or subtractive constant at all entered the primitive we are returning to, and then to annex to the bare integral thus found, a symbol (7, standing for a constant of indeterminate value. If, however, we can discover what particular value the constant ought to take, for any one particular value of the variable, then, since it is constant, that so-called particular value of it is the only value it can have. We shall now proceed to the elementary rules for integration; and, in general, shall frequently write for brevity Fx, instead of F\x). 535. Integration of the Form (Fxyd{Fx).—1t is plain that this differs from the differential of (Fa?)'^+i-f a, only in this, — that it is not multiplied by the factor (n + 1), for (p. 495), d[{FxY+'+a-\={n-^l){Fxrd{Fx), .-. f{FxYd{Fx)J£^^+C. Rule. — Increase the exponent of the function under the vinculum by unity : divide the power thus increased by the increased exponent, and annex the arbitrary constant C. (1) Integrate aa^da5. af;x?dx=!^^-ira .^^ -r ^dX (2) Integrate — . (3) Integrate {a-\-x^^xdx. Here the differential without the vin- culum, namely, xdx, would be the complete differential of the expression within, and therefore the above general form would be complied with, if it were multiplied by 2, seeing that of a+a;^ the differential is not INTEGRATION OF THE FORM [FxYd(Fx). 505 «dx, but lo^x : hence, introducing the wanting factor 2, and dividing also by 2, we have 1(0+^(0+^ 2~j— +C- g +0. (4) Integrate {6+ca!")'"ax''-'t?a!. Here it is easily seen that the differential without the vinculum requires to be mul- tiplied by — , .♦. — /'(6+c»")'"wca5'*-'c?x= a nc a(5+cic")'"+' + 0. nc{m-\-l) (5) Integrate {2ax—xi^(a—x)dx. Multiplying the differential by 2, lf{2ax-x^f {2a-2x)dx= i^ax-x^) + C. Note. — In some cases it may not be easy to discover by inspection whether a factor, and if it exist, what factor, is necessary to render the expression without the vinculum the differential of that within it : but we may always settle the matter thus. Taking the last example, assume its integral=^ (2aa!— a;2) + C, then differen- tiating, we must have 5 {2aa;— a;') (5a —x)dx, then, as before, assuming the integral to be 1 A{2ax—x^-\-Cf and differentiating it, we should have {2ax—x^(5a—x)= lA{2ax-x^\2a-2x), .'. 5=7A, 1=7A, 2 which are contradictory : we infer, there- fore, that the differential is not convertible into the proposed form. We shall add but one example more. (6) f{a-\-lxydx=-f{a+hx)%dx= (a-{-bx 36 -f C. But we may proceed thus : f{a+hx)^dx= f{a^dx-\- 2abxdx+l^x^dx) = a*x-^dbx^+ -- — \-C. Now there is an ap- o parent discrepancy here ; for, as it is easy {a+bx)^ . to see, — — — IS not equal to oo a^a;+dbx^-\ — —, and yet, whichever of o these unequal expressions we differentiate, the result will be the proposed differential. This fact alone would suggest to us that the two results can differ only by an addi- tive or subtractive constant, and accord- ingly we find that they differ only by the constant—. Hence, in the second mode oo of integrating, the constant C differs from the constant in the first result by 36" In (2ax—xY (a—x)dx= I A (2ax-x^)^ {2a-2x)dx, it .'. 7-^(a—x)=a—Xf .'. 7Aa=:a, 7Ax=x, which conditions agree in giving A=-, 1 I 80 that the integral is - {2ax—x^-}-C. But if the example had been It thus behoves the student, when he arrives at an integral different from the integral of the same differential as obtained by another person, not to conclude that either is incorrect : — both will be equally correct if they differ only by a constant. Examples for Exercise. either result the constant is arbitrary ; but if its value be fixed in one, it is fixed in the other, in virtue of this constant dif- ference between them. il)f^dx= {2)J^(a-^-^xhdx= (3) Qfs/{4^x^-]-B)xdx= 2adx dx= [See (536)] (6) r-^i^,: J (2ax—aPf XsJ{2ax—x^) xdx 506 INTEGRATION OF PARTICULAR FORMS. 7\ r ^''^^ — (3)/ N/(a'+^ cZx= 536. There is one case, coming under the general form above, which is not the differential of any power, and to which, therefore, the foregoing rule for the integration of it does not apply: it is the case in which n= — 1, the form being -—, which we know to be the differential not of a power of X but of log X (529) : hence /dX =log Z+ (7=log cX, putting c for the number whose log is O. And this formula is always to be employed when the proposed dif- ferential is a fraction whose numerator is the differential of the de- nominator, thus: — /^^='"«^(^+'''>- (2) /adx a p hdx a-\-bx~b I a-\-bx~~ -log C{a-\-bx). p bxfdx __ 12 logC(3x<+7). {x—ci)Hx --3axH h3a2loga;+C. /U X 537. Integration of the Forms --/\--, -~~-^, dx dx x^ib'x-a^/ ^{a^x-i^x^f dx Since (531) d wi-^x=-— -, .-. r d sin-i-a-=- -— , /dx 1 . _j h d COS"* x=- ■dx —dx '•/^{a- 1 , .h —dx -=— nr>a— 1 ■62x2) b dx d tan-' a;=— -— 2, .-. — d tan-* - «=— — -—„, 1+x^ ab a o?-\-b^x^ / ax _ —dx -T tan-* -a;. ab a —dx 1 b —dx dcot-»a;=— ; — 5, .-. — rfcot-»-a;=— 1 -J- nri atf a "^ ■\-Wx^ dx 1 ,6 INTEGRATION OF PARTICULAR FORMS. 507 Since (531) d sec-^ x— — 77-r— r:, .'• -d sec-» - x=. — 7--^^ — jr, dx 1 ,h ■f ■X, x>^{b'^x^—a^) a a' —dx 1 , ,6 —dx d cosec~* x=: — TT-^ — — , .'. - d cosec"* -x=- x^{x^-iy ' a a x^{p-x^—d^) •2\* ■f dx 1 ,6 =- cosec- ' - x^/ib-^x^-d') . , dx 1 , . , 252 dx d yeTsm"^ x=—7— —. .'. -d\eTsm-^—r, x ■/ ^/(2a;-x2)' " b '^ "^ " d' -" ^{d^x-b^x^ dx 1 . ,262 s/{a^x-b'^x^) b . , -dx 1^ . ,2ft2 -dx a coversin-i x=—j— -, .-. - d coversm-^ —^ x=- • — 7-— T-, .. — c*»;uvensiu t-m/ — ti^, ... ..v» /s/(2a;— jc2)' 6 d^ >^{a-x—b^x-) ■■/ dx 1 ,262 •=- coversin~i — x. s/ {a^x—b'^x-) b a Besides these forms, there are the following, deducible like them from the elementary dififerentials already given, namely, a* Since eia*=log^ a . a'cZic, .-. fa'dx-=.\ , .*. f edx=.iS'' log^ a -^ ,, d sin a;=cos a: dx _\ ( p dx p dx \ y{2ax-]-x^)}. These four integrals, we see, are arrived at by the aid of such algebraic expedients as are fitted to bring them under the known form -r^ for a logarithm ; and by means of similar artifices many other difi'erentials to which the preceding rules are not immediately applicable may be trans- formed into equivalent difi'erentials, which are integrable by the forms already given : but no general precepts can be laid down for bringing this conformity about. The following, however, is a principle of wide appli- cation. INTEGRATION OF PARTICULAR FORMS. 509 539. From the dififerential of a product, namely, d{uv)=udv -\-vdu, we have uv=.fudv-{-fvdu, .'. ftidv=uv—fvdu...[l'], SO that we can always integrate udv, provided we can integrate vdu, 1 /^dv V . r*v du If for u we put - tne formula is / — = — \- / . The following are examples of integration hy parts, as this method is called. (1) To integrate xe'dx. Put x for u in [1], and e'dx for dv, .'. e*=v, and fvdu=.fe'dx=.(^ \ and we shall have fx€'dx=.fndv=.uv—fvduz=.x^—^, (2) To integrate x log oadx. Put log a;=w, and xdx^=.dvy /, /a;~ (log xydx=fudv=uv—/vdu= ^~f~"^ 1 /a^"* (log «)**"* ^^a;. This last can be integrated by the above case if n=2, .*. the integra- tion can be effected when 7i=3, n=4, &c. : we thus have /..(log,)^d.=_ { (log .).-_ log .+ Ji}, /.".aogx)3..=^ {oog .)3--^ aog.)H^, log .-jll^,}, /a;'«(log a;)Ma;= If m= — 1, this cannot be computed, as all the denominators are then ; but the formula is not required in this case, for then /(iogx)"|=/aog.)Miog«=l!2«^'. 510 AREAS OF CURVES : DEFINITE INTEGRALS. x^dx (5) To integrate 5- (1— ^) This is the same as 1 d{l-x^ 2^* {l-x^y Put - a:=M, and d{\ {1-x^f x^) 1 1 --„=c??;, .-. d\i=- dx, and (535) v—- ; '^y 2 1—y^ /.rf«=iy^=iy(jij+jij)&=^iogi±2, hence /rrVa? - , 1 a; 1, 1+a; (1—^2)2 J -^2 1— a;2^4 ^1— a; Note. — It must not be overlooked that in all these integrations the arbitrary constant is suppressed : it must never be forgotten that the iii- troduction of such a constant is always necessary to complete the integral, and to justify us in equating integrals of the same differential arrived at in different ways, and appearing under different forms, as noticed at p. 505. Thus, in the first of the integrals at (538), if we make a=0, it /dot/ — , which, without the constant, is log a?, whereas the general integral without the constant, when a=0, is log 2x. But, supplying the omitted constants, the former integral in a complete form is log Co;, and the latter log 2C^a;, or log Cx, so that there is no contradiction. 540. Areas of Curves : Definite Integrals. — Let AB be a portion of any curve referred to rectangular axes OX, OY, and let it be required to find an expression for the area included between the ordinates aA, bB, which we shall consider as the first and nth ordinates, and shall de- note by 2/1, 2/„. Let ab be divided into n—l equal parts, each part being denoted by Ax, and take another part bb' also equal to Ax, so that there may be n equal parts in ab'. Then, drawing the parallels as in the figure, there will be n rectangles all of equal breadth between aA, and b'B\ and the area of the polygonal figure aB\ which is the sum of these, will be Area of Polygon=(y,-f 2/2+2/3+ +y„)^- It is plain that if Ax continuously diminish, the rectangles thus be- coming narrower and narrower as they increase in number, the polygon will continuously approach nearer and nearer to coincidence with the curvilinear area, which latter will be the limit of the polygonal areas : hence, Area of Curve=(yi+y2+y3+ +y„)^^-[l]- Again : complete the rectangle BO, the point O being on the curve : then corresponding to the increment Ax, the increment of the area of the curve is the curvilinear space bO ; and taking Ax, or bb\ sufficiently small, this increment will be between the rectangles bB\ bC^ in magni- tude, for that and for all smaller values of Ax, and y^ being any ordinate, the ratio of these rectangles is — = — '■ , which in the limit, that is, when Aa:=0, and .•. Ay=^0, is -=1. Hence A Area being always intermediate between AREAS OF curves: DEFIxVITE INTEGRALS. 511 ySx and {y-\-\y)^x, the ratio of A Area to either of these rectangles must be 1 in the .. ^ , ..,,.. A Area ^ d Area /■ j rm hmit; that is, m the limit, =1, or — ; — =1, .'. Area=/yrfx...[2j. yt^x ydx 541. If, instead of being rectangular, the axes are oblique, (p being their angle of inclination, then it is plain that we must replace y above by y sin ip, so that for rectangvlar axes, ATea,=fydx : for oblique axes, Area=sin (pfydx. Since y—F{x), it follows from [1], [2], that the Hmit of y;F{x)^x is fF{x)dx...[l\ where tF{x)^x stands for {F{x)-{■F{x-\-^x)-\-F{x-\-1^x)-\-...-\-F{x-\-n^x)}^x. This circumstance sufficiently explains why the long s is placed before a differential to indicate the integral of it : — the prefix is the initial letter of the word " sum," the integral being the sum of the elements or succes- sive increments F{x)^x, F{x^\-£^x)^x, , F{x-\-n^x)^x, which make up that integral, when these elements, by the continuous diminution of Aa;, become individually smaller and smaller, and at length vanish. It is thus said that " the integral is the sum of an infinite number of infinitely small quantities :" but as an infinitely small quantity, or an " infinitesimal," cannot have any magnitude, the statement is tanta- mount to saying that the integral is the sum of an infinite number of nothings. In either form the declaration is repugnant to the under- standing ; and conveys a meaning that is not in reality intended, and which is open to the same objection as the statement that nothing divided by nothing produces something. In the differential calculus nothings are not divided, as we have already shown (522), nor in the integral calculus are nothings added. In both cases general algebraic quantities are operated upon, and general results obtained ; these, in virtue of their generality, are open to any special interpretation it may suit us to give to them : such an interpretation may be given that if the same interpretation had been given to the general symbols under the sign of operation at the outset, the form - or x oo might have presented itself ; but so far from any use having been made of either of these forms in arriving at the result, it is rather the obtained result that has suggested them. In the matter before us, that particular case of a general result which in this way leads to X CO , is shown above to be J F[x)dx between assigned limits of x. 542. In the preceding diagram the curvilinear area considered is sup- posed to be bounded by the fixed ordinates aA, bB, corresponding to the abscissae Oa, Ob, which we may call a, b. It is within these limits, there- fore, that the integral fydx, representing the area indefinitely, must be taken : the definite integral, as it is then called, is expressed by the nota- tion flydx, or fa{Fa:)dx, which implies that the general, or indefinite integral, being found, we are to put for x the given values a and b succes- sively, and then subtract the former result from the latter. Each of the two results will of course involve the same supplementary constant, which will therefore disappear in the remainder. 512 AREAS OF curves: definite integrals. (1) Required tlie area of the parabola y^^ax, or y=i{ax)^. Here, between the limits, a;=0, and x=x, Axea,=fl ydx—ahfl x^dx=ah — =- x{ax)^, j y^ that is, Area ABX=- ary, since for x=0, the integral vanishes. A ■ ^ 3 The area here taken lies between the limits AY, XB, the corre- sponding values of x being x=0, and a;=a;, that is, any value AX. (2) Required the area of the circle y^=r^—x% or y={r^—x') «, /. fydx=/{r^—x^dx, or representing the radius by unity, i sc^ X* a^ 5a;8 /yd»=/(l-a^*/(«^+«^)+i a-\-2x-\-2^{x^-\-ax) a Suppose, for example, it were required to find the length of the para- bolic path of a shot, of which the horizontal range is 4800 feet, and its greatest height {x) 1600 feet. Here x=1600, «=2400, and a=7-=900, 4a; /. «=^(160024-900xl600)+4601og 900-f-3200-i-2V(1600i'-f900Xl600) 900 =2000-f 450 log 9=2000-i-450x 2 •197225=3735 -9 feet. (2) To find the length of an arc of a circle. According as the rectan- gular axes originate at the centre or at the circumference, we shall have 3^=^-x', or y^=2^x-^, ,. ,=rj-j^--^, or ,=y2_|_, and the integration in each case will itself involve a circular arc (637) : LL 2 516 SURFACES OF SOLIDS OF REVOLUTION. the length cannot, therefore, be found in finite terms. By (531) the differential of an arc whose tangent is t and radius 1, is .-. tan-i t=t -\ <3 -\\ i^ -1 174. ... o o 7 Since, when the arc is 0, t is 0, the correction C is 0, and the series is the same as that otherwise determined at p. 217. If, however, we develope (1 — ic-)~*, the coef. of dx in the preceding dif- ferential, when r=l, we shall have which, when a5=-, or the arc=:30°, gives .'. 30°x6=«-=3-141592... In like manner, the expression for an arc of an ellipse involves a dif- ferential that can be integrated only in a series. 545. Surfaces of Solids of Revolution.— Let S be the sur- face generated by the revolution of the curve s=AB about the fixed axis of a;, OX, and let BC be any increment As of s, AS being the corre- sponding increment of the surface. Draw Be, Cb, parallel to the fixed axis, and each equal to BC or As : then, since every point in BC, with the exception of the point B, is farther from the fixed axis than the corresponding points of Be, that is, than the points having the same abscissa, the surface generated by BC must exceed the cylindrical surface generated by Be; and since every point in Cb (except C) is farther from the fixed axis than the point having the same abscissa, of CB, the cylindrical sur- face generated by Cb must exceed the surface gene- rated by CB ; hence, AS always lies between 27ryAs, and 27r(2/ + Ay)As, provided, at least, that BC be so small that the curve has no sinuosities or bends in that interval, a condition that may always be satisfied by taking BC sufficiently small. Now the ratio — -- — 7 — — -= — — —=1 m the limit, or when Av=0. 2^{jt/-\-Ai/)As y-\-Ay ^ Hence the ratio of AS to either of these cylindrical surfaces is 1 in the limit, .*. =1 in the limit, that is, =1, 2iryAs ' 29ryds ' A dS=2^yds, bnt (544) ds=:y/ 1 l+(^y\dx. SUEFACES OF SOLIDS OF EEVOLUTION. 517 .'. S=2tffyds, or S=2^fy^ll-\-(-^J \dx, between the assigned limits. (1) Kequired the surface of a sphere of radius r. The equation of the generating semicircle is ccP-\-y'^=r^, .'. 2xdx+2ydy=0, "dx y' V V^\dx/ ) y y' y ' .'. 2^fyds=2*frdx=2irrx+C, .'. S=2*fl=2*A This is the surface of the hemisphere : hence, for the whole sphere, the surface is r^x, is tilled up (541). The limit of the sum of the terms, following one after another according to this law of succes- sion, is what that sum actually amounts to only when, by continuously diminishing the arbitrary interval ^x between every consecutive pair of terms, each interval becomes reduced to zero, and consequently the number of them becomes infinite. It is the special office of the integral calculus to furnish us, in all cases, with this limiting value, to the exclu- sion of all the other values, in the continuous series of values which this terminates. The differential calculus and the integral calculus are both occupied exclusively with the limiting values of continuously varying quantities : the former calculus furnishes us with the exact interpretation, as to value, of what, in the absence of such interpretation, would 518 AREAS, SURFACES, AND VOLUMES. be merely the vague symbol - : the latter supplies the exact interpreta- tion of what would otherwise be merely the equally vague symbol x oo . 647. From thus regarding an integral as the limit of a summation, or what the series becomes when Ax becomes da;, the foregoing general ex- pressions for areas, volumes, &c., may be deduced with great facility: thus, since (see fig. p. 510), 1. Area=limit of X « (y^*)> •'• Area=y * ydfar. The parallelogram yAx, or rather this in the limit, namely, ydx, is called an element of the surface ; the surface being made up of an infinite number of these elements. 2. A solid of revolution may, in like manner, be regarded as made up of circular slices, each slice of thickness Ax, and all perp. to the fixed axis. If, however, we consider first — not the solid itself generated by the curved area above (fig. p. 510) — but that generated by the assemblage of parallelograms, the result will be a series of cylinders ; and y representing the radius of the base of any one indifi'erently, the sum of all will be ex- pressed by 2*+^^ {iry'^Ax), which sum, in the limit, or when the thickness Ax of each cylinder is reduced to zero, accurately gives the volume of the proposed solid, that is, Volume=lmiit of t^\^ (rfAs:), .-. Volume=T/Jy«fia;. The cylinder wt/^Aa?, in the limit, that is, Trirdx, is the element of the solid. 3. In like manner, the element of a surface of revolution is 2TT/^{{Axy-{-(Ai/y} in the limit ; the chord of the arc As being in a ratio of equality with that arc at the limit, .-. Sur£ace=liinit of t^\^2^>^{{Ax)^-{-{Ayy}= limit of 2*r2«^^{l4-(^^y}A., that is, STirface=2*/Jyy|l+(^yW. The student will do well to consider attentively the train of ideas which lead to these several results. In the case of a plane area included be- tween two bounding ordinates corresponding to x=a, and a:=b, and between the portions of curve and axis of x intercepted by these ordinates, we mentally proceed thus. Commencing with the first or initial ordinate, we conceive it to move parallel to itself, preserving its original length, towards the final ordinate, till a parallelogram of breadth Aa; is generated, Ax being entirely under our control as to degree of smallness : the moving ordinate then stops, and increases, or diminishes, as the case may be, till it again reaches the curve : thus modified, its motion recommences, and another parallelogram, of the same uniform breadth Ax, is generated; and in this way a succession of parallelograms, with one angle of each on SUCCESSIVE DIFFERENTIATION. 519 the curve, is formed by intermittent motion, the last parallelogram ex- tending beyond the final ordinate. We then imagine Ax to continuously diminish, and thus the number of parallelograms, each getting narrower and narrower, to increase, the in- termittent motion above alluded to approaching nearer and nearer to a continuous motion, which, when actually attained, the continuously in- creasing or decreasing ordinate generates, by its uniform motion — not a series of inscribed parallelograms — but the curve surface itself, which is the limit to the sum of that series when it consists of an infinite number of terms, and of which limit we have seen that the definite integral J'lydx expresses the exact value. In like manner in reference to a solid of revolution, bounded by two circular sections, through the centres of which the fixed axis passes. Commencing with the first bounding circle, we conceive it to move parallel to its own plane, preserving its original magnitude, towards the final circle, till a cylinder of breadth Ax is generated: the circle then expands or contracts till it again becomes a section of the solid ; and then a second cylinder of breadth Ax is generated ; and so on. The limit of the sum of this series of cylinders, which limit is approached to closer and closer as Ax diminishes, and which is ultimately reached only when A:k=0, is the proposed solid, and we have seen that its value is expressed by the definite integral irj^ydx. And a similar process applies to the curve line, and to the curve surface. 548. Whatever function of the single variable x, X may represent, it is plain that y=X will be the equation of some plane curve; so that we may regard every definite integral flXdx as a case of quadrature, or as a plane area taken between the limits x=a, x^=c. If b he an intermediate value of X, between a and c, the area will consist of the portion between the limits x=a, x=:b, and of the portion between the limits x=b, x=c, as is obvious : hence and in this way may any definite integral be cut up into component definite integrals, as many as we please. Since the notation /^ denotes the general integral when x=c, diminished by the general integral when x=ia, it follows that flXdx=.—J'lXdx, and from this we may conclude that fl(Fx)dx is always the same as flF{a—x)dx. For, in the former, put a—z=x, then dxr=—dz, and the integral is the same as —JlF{a—z)dz, because, when z=^0, x=a, and when z=.a, x=0 : but it was before seen that -flF{a-z)dz=flF{a-z)dz, .-. flF{x)dx=flF{a-x)dx, the last definite integral being, of course, the same, whatever letter whether z or x, the constants and a replace. We shall now return the differential calculus. 549. Successive Differentiation.— Since the diff*erential co- efficient, derived from any function of a variable, may itself also contain that variable, or be a new function of it, this new function may, in like manner, be diff'erentiated, and thus a second diff'erential coefficient be ob- tained. If the variable appear also in this, we may obtain a third differential coefficient, and so on, till we arrive at a differential coefficient 520 THE THEOREM OF LEIBNITZ. into which the variable does not enter, and at which, therefore, this process of successive differentiation must stop. For example, taking the function y=ax^-\-hx^-\-c, we have First differential coefficient — =4aa;^-f 26a;=/,(a;). Second Third Fourtli dm dx =12ax'-^2h=Ux). ax df (x) J^ =24a=/,(a;)=constant But the usual notation for successive differential coeJB&cients is this, namely, dy d^y d^y d*y d'*y ^' d?' dc^' rf^' dx^* where the small figures over the d denote not powers, hut repetitions of the operation of differentiating ; and where the exponents over the a apply to dx, that is, da;'' is merely a shorter way of writing [dxy. When- ever, hereafter, a differential of the power of* is to be indicated, a point will be interposed, thus, d.a;"\ We shall give an example or two of suc- cessive differentiation of explicit functions. (1) y=a;'*, . dy da ' d^if d'y dx ^=%(7i— !)...[%— (r-l»«-* (2) y=loga;, dx ' ^=(-1)^2.-3 g=(-l)-i2.3.4...(r-l)^ r. (3) u-=yzj where y and z are both functions of the independent variable a?, du dz dy dx dx d£ d^u__ ^ dy dz +^ V* that is, arranging the terms according to the differentials of z, dr+hi_ d^+h dy £z (r-\-l)r^ d^-^z dx' +i~^ rf^' H-*"^^^"^ ' dx ' dx^"^ 2 dX' ' dx^^'^ (r+l)r(r-l) dhf d^-^z d'+^y 2.3 dx^'dx'-^'^'"'^dxr+^ ^•••^ -■* Now this series is what [1] becomes when r is changed into r+1 5 so that if [1] be true for any value of r, it is necessarily true for the next value r-h 1, and consequently for the next r-f2, and so on. But, as shown in last page, it is true for r=3, hence it is true for r=4, for r=5, and generally for r= any positive integer whatever. Examples for Exercise. To prove the following results, the first three by successive differen- tiation, and the fourth by the theorem of Leibnitz. d«y 2.3.4.6 (l),=log.,.-.^-|=~^j^. (2) y=(a:«+a«;taii-»-, ft tPy 4a3 (3) y=a% d"y „ . .*. — -ry=constant=(loga)'". dx d^u (4) it=««-y, .-.--= dx'' dx Mn-1) ^_,^ «"-' T^+- + 2 dx^ dx"")' 551. Development of Functions: Maclaurin's Theo- rem. — If 2/ be a function of x, which it is possible to develope in a series of ascending powers of that variable, then will the development be where the brackets are intended to imply that, after the differential co- efficients which they enclose are obtained in a general form, a; in each is to be put =0. Or, if instead of representing the function by ^, we write it /(^), and as at (549), put f^i^), /2W, &c., for the successive derived functions, or differential coefficients, the development may be written thus : M=m+U0)^-\-lM0):^-\-^M0)^+ [2]. Since, by hypothesis, the function is developable in a series of ascending powers of x — of course with coefficients finite and independent of a — we may assume 522 DEVELOIMENT OF FUNCTIONS I MACLAUEIN's THEOREM. y=zA,-\-A,x-\-A^^-]-A^-]-A,x*-^..., /. [y]=A^, g= 2A,+2.SA^+SAA,x'^^+..., .: ^^=2A^ g= 2.3^3-1-2.3.4^,.+ ..., .-. [3]=2.3^3, ^= 2.3.4...n^„+..., .-. rpr\=2.3.L..nA^. The values of the assumed coefficients are, therefore, .„=[.M.= [|]...^[g],.3=^3[S]--- thus establishing the truth of [1], which is called Maclaurins Theorem, although it appears to have been first given by Stirling. 552. It is of importance that the student be not betrayed into attri- buting to this theorem an extent of generality not justified by the hypo- thesis upon which the reasoning that leads to it is founded. He must take notice that the original assumption is not only that y=AQ+A^x-\-A^'^-]-A^:^-\-kc., but also that the quantity comprehended in the " &c.," whatever it be, must, equally with the terms following the first term of the prefixed series, vanish when a?=0, and moreover, that all the successive differential coefficients derived from this quantity must vanish when a;=0. In all those cases in which the series [1] terminates, there is, of course, no sup- plementary correction ; the " &c." is then superfluous, and there is com- plete harmony between the premises and the conclusion. It is only when such strict equality exists that the second member of [1] can ever be regarded as the complete development of the first, and we have accordingly suppressed the " &c.," which most writers append, and which is only an unintelligible symbol, covering a meaning of which, in most cases, we know nothing. It will be remembered that when we equate a function with its develop- ment, we express merely an identity : the two expressions equated are different in form, but always the same in value. If the second member of [1] is the development of the first, the equation is true, whatever par- ticular value be given to x : such a value may be given to it as that the more of the terms of the series we sum up the closer do we approach to the exact value of the undeveloped function, and .*. the more insignificant do the terms neglected become: the portion of the whole thus neglected approaching continuously to zero, as the terms retained become more numerous. In every such case the " &c.," which, in different circum- stances, implies a correction, is, of course, superfluous : it is to such cases alone that the theorem rigidly applies. Maclaurin's theorem, then, always gives the development of a function in the declared form, whenever, by successive differentiation of that func- tion, we arrive at a diff. coef. which is constant, that coef. being the last of the differential coefficients in [1] : the development is then general and finite. APPLICATIONS OF MACLAURiN S THEOREM. 523 The theorem always gives the development for a particular value of the variable x, even when the series is interminable — from a constant dif- ferential coefficient never occurring — provided that for the particular value of X assigned, that series is convergent, or that there is no supple- mentary " &c.," and provided, moreover, that for the proposed value of x, none of the differential coefficients become infinite. In this latter case, we should, of course, conclude that the development in the prescribed form with finite coefficients is impossible. We say " of course," because, if finite coefficients were possible, the above process would supply them. [The student may re-peruse the observations on general developments at pp. 77, 83, and 167.] 553. In certain applications of Maclaurin's theorem to particular functions, the successive differentiations may become laborious : but when dv the first differential coefficient -^ is obtained, we may frequently deduce all the others by simply developing j- by common algebra. Thus, by the QiX dii theorem, since -:;^ is a function of a:, we have dx and comparing this with the development of y, as exhibited in the theorem [1], we see that [1] may be deduced from [3] by adding to [y] the several terms of [3], when they are respectively multiplied by X, \x, \x, &c. Hence, we have this rule to find the terms after [t/]. Develope ;^ in a series of ascending powers of x, whenever this deve- (tx lopment is easy by common algebra. Increase the exponent of x in each term by unity, and divide the term by the exponent thus increased. An example of the facilities furnished by this rule is given in the de- velopment of tSixr^x in next page. 554. Applications of Maclaurin's Theorem.— (i) To de- velope (a+a?)\ Putting y= (a 4- a;)", therefore [y]=a'*, .•.|=n(a+.)«-i, „ [|]=m«-i, ^=«(»i-l)(n-2)(a+;r)«-3 „ ^^=n{n-l){v,-2)a-^, &c. &c. Substituting these values in the general theorem [1], we have 2 "^ "^^ 2.3 (a-H;r)«=a«4-^a«-iz-f!^i:pL^a«-»:r>+!li:^ which is the well-known binomial theorem. 524 APPLICATIONS OF MACLAURIN S THEOREM. (9) To develope log (a+a;). Putting y=log («+«), •*• [y]=log «, and -dy- dx^ g=2(«+.)-3. &c., &c., ... log(a+^)=loga4-~2^,+3^-^,+ ... If a=0, or the function be log x instead of log {(i-^x)^ then [y], ^ , &c., "would all be infinite : \ve infer, therefore, that the development of log x in the proposed form, with finite coefficients, is impossible. But we know that \ogx={x-l)-\{x-lf^^{x-lf-... : we may infer, therefore, that — (l+i+i+...) is infinite. (3) To develope sin x. Putting y=sin x, .'. [y]=0, -I— -eg-- s— ••[§>. g=-c„.....g]=-.g=s.....[2>0. a? a^ The preceding values now recur, /. sin x^x—t—-^' 2.3 ■ 2.3.4.5 the arcual development of the sine ; and in a similar way may the development of cos x be obtained: but, by differentiating the above, we get cos x=l—--\-——-— 2^2.3.4 (4) To develope tan" 'a?. Putting y=tan~*a;, we have [y]^±n«r, where n is any integer. ^^ ^-1+ar^' • • \_dxA ' cia;2-~(iq:^2' • • L^^J-"- Now it is easily seen that the continuation of this process would soon become laborious; we shall therefore avail ourselves of the rule given in last page. Thus, developing the value of j- by common division, we have DEVELOPMENT OF FUNCTIONS I TAYLORS THEOREM. 525 ^=l—3iP-{-x*—3fi-\-ofi—z^-\-kc.j and consequently, by the rule referred to,* tsiU-^x=±nir-\-x—-a^-{-zsfi—=x'^+-^x^—-'-) as at p. 220. (1) To develope a*. (2) „ sin- (3) „ cos- EXAMPLES FOR EXERCISE. (4) To develope (tan x)*. (5) ,, {cosa;+/v/— 1 sin«)» (6) „ {a-\-bx+caP+...)\ 555. Development of Functions: Taylor's Theorem. — We have seen in the preceding articles that whenever a function /(a?) is developable in a series, proceeding according to the ascending powers of a, Maclaurin's theorem will always enable us to determine the coefficients of those powers. We have now to show that whenever a function of the form f(a} + h) is developable in a series, proceeding according to the ascending powers of h, another theorem, that of Taylor, will, in like manner, always enable us to determine the coefficients of the powers. But before entering upon the investigation of Taylor's theorem, it will be necessary to establish the following principle, which is all but self- evident. Lemma. — If, in any function of p-\-q, it be known that one of the quantities p, q, is constant, and that the other is variable, we may derive the correct differential coefficients from the function J{p-\-q), without inquiring which of the two is the variable, and which the constant; for the several differential coefficients will be the same, whether we dif- ferentiate y(jo+^) on the presumption that p is the variable, or on the presumption that q is the variable : that is, we shall have j{p-\-q) _, J(P+q) ^^^ .^ ^^^ following coefficients also equal. dp dq For since the function contains but one variable, we may put p + q=f{a!) ; and whichever of the parts p, q, of x^ takes the increment k, the result f{x + k) is necessarily the same, f(x4-lc)—f{x) and .'. -~ — - the same (whatever be h), whether h be regarded as Ap, or Aj^ : , ,. . df{x) . , df{x) , . hence, passing to the Jimit, —^ — is the same as — ; — \ that is, dp dq mp± u the same as S^S±^, ,. ^JiP+l) J'AP+i) _ dp dq dp dq For example : let y=^^{p-\-qf—Q{p-\-q). Differentiating with respect to J3, -^=6(2)+2)2-6 ; and diff. with respect to q, -^=Q{p-\-qf-Q. dp dq Again : let y=.\og (i3+g)+sin {jp-\-q)y .'. ^= ^^ + ) "^^^^ (i'+2)=j'> &c- This principle being admitted, let f{x-^h) be developable in a series of the form * This simple rule, which is to be found in most recent works on the calculus, the author believes to have been first given by himself in 1831, in his "Integral Calculus," p. 82. 626 DEVELOPMENT OF FUNCTIONS: TATLOB's THEOREM. f(x-\-h)=A,-\-Ath-\-AJi''+A,h^-^A,h*-\- [1], . df(x+h)^dA^o.dA, dA,j^,d_A, " dx dx^ dx dx dx ■■ ■" ^d.^-^^^=A,-^2AJi-\-ZA,h?^iAji^+ [3]- dfi Bj the lemma, the series [2], [3], are equal for all values of /i, _JA^ _l dA ,_1 cZMq ='~3 da;~2.3 rfx^' ^ 4 c^o; 2.3.4 rfa;* ' Now, for ^=0, in [1], the first member isf{x\ and the second Aq, ... /(^+A)=/(:r)+/.(x)A+/,(x) f +/3(^)^^-/4(^)2X4+••• Or, putting 2/ for /(a;), which development is Taylor's theorem, h being either positive or negative. From this theorem of Taylor that of Maclaurin may be easily deduced thus : — After the general differential coefficients have been obtained, and the above general identity exhibited, let a take the particular value as=0, then /.)=.H[|]..[0]|%[g]^3.g],4,.... Now each of the bracketed coefficients contains constants only; they are, therefore, all independent of the value of h, which value may .*. be any whatever, without affecting these coefficients : we may, therefore, replace it by x, or by any other symbol, so that A.)=.=E.H[g.4[g]-+.-^B]-+ which is the theorem before established. We shall now proceed to apply Taylor's theorem to one or two develop- ments in different cases of the function y(d;+/i) : but the principal uses of the theorem will be shown hereafter. Its importance in effecting algebraic developments is small in comparison with its value in suggesting geome- trical and analytical theorems by a discussion of its coefficients. (1) To develope sin (x+h) in a series of powers of the arcual measure of the angle h. ^ . . dy ^y . ^ Put 2/=sin X. .*. -T-=cos X. ^r-o=— sm x, -t-;=— cos x, &c., dx dx^ dot? ^2 1^ J^K .*. sin(a54-^)=suia;-|-cos 05. A— sin a;— — cos* — - -f-sina; ^ i ~'-- ^ 2.3 2.3.4 =sm,a--+2-3^-...)+eosx(i--+j^5^-...), LIMITS OF Taylor's theorem. 627 which merely brings us back to the known property sin (a;+A)=sin x cos ^+cos x sin Ti. (2) Develope tan"' (;»+/») by Taylor's theorem. Put y=tan"'a;, .-. -j-= — _=cos2y, i4,= — 2 sm y cos y — =— sm 2y cos^y, dic sec^2/ <^^ '^^ -^=—2 (cos 2y cos^y— sin 2y sin y cos y) ^=—2 cos (y+2y) cos y — = —2 cos 3y cos^y, —^=2. 3 (sin 3y cos^y+cos 2>y sin y cos^y) — =2.3 sin iy eos^y 3^=2.3 sin iy cos*2/, &c., „ , sin 2y cos^y , „ cos 3 v cos% , „ . sin 4y cos*v , , .-. tan-» {x-^h)=y-\-oos^y.h 1 ^^2 1 ^^3^_ 1 ^A*+... For exercises in Taylor's theorem the student may develope the func- tions cos (a;-\-h), log {x-\-h), and tan (x + h). 556. Limits of Taylor's Theorem. — From attentively ex- amining the reasoning by which the theorem of Taylor has been established, it will be seen that there is strict algebraic equivalence between f{x-\-h) and the series which that theorem affirms to be its development, only on one or other of two conditions. 1. The series must spontaneously terminate, in consequence of a con- stant differential coefficient being arrived at. '2. But if no such constant differential coefficient can ever be reached, and the series be therefore interminable, there can be strict equivalence between the infinite series and its invelopment /(x+h), only for such values of h as cause that series to be convergent, either from the beginning, or after a finite number of terms. As to the cases in which, for particular values of x, any of the dif- ferential coefficients become infinite, they are, of course, excluded from the theorem ; the development in the proposed form, viith. finite coefficients, being then impossible ; as in the analogous cases of Maclaurin's theorem. Such are improperly called the failing cases of Taylor's theorem : they will be examined into hereafter ; our business, at present, is with Taylors theorem, and not with cases that do not belong to it, and which are expressly excluded from it by the hypothetical conditions on which it is founded. The purport of the present article is to show how the Bemainder, or sup- plementary correction of the series [1], p. 526, may be estimated, when we stop the summation of it at any particular term, thus neglecting all that follow, inclusive of what may be implied in the '* &c.," when the series is not convergent. Let /(a?) be a function of x, such that it and its several differential co- efficients are finite and continuous between the limits «=«, a}^=a-\-h. And in the expression ■J.2 ~a ~.M— 1 ,y.n /(«+x)-/W-/.(a).-AW --Ma) 5;3-...-/.-,W 5X7(^1=1)-^ ^JTn-^'^' let R be such a finite quantity, not involving x, that when iB=h, the ex- 528 LIMITS OF Taylor's theorem. pression [1] may =0 : then, since the expression is also when a?=0, it follows that the function [1] becomes zero for two different values of the variable x. Consequently it cannot continually increase, nor continually diminish, as x passes through its intermediate values, from a;=0, up to x=h: — it must somewhere, in the interval, change from an increasing state to a diminishing state, or vice versa ; that is, calling the function [1] F, — , and consequently -7-, must change from + to — , or from — ijkX dx to +, at least once in the interval. But, for a continuously varying finite quantity to thus change its sign, it must first pass through the value zero, .-. there is some value x^ of x, between and h, for which the differential coefficient of [I] is zero, that is, for which fM+^)-A{(^)-ua)x-fM~----ii^^y-m is zero. But [2] is also zero, for x=0 : hence, as before, there must be some value x^ of x, between and h, for which the differential coefficient of [2] is zero. Continuing this reasoning up to n differentiations of [1], we have finally /„ (a -fa;)— R=0, for some value x,^ of x, between and h. Let this value be x—^h, where 9 is some proper fraction ; then R=zf^{a + Qh). Substituting this value of R in [1], and remembering that [1]=0 when x=h, we have f{a+h)=f{a)-{-A{a)h-\-f,{a) ^-\-Ma) |1 +...+/„_, (a) ^^^^^^+ M<^+eh)-^ [11 Z,o...n which is Lagrange's theorem on the Limits of Taylor's theorem, to which it gives the necessary completion ; for it is plain that the constant functions f(a), fj{a), &c., are no other than what f{x), -^y— , &c., become when X takes any value a : the letter a may therefore be replaced by x* The symbol denotes some fraction, about which all we know is, that it must be less than unity, that is, it must be between and 1, so that when Taylor's theorem, in any particular case, is interminable, and we take n terms of the series for the development of the function /(a + /i), we know that the Remainder, or supplementary correction necessary to complete that development, will be a quantity lying between the limits * The foregoing investigation of Lagrange's theorem is due to Mr. Homersham Cox, who published it originally in the Cambridge and Dublin Mathematical Journal, and afterwards in an appendix to his rudimentary work on the Integral Calculus, in Mr. Weale's well-known series. It is this latter version of Mr. Cox's method of proof which, with slight modifications, we have adopted above. For a different way of establishing the theorem, the student is referred to the large work on the Calculus, by Professor De Morgan, a work which every student of the Higher Analysis ought to possess, as it not only comprehends a more extensive amount of information on the Differential and Integral Calculus, than any other English publication, but being the work of one who has evidently examined Qvery point of diflS.culty for himseK, and has exercised an inde- pendent judgment upon all, it is replete with matter for profitable reflection, even to the advanced analyst. LIMITS OF MACLAURINS THEOREM. 529 /«(*) ^Tq ' ^^^ A(«+^) 17-5 • If ^ ^G so small that 2.3.. .w' •' ^ ' ' 2.3.. .» 2.3...% tends continuously to zero as n increases, becoming confounded with it when nz=z CO J in other words, if the series be convergent, this correction vanishes ; and the series is complete in the form in which it was estab- lished at (555). Let us apply Lagrange's correction of Taylor's theorem to an example or two: and first for the development of {a-^hf. Here we have (a+A)»=a«+w(a+0A)»-'A, or =a"+wa"-» A+w ^^ (a+^A)»-2 A, or =a«+wa'»-» h+n ^^ a»-2 h^^n ^^ . ^^ (a+^A)«-37i3, 2 2 3 and 80 on, where it must be observed that although 6 is less than unity in every one of these expressions, it is not the same in all. Again, for the complete development of log {x-j-h), we have log(x+A)=logx+---^+-^-...-^^3P —-^——^--^^. 557. Limits of Maclaurin's Theorem.— Putting a=0 in [1], aad replacing h, which may be anything, by a?, we have /(^)=/(0)+/,(0)a:+/.(0) f +/3(0)|^+...+/«-i(0) 2:3^-1^+ /'.^^-)2-£r. t^^- For example : let /(x)=sina;, .-. /(0)=0, /i(0)=l, /8(0)=0, /3(0);=-l, &c., as at p. 624, .*. sin a?=04-co3 Ox . «=«;— sin Ox . — =»— cos Ox . — - 2 2.0 =«:—--+ sin 0a;. - - =a;— — -4-cosga;. 2.3" 2.3.4 2.3' 2.3.4.5 =^-1^3+2X15-^^^ '^- 2.3.4I6.7 - "^^'°"''- And, in like manner, for the complete development of e' we have 2 2.3 ' • 2.3.. .(71-1) ' 2.3.. .» Since G lies between and 1, e^ must lie between 1 and e, so that the remainder after n terms lies between and e*- 2.3. ..« 2.3. ..TO It may be remarked in reference to the development of an exponential function, that the series gives but one value — the arithmetical value as it is usually called. If x were =^, the series would not supply the negative square root of e, though .^e^ would enter it, thus seeming to provide for the double value ; for tbis negative root is minus the entire series. It must be remembered that differentiation is applicable only when the variable to M M 630 DIFFERENTIATION OF COMPOUND FUNCTIONS. wbicli it is applied varies continuously: if x vary from a;=0 to a?=l, the interval e° e' must be capable of being filled up by a continuous series of values, uniting these two extremes : isolated values of e', occurring within the range a;=0, a?=l, and which fall out of the continuous series, are unrecognized by the calculus ; and we must not look to its results to account for them. It is, no doubt, common for functions of x, during the continuous progress of that variable from one value to another, to change from positive to negative ; but the change is from a continuous series of positive values, to a continuous series of negative values ; and each series may be regarded as distinct, and the interval be accordingly cut up into partial intervals, as instanced at p. 519. But in neither series can a value be accompanied by what may be called its congeneric value, of opposite sign, as is plain ; for such a value would break the continuity. If the multiple values of e'' are required to be symbolized in its development, we must connect with that development the factor 1"^. In the remaining portion of the present work we shall have but very little occasion to employ Lagrange's correction of Taylor's development, which correction, as we have seen, is necessary only when that develop- ment is a diverging infinite series, and such series, being useless for the purposes either of practical calculation, or of theoretical research, will always be rejected (see Note, p. 83). Whenever the development is a converging infinite series, that series alone, taken, of course, in all its infinity of terms, is the complete de- velopment, no correction of it or supplementary addition to it being requisite. It is true that even in such series, Lagrange's correction will exhibit the limits of error committed in summing up only a specified number of terms, and rejecting all the rest, and so far it may be useful in numerically evaluating functions approximatively, in particular cases ; but the doing this will frequently involve the necessity of actually computing the invelopment itself, the labour of which it is usually the object of the development to avoid, even when that labour is practicable. For instance, in calculating sin x in terms of the arcual value of x (p. 529), if we wish to know the numerical limits of error committed in stopping at the third term of the series, we must actually know the value of sin x, which is supposed at first to be unknown. Nevertheless, it is of interest to learn that whatever the correct value of sin x may be, the sum of the three terms mentioned differs from that correct value by less than the - 2.3.4.5.6.7 part of the whole. And, in like manner, by taking n terms of the series for e', the sum of them will difi'er from the correct value by less than the —r part of the whole ; but to find the actual numerical value of 2.3. ..w this part, for a proposed value of x, would require that e' be independently computed. 558. Differentiation of Compound Functions. — Let du dx du __ i*=F(2/), where yz=f[x); and let it be required to determine — . Here we have Ay=f{x+Ax)-f{x), and Au=:F{y+Ay)-F{y). DIFFERENTIATION OF COMPOUND FUNCTIONS. By Taylor s theorem, ^y-dx ^^d^' 2 ^d^ 2.3 ^ Also, by the same theorem, ^ du^ ^d^'uiAyY^d^uiAy)^ 531 dy dy^ 2 ' dy^ 2.3 =^k''^+^-2" + ---|+2^4rf-^^+^a;-^'T^+-'-/ Ji=|f.^_,PA.+ .(A^^+...,...|=|.|...[l]. Ax (2y dx du Hence the dififerential coefi&cient — is found by differentiating w, on the hypothesis that y is the independent variable, and then multiplying tne coefficient thus obtained by that derived from i/, considered as a function of X, Whenever y varies uniformly with x, — must be constant, for then -^ is always the same : dx Ax hence, in this case, ^-^=0, .'. -7"=^" • r~---L2J' ' ' dx* ^ Ax dy Ax ^ -^ The following examples will illustrate the application of [1] : — (1) u=:a^, where y=J du -=ay log a, -f =5* log 6, ay ax du du dy i' ,,, , , .*. -r=— . ^=a . 6' log a. log 6. dx dy ax (2) w=log y, where y=log x, du__l dy_l dy~~y* dx~~x* du du dy 1 " dx dy' dx a; log a;' (3) M=cot a^f where y=log du ^{a^-\-x^} — =— cosec^oy.ayioga, —= . g , os ) dy da; x{a^^ar) du « , a^ ..._=-cosec»a..avloga.^-^^,^^. (4) u=8m(-^y. Put(~^y=:y, ^ ' \a-{-x/ \a+x/ du_dsva.y_ dy_ 2ax •*• dy~~df~^^^y' di~{a-i-xf du_ / ^ V ^^^ 659. Suppose now that u=F{y, «), where y and ;? are both functions of x, du and that it be required to determine dx' Then Au=F{(y+Ay), {z-^Az)}-F(i/, z), which difference of u, by adding and subtracting the same quantity, as below, may be split into the two differences, Au=iF{y, (2+A0)}-i?(y, z)]-\-[F{{y-[-Ay), {z-{-Az)}-F{y, {z+Az)}]. Now the first of these two differences is got by changing the « in w into z-{-Az, and leaving the y untouched, as if it were a constant: the second of the differences is got by taking the changed function, namely, F\y,{z-]-Az)], as an original function, and changing y into y + Ay, leaving the z untouched as if it were a constant, and diminishing the result, as M M 2 532 DIFFERENTIATION OF COMPOUND FUNCTIONS. before, by that original function. Hence, so far as the first partial difference is concerned, Ar^ AF(y, 2), when 2 alone takes an inc. A3 Az — = — :^ii_!± .^ , — . ^nd so far as the second is concerned, Aic A2 Ax Aw ^F{y, (z+ A2) } , when y alone takes an inc. Ay Aii/ Aaj Ay * Ax' Hence, in the limit, when Aa?, and consequently Ay, As?, each become zero, we have for the complete differential coeiB&cient -r-, ax du du dz du dp dx dz dx dy dx .[3], that is, we are to differentiate the compound function u with respect to each of the component functions y, z, as if the other were a constant, and are then to add the two results together, each result being of the form [1] given in last article. If z is simply the independent variable a:, the expression [3] becomes J du\_du dM dy Xdxi^dx^dydx '- -■' where the total differential coefficient is enclosed in braces to distinguish it from the partial diff. coef. on the right, which is the diff. coef. derived from u by treating y as if it were not a variable, but a constant. We shall give an application or two of these forms, y being always a function of OS, (1) u=cotxy. By [4], (du]_dududy \dx)dx dy dx' Now -r-= — cosec^ xv —r- dx dx =— cosec'^ajy. yas-v-i, — = — cosec^a;^ —r- = — cosec^xJ' . xv log x, dy dy ... !*}=_.. oosec=^(^+| log.) dy where the operation -— • must remain unex- ecuted till the function y is assigned. (2) w=log tan -, d tan - du y dx~, X , tan -an. y X 1 dvt._ dy' X d tan - tan-( y V V" X tan \dx) -, x\~ dx/ s'-^rfS ^^tan - y dy y-^T. 2 . ^ ic 1 a; y^ sin - cos -?/* sin 2 - y y ^ y (3) w=tan~* {scy), where 2/=^» du y du x dx l-\-{xyY ■l + (xi/)«* Also -;^=e', dx 1^1 — •'• \dx\~ y+x^ (H-a:)e* ■l+(x2/)2 l-^-x^t^" (4) u=^z'^-\-y^-\-zy^ where 2=sin x, and y =e*, du ^ , du , Iz^^'^^y^dy- ^+2. Also ---=cos X. and -— =e*, dx dx •.[3], |^|=(22+y)cosx+(3/+s)« = (2 sina;+e*) cos x+(3e^+sin j?)(f =sin 2x-^e' (sin x+cos a?) +3c^'. DIFFERENTIATION OF IMPLICIT FUNCTIONS. 533 Note. — It is sometimes easier to replace y and z by the functions of X which they represent, and then to differentiate u directly : thus, in the above example, M=sin2«-f-e^-}-6* sin a;, which differentiated, gives -—=2 sin X cos x-\-%^-\-e' cos a?4-sin a?e'=sin 1x-\-^ (sin a;+cos a;) + 36^*, dx the same, of course, as before. Examples for Exercise. (1) u—9?-^%axy-\-f. (2) uz=.x\ogy. (3) w=log{x-a+V(a^J-2aa:)}. (4) w=(cosa;)y, where 2/=sin a;. (5) M=^4-z, where yz=.sj{\.—i?\ and 2=sin~'a;. 560. Differentiation of Implicit Functions.—If the vari- ables x,y, are connected together in an equation i^(a;, ^)=0, then the function that ?/ is of a: is not explicitly given : it is only implied in this expression of the relation between the two variables. I3ut to obtain the differential coefficient ■— there is no necessity to solve the equation for y, dx and thus to determine what function of x, y really denotes ; the formula [4] suffices for this purpose : for putting „, , ^ {dxi\ du du dv „ dv du du we have only, therefore, to differentiate the several terms of the expres- sion, first with respect to x, as if y were constant, then with respect to i/, as if X were constant ; to divide the former result by the latter, and to prefix the minus sign. Or, without any special rule, it is enough to dif- ferentiate the several terms of,?^ in the usual way — remembering the form [1], since ^ is a function of a?— and then, by equating the result to 0, to determine --. ax (1) w=a;H3aan/ + 2/5=0, ^^ o o ■ o ^w « . « o cty x'^-\r -=Z7?^Zay,-=Zax+Zf, .'.■£=■ dx ^^ dy ^ ^ " dx ax-\-y*' Or, differentiating in the usual manner, ZaP+Zay+Zax ^+Zf -^-^=0, ux dio .•.(-+^|=-(.+«.)..-.|=-S^- If the second differential coefficient be required, we have ^= ;:S+;3J5 (^Or^bstitutogfor-^, 534 VANISHING FRACTIONS. _ (aa;+y2)(2aa;2+2a;y2-aa^-a^y)-(a-.^+ay)(tt^^+ay'-2xgy-2ay^ __ 2a^+ 6as^y'^+2x*y-2a^xy _ 2xy(y^-\-Saxy-\-x^-a^) „ „ n ^ cPy 2a^xy Buta^+3«j,+j3=0,.-.J=^J.3. (2) Given i/'— 3y+a;=0: to develope j/ in ascending powers of a. ey^ ^2^ da: ""dx"^ ' ••rfx~3(l -2/2)' cte2-9(l -2/2)2-32(1 -2/2)^' ^3, 3^(l-,2)3.2|+2y.33(l-,2).2,| 2(1+5,^| ^- F(r^?p 32(1-2/2)* ' ^^^ Now when a?=0, y=0, or y=^^; taking only the first value, we Hence, hy Maclaurin's theorem, we have, as far as the first three terms, the second being 0, ,=|+^+&e.4{.+ (|) +&C. } [1]. It has not been thought worth while to carry on the differentiation further: the example is given merely for the purpose of showing how Maclaurin's theorem may be applied to the development of an implicit function of an independent variable. The process here exemplified has been affirmed to be applicable to the solution of cubic and higher equa- tions, by means of infinite series ; but its efficiency in this way is so ex- ceedingly limited, that it is all but valueless for the purpose ; since, to be of any use, the resulting series must be convergent. In the present example, for instance, if any given number a be put for ^ in [1], a root of the equation i/"*— 3i/ + a=0 might seem to be given by the series, what- ever that number a may be ; but if it be such as to render the series divergent — which is more likely than otherwise — the expression is useless, on account of the unknown value concealed under the " &c." Examples for Exercise. t2 dx (1) 7/2+2a:2/+JB2=a2, to find dv dN (3) sr'-ary+o»=0, to find ^, g. («— a)2 " x—a dx Log — = , y a dy to find , . dx (6) y^—xy=lf to develope y in powers of Xj as far as a;'. 561. Vanishing Fractions. — Expressions which have this desig- nation have already been discussed in the Algebra (p. 144) ; and the VANISHING FRACTIONS. 535 student will have perceived by this time that the process at (190), for finding the value of any such fraction, is virtually a process of differentia- tion, though by aid of the simple theorem given at (184), common algebra sufficed to suggest and carry on the operations there necessary. Fix) The determination of the true value which -—-7 represents, when for a particular value a of x the undeveloped fraction assumes the form -, can always be effected whenever F{x-^h) eindj{x-\-h) are developable in a series, with finite coefficients, proceeding according to the positive powers of h, whether integral or fractional. This it is our present purpose to show, explaining first how the evaluation of the vanishing fraction is to be determined when the developments spoken of are given by Taylor's theorem, and then liow the value is to be found when that theorem does not apply, in consequence of the occurrence of a differential coefficient, which becomes infinite in the proposed hypothesis of x=a ; such occur- rence being a sure indication that, for the assigned value of x, the deve- lopment is not according to the positive integral jewel's of h (556). The following lemma must be premised : — Lemma, — In the general development F{x-\-h)=F{x)-{-F,{x)h+F,{x) ^+..., it is impossible that any particular value given to x can cause F{x\ and at the same time all the differential coefficients Fj{x), Fjx), &c., to vanish. [It is, of course, implied, from the absence of the ** &c." after the series, that h is always sufficiently small to render the series convergent for the proposed value of x.] For if such could be the case, then, for that particular value a of x, we should have F{a-\-h)=0 ; but by hypothesis F{a)=0 ; hence, whatever values put for a satisfy this equation, the same values put for a -i-h must satisfy F(a-{-h)=0. In other words, a=a-{-h, which is impossible, how- ever small h may be short of h=0. Fix) This truth being established, in the proposed fraction — — , which takes the form - when x=a, let x be changed into x-\-h, then, by Taylor's theorem, •-[1]. In this identity put a for x, then, since by hypothesis F(a)=0, and /(a)=0, Fia^k) ^^(«)+^^»^+2-3^3(«)^^+ ^^""^^^ Uo)+\Ma)h+^^f,{a)h?-^. .[2]. And this, when A=0, becomes —~=~=-i)-{. f{a) /,(a)* 536 VANISHING FRACTIONS. and, therefore, if FJa), f^{a), do not themselves both vanish, the value of the fraction, in the proposed hypothesis, is determined. But if Fi(«)=0, and f^{a)=0, then [2] becomes simply ^(„^,>_W3^W*+. . P(a)^PM ^(^+*' /.W+i/3(<.M+ '"^W ^'<''>' In like manner, if this equivalent of the proposed vanishing fraction should itself also be a vanishing fraction, we should have to equate the former to 7^; and so on, till, by successive differentiation oi F(a:),J[x)^ fir} we at length arrive at results which do not both vanish iox x-=a : — results which must inevitably be attainable in virtue of the preceding lemma. All this is in complete consistency with what is established at art. 190, where the process here described has been sufi&ciently exemplified in the case of a common algebraic fraction : we shall here, therefore, give a few examples of its application to transcendental functions only, that is, to functions involving exponential, logarithmic, or trigonometrical expres- sions. (1) Required the value of -, when a;=0. -log 6. 5^ .•.^=loga-log6=log-, the value required. (2) Required the value of , when a;=0. sm X fl{X) COS X *'/i(0) 1 ' the value required. (3) Required the value of 1— cos X x" , when xz=0. „ F[(x) sin a; , ^^)__COSW . •^2(?)_.l Mx)- 2 ' ''' U0)~2* the value required. (4) Required the value of 2zsinx—tr . tt , when x=-. COS a; 2 „ F, (x) 2 sin ar+ 2x cos x Here 7^"^= : , /,(») —sin a; =—2, the value required. (5) Required the value of (a;— a)" , ^ , when x=a. Here^="<"-->"". Hence, according as «'>1, <1, or=l, we have > / \="' or 00, ore . If, in seeking to determine the value of a vanishing fraction by the above process, we should arrive at a differential coefficient which, for the assigned value of the variable, becomes infinite, we shall be apprized that, for that particular value of the variable, the development of the function, according to the positive integral powers of a?, is impossible. But a method analogous to the foregoing is still to be applied ; thus : Let -r^- FRACTIONS OF THE FORM — . 537 00 become- when x=a'. substitute a + ^ for x, and let the terms of the fraction thus changed be developed either by involution, the extraction of roots, or any other algebraic process, so that we may have f{a-\-h)~A'ho.'-\-B'Jtfi'-\-...' the terms of each series being arranged so that a and of are the smallest exponents in the respective series, fi and /3' the next in magnitude, and so on : one or other of these three cases will present itself ; either 1. «>«', or 2. «=«', or 3. «<«'. In the first case, by dividing numerator and denominator by h/, and then putting h=0, the result wiU be ^=^=0, fifl) ^ In the second case, the result of the same operation will be -^=—,. In the third case, by dividing the num. and den. by ^% the result for A=0 will be -rf-r=— =«> . It appears, therefore, that the development of numerator and de- nominator of the fraction, after a + ^ is put for a?, need not be extended beyond the first significant term, or that containing the lowest exponent of h : and that, according as this leading exponent in the numerator is greater than, equal to, or less than, that in the denominator, will the true value of the fraction be 0, finite, or infinite. And the finding these leading exponents by common algebra will sometimes be an easier way of arriving at the value of the fraction than by the process of differentiation. The examples at p. 146 will illustrate this method, to which the two following are added. «v «. , iJx—>Ja-\-s/(x—a) . , , (1) Since when x=a, ., ., -■ -=-. put a-\-h for x, then F(a-\-h)_{a+h)^ -ak-{.hk_ hi + ... . i^(a)_ 1 /(a+^) A4(2a+A)4 (2aA)4+ ... /(«) (2a)i (2) Since when x=a, ^— — --^ — - =-, put a+h for a?, then F(a+h) _ 7?(2a+7if+h _ h+ ... ^ H^_l /{a+hy (1+A)'-1 ""3A+...' •'• /(a)~"3' 562. Fractions of the Form — .—When this is the form CO which the general fraction assumes for a particular value of the variable, it may easily be converted into the form - just discussed. For if 1 F{a) 00 . , F(a) no) -—:'=—, then also —4' =*—•=-... [1 J. /(a) 00 ' f{fl) 1 F{a) 538 FRACTIONS OF THE FORM 00 l^ovr d W)/ ^"^K^))=TfW. .: -— — -= ■' . , as in the other form. SO that the methods already explained equally apply, whichever of the two forms the fraction assumes : thus : — FJoo) ^ . .'. --Y — r =0, since, n being a finite /«(« ) number, the numerator is finite, and the denominator infinite. (1) Required the value of F(x) log sin a; , « iJ_''=— -2 , when x=.0. f{x) log sin 2x Here 77^;=—. And ^,(a;)_cos X sin 2x /j(ar) sin X cos 2a: 2 sin X cos^ a; 2 cos^ ar sin a; cos 2x cos 2x * .«, —^^=2, the value required. (2) Eequired the value of — when a:=oo , and n a finite positive integer. Fj{x)_nx'*-^ ^ Fi(oo ) _oo ^^(^_w(w— 1)^^ ^ . /^^(OP ) _00 i^>,(a?)_ n(n-l)(7t-2)...2.1 (3) Required the value of ^_^, » being a positive integer, a;2» when x=0. Put - for a:*, then the fraction is — V-^=— =— , which has the form f{z) 1 e-' when 2=00 , that is, when x=0. J'„(2)_ w(ro-l)...2.1 i?^(oo )_00 =0. [Note. — It has been remarked of the function e ^ that " although it vanishes when a?=0, yet a is not a factor of it," and in reference to this circumstance, a recent writer of great ability makes the following obser- vation : — " The statement made in most of the ordinary text-books on the Theory of Algebraical Expressions, that if F(a-)=0, when a?=0, a; is a factor of F(x), or (what is equivalent) that if F{x)=^0, when x=a, x—a is a factor of F(x), is only one of too many universal propositions which are unduly assumed." Those who make this statement, if any there really be,* need not go to transcendental functions to discover their error. Common algebra will supply functions, as many as we please, each of which, like * De Morgan, in alluding to the function here commented upon, has more accurately represented '* the statement made in the ordinary text-books." He says : — '* Until veiy lately, all analysts considered functions which vanish when x^a as necessarily divisible by some positive power of x — a" [not necessarily by some positive integral power of x—d\. "This is only one of a great many too general assumptions which are dis- appearing one by one from the science." It may be proper to add here that the author of the present work has not had an op- portunity of consulting Sir W. R. Hamilton's paper on this function in the Transactions of the Royal Irish Academy : the assertion above, that x is not a factor of it, must therefore be regarded as a quotation, at second hand, from that eminent analyst. FORMS X 00 , AND 00 - 00 . 539 that here noticed, vanishes when a;=0, and yet is not divisible by x : for instance, \^x vanishes for x=0, but is evidently not divisible by x. In the theory of equations it is affirmed, and proved, that if the polynomial which forms the first member vanish for x^=a, the expression is neces- sarily divisible by x—a\ but then it is always premised that the poly- nomial is rational and integral (see also art. 190).] 563. Forms x oo , and oo — oo . l. Let jP(a)=0, and J{a)=oo , then i^(a)x/(a)=2^=?=^|, where /'(a) stands for ^. 2. Let F(a)= 00 , and/(a)=oo , then ^ ^ J^ ^ \fia) F{a)J ' Fia)xf{a) 0' (1) Required tlie value of (1— «) tan — wlien a;=l. 2 Here the first factor becomes 0, and the second oo ; also tan —-=«—■, 2 9rx cot- .*. the expression is l-» cot- which is - when a;=l, — cosec^ -— 2 2 F,(l) 2 •*• ^/ )-. N =~ the value required. (2) Required the value of X tan X—- sec x, when a;=90°. A Here the two terms are both oo in the proposed hypothesis, _1 1 _ 1 1 f{x) F{x) "~4 «■ sec a; a; tan a;* F{x)xf{x) a^tan^x^a-sec* Dividing the first by the second, .1 . . xsinx—h* the expression is ^^— , and cos X differentiating num. and denom., which each become when aj=90°, we have —sin X .-. i;'(90°)-/(90°)=^=-l. Otherwise. Since tan x=. COBX* and sec x=- ir asinx— hit .'. X tan X—- sec xz=z 2_ 2 cos a; which is of the form - when a;=90° : and to this the form oo ■.— oo may often, in like manner, be reduced without recurring t the general formula. Examples for Exercise. ,^. oc^—ax^—a^x-^a^ (1) 5- -^ , when x=a. (2) — -L^- — '- , whenaj=l. (a;2_i)i_(a;_i) (3) -1^, when a;=0. ' cot a; (4) r, when x-=\. \-i^x-x^\. ,^. tan ic — sin x (6) : 5 — , when x=.Q. sin a? (6) X^ log oj, when x=zO. (7) sec a;— tan x, when a;=90°. X 0' 540 TO FIND -=p:, WHEN y IS AN IMPLICIT FUNCTION OF X. (8) flj^, when x=0. (9) — -—- — , when x=l. a—1 logx (10) ; , when rc=l. logx logx g' gsinar (11) : — , whena;=0. a;— sin x (12) {l-\-axy, whena;=oo, V 564. To find -=-, when y is an Implicit Function of X. — Let F{x, 2/)=0 : it is required to determine -=;p, when a;=0, y=0. X (XX This may he effected without differentiation, as in the following ex- amples : — (3) 2/*-96aV+100a2x2-x*=0. Divide by a^, then 2/2 ^|Y-96a2 (^|y+100a2=0...[l], (y\— /«^— 100a^_ /25_ •'• \a/""V 3,2_96a2 ~V 2i~" The last term of [1], — «*, was omitted because a:=0 ; and if we could have fore- (1) Given ay^-^-lst?=Si to find y dy - or -T-t when a;=0, and y=0. a; ax Divide by a? : then a f-f —x—h=0, ,'. when x=zO, and y=0, the brackets implying that x and y are each =0. (2) x*-\-af-2axy^-Bax^y=0. Divide by x% then which, since ic=:0, is the same as And solving this quadratic, we have /y\ 2+4 and since [1] was divided by ^-^ to get [2], .-. another value is (|)=0:henoe(|)=0,3,or-l. seen that no infinite value J ^^ above : the values of this will evidently be those of ^ in the proposed hypothesis a?=a, ?/=&.* Examples foe Exercise. (1) Required -^, when a;=0, 2^=0, in dy ired --^, when a=0, dy (2) Eequired — , when a=0, y=0, in (3) Eequired j- , when a;=0, y=0, ii dy lired j-, when x=.0 (4) Required j-, when a;=0, 2/=0, m 565. Maxima and Minima: Explicit Functions.— This subject, like that of vanishing fractions, has been partially discussed in the Algebra, p. 147 : but its complete theory requires the Differential Calculus. An explanation of the terms maximum and minimum has been given at the page referred to : it is equivalent to the following : — In any func- tion y=.F{x), let the independent variable x take a particular value, as a/=a, as also a preceding and succeeding value, x=a—h, and x=a-^h ; then the corresponding values of the function being Fia-h), F(a), F{a-\-h), if a be such a value that for any finite value of h, however small, and for all intermediate values between and this, the middle value F{a) of the three functions exceeds that on each side, the value x=a is said to render the proposed function a maximum; but if the middle value continue less than that on each side, between the same limits of h, the value a;=a is said to render the function a minimum. In other words, if a?, starting from any value, by continuous increase, causes F{x) to increase till x arrives at a certain value a, and then causes it to decrease, F{a) is a maximum value of the function. But if x, starting from any value, by continuous increase, causes F(x) to decrease, till x arrives at a certain value a, and then causes it to increase, F{a) is a minimum value of the function. In order to discover what the values of x are that render the function yz=zF(x) a maximum or a minimum, let x be changed into x:hh; then, by Taylor's theorem, h being so small as to render it convergent, * The above method, which supersedes the more operose process by differentiation, is due to Mr. Woolhouse (Differential Calculus in Weale's Series, p. 95). The rule in which we have embodied it may perhaps be acceptable to the student 542 MAXIMA AND MINIMA: EXPLICIT FUNCTIONS. Now, if ^=a make the proposed function a maximum, there must exist for h some finite value, such that for all intermediate values between and this, we must have F{a)>F{a±.h) ; and consequently we must have the condition when a is put for x in the coefficients. But if the value aj=a make the function a minimum, then the condition will be F{a)^h)—F{x) and F(x—h)—F{x) will both be negative; and in the de- velopment of these dififerences by Taylor's theorem, h may be taken so d'*y /t" small, that the first term that does not vanish — say the term -— •^ dx"" l,^...n* may have the sign which belongs to the entire series ; that is, the sign of the whole difference : but, as just seen, the sign of the whole difference the dy_ dx does not vanish must be of an even order (or belong to an even power of zt.h), and be negative. Applying similar reasoning to the point Pg, we conclude, in like manner, that at that point, where the ordinate F{x) is a IS negative. Hence, at maximum, we must have point Pj, where the ordinate F(x) is a :0, and the first differential coefficient which . dy minimum, we must have -,-=0, ax and the first diff. coef. which does not vanish, must be of an even order, and positive. It might, however, happen that the points P^ Pg, are on such curves as in the annexed diagram, the prolongation of the ordinate being a tangent to each of A}— A two miiting branches. In such a case -f- will not dx be 1 but 00 , and this illustrates the case last considered in the above analytical investigation. MAXIMA AND MINIMA : IMPLICIT FUNCTIONS. 545 568. Maxima and Minima: Implicit Functions.— Let now the function that y is of x — which function is to be rendered a maximum or minimum — be only implicitly given, that is, let „=^(x,,)=0, then (660), |=4:^g...[l]. Hence, when -r=0, we must have -—=0, and conversely ; so that the dx dx values corresponding to maxima and minima, as determinable from the dt/ condition t^=0 of the preceding article, are now to be determined from (tx the conditions u=0, — =0 [21. dx The values of x, and the accompanying values of y, being found from these equations, we must substitute them in succession for x, and y in d^y ~^, when those values of y will be maxima values, that render this co- efficient negative, and those will be minima that render it positive : and so on, as in last article. In order to derive the second diff. coef., put for brevity — =—-— for [1] above, then /dM dMdy\ ^^/dN dN dy\ \ dx dy dx/ \ dx dy dx/ dx^ N' and, therefore, dividing num. and den. by iV, for those particular values of X which satisfy the condition M dy ^ ^ , d^y d?u du _„^ -^=0, we must have -r4=— j-j-r-^r ...[3], N' "' dx d^ Note. — For examples the student may return to those at pp. 150 and 280-4 ; and in solving the additional examples given below, he must keep in remembrance the precepts 5, 6, at p. 280. (1) y=a+V(a3-2a2a;+ax2). Here the additive constant a may be sup- pressed, the remaining expression may be cubed, and then the constant factor a re- moved ; hence y is a max. or min., if a?—2ax-\-x^ be so, or suppressing the a?, if X'—2ax be so : putting, therefore, du u=^x^—2ax, we have — =2a;— 2o=0. dx Also j-^=2, a postiive quantity ; .: y is& minimum when x=zaf and the expression has no maximum. (2) y=b-\-'iy{x—a)^. Omitting 6 and the radical, and then taking the 5th root. we have u=x—a, .'. — =1, which being dx constant, the function has neither a maxi- mum nor a minimum. (3) y=x-, .-. ^=x-(l-f loga^)=0, The factor of can never become 0, .*. 1-f log 05=0, .'. loga;=— 1, /. a;=e-', and .*. dy /1\- T-2=( - y e,a positive quantity, N N 546 MAXIMA AND MINIMA*. IMPLICIT FUNCTIONS. .'. when x=.~, ic* is a mimimim, e but it has no maximum. (4) y=x-s/{o?-^'). .-. ^-^=H-(a'-a;2)-ia;=0, dx .-. a;+(a2-x2)*=0 [1], /. x^=a?-x^ [2], aN/2 For this value of x the second differential coefficient will be found to be plus, so that we might be inclined to infer that for this value of X the proposed function is a minimum, which, however, would be a mistake. In fact, the value of x which satisfies [2] does not satisfy [1] : the former equation is equivalent to {x+{a?-x')^ {x-{a?-x'')^}=0, and it is the second of these factors which becomes zero for x^=—^ , and not the first, 2 whether we take ^2 plus or minus : hence the proposed function does not admit of either a maximum or of a minimum value, for it is plain that, for the first member of [1] to become infinite, x must be oo (see the remarks at p. 285). But it is easy to show, from other con- siderations, that the proposed function can- not be either a maximum or a minimum : for let a=l ; then we may replace the fimction by sin ic'— cos a;', where from the above determination sin x'-=.-s/% ,'. x'=45°. Now, however small h may be, sin(45°+^)-cos(45°-fA) = -f, and sin(45°-/0-cos(45°-A)=- ; results inconsistent with either a maximum or a minimum. (5) Divide a given number a into so many equal parts, that .the product of them may be a maximum^ Let X be the number of equal parts ; then G)= a maximum, a cb .*. a; log -=max., /. log 1=0, X X Hence each of the equal parts must be e=2'l7l828..., and the number of them -, in order that their product, which will e a be (e)% may be a maximum. (6) u=y?—Zaxy->ry^=:0, du Tx" :3a;2- -Zay, p=Zf-Z •■• [2], x^-Zaxi/+y^=zO 3»2_3ay=0 •••2/ x^ .-. a^-2a^x^=0, .'. X =0, x=a\/2, . rsi d-y__^ dht, ^ du \n dx^ di? ' dy n o/^* \ 2a2 2 2 =-, or . a a Hence, when a:=0, y=0, a min. ,, x=al/2, y=a^4, a max. (7) Given the length of a man's foot, and the distance between his heels, to find the position of his feet when he stands the firmest. Let AB, CD, re- present his feet, and A, C, his heels : then he will stand the _ firmest when the area ABDC is the greatest (419). Draw AE perp. to BD, and put AC=a, AB=.h, BE=.x, .'. AE=^{lr—x^), and the area ABDC={a-\-x)>^{b'2-x^=m&x., ..x^{0 X) ^^j2_a,2) "' .-. &2-x2-aa;-x2=0, or 2x^-\-ax -11^=0, _ -a-fV(8&2-fa2) If a=0, that is, if the heels touch, x=:h.l^2, but iv^2=sin 45°, 2 ^ /. BE=AB sin 45% .-. the angles BAE, DCP, are each 45°, so that if the heels touch, the position will be the firmest when the feet are at right angles to each other. (8) To cut the greatest parabola from a given cone BA C. MAXIMA AND MINIMA : IMPLICIT FUNCTIONS. 547 Let AGO he the diameter of the base, perp. to DGF. Put AC=za,AB=h, CG= X, then A G=a—x, and since EG is parallel to BA (403), we have by the property of the circle, I)G=^{ax-x^, .'. DF=2^{ax-x^) ; and by sim. triangles, , ^„ bx a:b: :x : GE= — . a Hence (p. 512), the area of the parabola is -—>s/{ax-a?)=maji., .*. x^{ax—x^)=m&x., .'. 3aa?-iaf^—0 [1], 3 .•. x=0, a;=- a. The second value 4 substituted for x in the differential of [1] gives a negative result : hence, for this value of CG, the section is a maximum : for the other value, the section is merely a point : it corresponds to neither a max. nor a min., since, for a;=0, the second diff. coef . vanishes, but the third does not (566). (9) To inscribe the greatest rectangle in Let (x, y) denote one of the vertices of the inscribed rectangle : then 4a;y, or xp, and .•. x^y^ is to be a maximum ; that is, since y2=i {a^-x% x%a^-a?)=mB.x., ,'. 2a^x-ia^=0, /. 2a2-4a^^=0, , 2a2 1 ,„ .-. a^=-^, .'.x=-a^2, the area, or 4a^=:2a6=one half the cir- cumscribing rectangle, or of any circum- scribing parallelogram, whose sides are parallel to conjugate diameters (p. 319). Examples for Exercise. (1) The greatest rectangle that can be inscribed in a parabola is that whose height i« I the height of the parabola : required the proof. (2) The greatest parabola that can be inscribed in another parabola, the vertex of the former being at the middle of the base of the latter, is that whose height is § the height of the given parabola : required the proof. (3) Prove that the greatest parabola that can be inscribed in a given isosceles triangle is that whose altitude is | the alt. of the triangle. (4) Given the base b of an inclined plane, to find its altitude a, so that the horizontal velocity which a body would acquire by descending down it may be the greatest possible. (5) Determine a number x such that the ajth root of it may be the greatest possible. (6) Determine those conjugate diameters of an ellipse which form the greatest angle. (7) Given the volume of a cone : what must its base and altitude be, in order that its whole surface may be a minimum ? (8) What radius must a circle have, in order that an arc of it, of given length a, may belong to a segment of the greatest area ? (9) A conical ale-glass is 6 inches in depth, and 5 inches in diameter at top : what must be the diameter of a heavy sphere which, dropped into the vessel when full, shall expel the greatest quantity of the liquid ? (10) Required the angle of inclination of a pair of gates for a lock on a canal, so that the resistance to the fluid pressure may be the greatest possible. (11) Find for what angle, less than 180°, of arcual measure 6, the product fi sin ^ is a maximum.* (12) Given y^-{-2x^y-\-S2x—4t8z=zO : required the values of x for which ^^ is a maximum or a minimum. * This will lead to the equation tan a;+l=:0, an equation which may be solved by the method of Double Position, as explained at (181). N N 2 548 FUNCTIONS of two independent tamables. 569. Functions of Two Independent Variables.— Every function hitherto considered has been regarded as passing through its various changes, solely in consequence of the variation — the uniform variation — of but one of the quantities composing that function, all the others remaining constant : in other words, we have treated, as yet, only of functions of a single independent variable. We shall now briefly examine functions of two independent variables, or those whose changes take place in consequence of independent changes in two of the quantities which com- pose the function. As an example of a simple case of such a function, let V be the volume of a parallelepiped of length x^ breadth y, and there- 73 fore of depth — . The surface 8 of this solid will evidently be 8 being a function of the two independent variables x and y : a change in X, the length, does not affect y, the breadth ; nor does a change in y affect Xy though a change in either, or in both, affects 8', there is evidently, therefore, no necessity that these independent changes should take place simultaneously. Let u=F(x, y\ and let x become x-{-h, y remaining unchanged; then, by Taylor's theorem, observing that y undergoes no change, n^+k, ,)=»+^ ,+__+_ —+...[1]. But if y also change, becoming y + k, then, in [1] «ml] beooMe»+-i+-,-+-3j^3+..., du d^u dhi du du dy , , dy^ P , dy^ F •'• dx " diir dx "*" dx l72i dx 1273"^ *■■* ^^ «^ ^d?u . ^ „ d% /* dy "" )+...fi3, 2.3\c?x3 dx-dy dxdy^ dy^ where the partial differential coefficients in each term are the differentials of those in the immediately preceding term : thus, du du du=—- dx-\--r- dy- dx dy .[2], in which du on the left is the total differential made up of the differential of u with respect to a?, and of the differential of u with respect to y. In 550 FUNCTIONS OF TWO INDEPENDENT VARIABLES. like manner, differentiating again, remembering that -—, -;-, are also f unc- age a?/ tions of the independent variables x and y, we have , du ^ ■, ,du d?v, d?u dx dx^ dxdy •=" dy dydx dy^ ^ In like manner, these coefficients being functions of x and y, we shall find, by differentiating each separately, and then, as before, uniting the several differentials, that and so on ; the numerical coefficients agreeing with those in the corre sponding powers of the developed binomial. The development [1] may be regarded as the extension of Taylor's theorem to two independent variables, each of which receives increments ; and we may easily deduce from it a similar extension of Maclaurin's theorem : thus, supposing x and y each to be 0, [1] will be the develop- ment of F{h, k) ; and writing x, y, for the general symbols h, k, bracketing the coefficients as at (551), to imply that, when they are computed, x and y are to be made zero, we have 571. We may now extend the theory of Maxima and Minima to func- tions of two independent variables, the condition being that if the function u=JE'{x^ y) become a maximum or a minimum for particular values of X and y, then h and k must be finite increments, sufficiently small, that between and those small values, we must always have, for the aforesaid particular values of x and y, ^(«» y)'>F{x±h, y+Jc) for a max., and F(Xf y)<-P(x±A, y±i:) for a min., and consequently [1] last article, must be negative in the former case, and positive in the latter. But since h and k may be taken so small that the sign of [1] may be the sign of its first significant term, and that the term which now appears first changes its sign with changes of sign of h and A, this first term must be 0; that is, h and k being independent, we must have — =0, and -;-=0, dx dy for those values of x and y which make the proposed function a maximum or a minimum. And it will be the former, or the latter, according as, for the substituted values of x and y, the next term, or its double, cPtt-- ^ d^u ,, . d^u^„ dx^ dxdy dy^ + .. FUNCTIONS OF TWO INDEPENDENT VAEIABLES. 55) is negative or positive. For brevity, write this, after dividing by /c*, thus: -ay±^KD+^ ^^y we have then to ascertain the conditions which A, B, C, must satisfy, in order that this expression may be always of the same sign, whatever be the ratio - ; and it is plain that it will be of the same sign, whichever be the sign of the middle term, only when one, at least, of the quantities A, C, is of that sign. But (68) we know that it will be of the same sign, h whatever be -, only when AC—B^ is positive^ and that when such is not the case, the expression will be positive or negative, according to the value of T ; so that then there can be neither a maximum nor a minimum. k Hence, for there to be a maximum or a minimum, we must have AG—B- positive, and consequently A, C, must be both of the same sign. If this k sign be + , then [2] will be wholly + , whatever be - : if the sign common iv to A, C, be — , [2] will be wholly — . Hence, for a max. or min. to exist, we must have and these conditions being satisfied, the function u=F(x, y), is a maximum for such values of the variables as make-—., -r-:;, hoth. negative ; dx' dy^ and a minimum for such as make them both positive. The condition [3] having been first shown to be necessary by Lagrange, is usually called Lagrange's condition. If it fail, there cannot be either a maximum or a minimum. Ex. 1. Among all rectangular parallelepipeds to determine that which, having a given volume d\ shall have the least possible surface. Let a, y, z, represent three contiguous edges, then o' Mz=2a;y+2flM!+2yz=iiunimuin : but since xi/s=:a^, ,'. z= — , xy a* 2a* .♦. w=2xy+2 — I =minimum, y X du ^ 2a3 ^ du ^ 2c' -^=2y 2=0, — =2x . dx x^ dy 2/2 ••• -=22^-~2=0. X=2«^-^=^' •*• «^=2'=«' ••• «=«• d'u dru Now, for these values, -r^, and — -, are evidently both positive ; and dx' dy Lagrange's condition holds : hence the surface is a minimum when the figure is a cube. 552 CHANGE OF THE INDEPENDENT VARIABLE. (2) For what values of x and y is the expression x'y-^-xy^^axy a maximum or a minimum ? Here, calling the expression «, — =:2a;y+2/2— ay=0, and — ^a;2+2a;y— aa:=0 : ttese are satisfied by dx dy a;=0, 3^=0 ; x=ay y=0 ; x=0, y=a ; x=-, y=- ; the last of which is the only pair of values which satisfy Lagrange's condition : d^u dPu these make -m, and -—., positive, and therefore, for dot^ dy^ a ^. .... - /aV a5=y=:-, the expression is a minimum, namely — ( r I • (3) Prove that a rectangular cistern, open at top, in order to contain a given quantity of water under the least internal surface, must have a square base, and its depth equal to half a side of the base, that is, the cistern must be half a cube. du du Note. — If the values which satisfy — =0, — =0, be any of them such dx dy as to cause the next following partial differential coefficients to vanish, the character of such values can be ascertained only by an appeal to the coefficients still more advanced, as in the analogous case at (566): the general tests would, however, be very complicated. We may notice here that, although we have designated the differential coefficients derived from u as partial differential coefficients, yet there is no total differential coefficient furnished by a function of two or more in- dependent variables : there is a total differential, but as there are two or more independent increments, there can be no total coefficient to either.* 572. Change of the Independent Variable. — Every inde- pendent variable is always considered to vary uniformly ; but if we wish to introduce the hypothesis that this variation is not uniform, we may as- sume the variable to be itself dependent upon a new variable, which does change uniformly : this change of the independent variable may be effected as follows : — Let y=zF[x), and let w=f{t), the variation of a depending upon the uniform variation of t, then (558), dy_dy dx ^ dy_dy _ dx ^y _dhj dj? dy d^x dt~dx H' ' ' dx~di '^~di ' dl~d£^ d^ dxdF' Hence, putting for brevity, —— for — -i- — -, we have {ax) dt dt • In connection with the preceding article, the student may consult, with advantage, the work of Ramchundra, referred to at p. 280 (Chap. IV.). On the same subject, see also Jephson's **Fluxional Calculus," Gamier's "Calcul Differentiel," and Puissant's " Problemes de Geometrie." FAILUEE OF TAYLOR's THEOREM. 553 dy_{d^ dhf_ {d?yXdx)-{d?x){dy) dx~(dzy dx'~ {dxf ' ^ -•• If we make t=y, the hypothesis is equivalent to making y the inde- pendent, and X the dependent variable. In this case (rf2^)=|=l, {d?y)=% W=|, W=g, .-. [1], dy . dx ^y_ {d?x)_ dPx ^/dx^ Tx~ ^dy dx'~ {dxf dy'^'^Kdy) ^^' In certain investigations it is sometimes found necessary thus to alter the hypothesis respecting the variable originally regarded as independent : in such a case the above substitutions may be made at any stage of the inquiry. 573. Failure of Taylor's Theorem.— This, although the authorized phrase, is a very improper designation of the cases we are now about to examine : — the cases, namely, to which Taylor's development is not applicable, simply because they do not conform to the conditions for which that theorem stipulates : — it is not the theorem that fails, but the function; which, in the cases mentioned, is deficient in the requisites demanded by the theorem. The theorem fails in the same sense that the 47th of the first book of Euclid fails, when the triangle submitted to it has no right angle. The indispensable condition that Taylor's theorem shall give the de- velopment of F{x-\-h), whatever particular value be substituted for x in that development, is that such particular value never causes any of the coefficients to become infinite : it is only for such values of x as violate this condition that the theorem is said to fail, that is to say, it fails for such values of x as are roots either of the equation F{x)=:co , or •——=0, Ji{x} or else roots of the equation =-—=0, where FJx) stands for any one of the differential coefficients. Take, for example, the simple case F(x)—{x—a)~"': then by Taylor's theorem, the first term of the develop- ment of F{x-\-h)={x—a-\-h)-"\ namely (x—a)-"", would be oo for x—a : the inference is that h~"' cannot be developed in the form that theorem assigns. Again : suppose we have a function of the form p F{x)=X-\-{x—a) 9 f(x)y in which neither X nor f{x) involve the factor x—a, m being a positive whole number, and - a proper fraction : then, as soon as by successive differentiation we have exhausted the integer m, the dif- ferential coefficients which follow will each have a fractional power of x—a entering its denominator as a factor ; and .*. for x=a, all the co- efficients, after the coefficient F„lx), will be infinite, so that, from that term, the development fails. Up to the point of failure, however, the de- velopment, as furnished by Taylor's theorem, is always true ; for we know, regardless of these so-called failing cases, that 554 FAILUEE OF TAYLOR's THEOREM. ^2 Am F{x-Vli):=F{xnF,{x)h^F,{x)-^+...^Fn{x^6Ti) ^-^^ is always true (556), provided the failure do not occur previously to the introduction of Lagrange's correction. 574. The usual way to obtain the true development of the function, for those particular values of the variable for which Taylor's theorem fails, is to have recourse to the ordinary processes of common algebra. For example, suppose the function were F(x)-=.'ilax—x'-\-as/{x'~d% and that we required the development of F{x-^h) for x—a. Differentiating, we have dy ., , , ax , . , - -=2(a-«r) + -^;^-^— ^, which, for x=a, is oo . As the theorem thus fails to be applicable after the first term, the true development, when the terms are arranged according to the ascending powers of h, must have a fractional power of h in the second term. De- veloping (^a-\-}if by the binomial theorem, we have =a2-|-a ) (2a)V 2(2a)5 2.4(2a)i A5 /i2 /J + ... 2(2a)2 « 2.4(2a)^ Again : let the function be F{x)— ^x-\-{x—ay log (ic— a), and let it be required to develope F{a-\-h). Since here F(a)= Va— x oo , it becomes necessary to ascertain whether the expression which assumes this form is really finite or infinite : applying then the method explained at (563), we find that when x=a, the true value of {x-a)nog{x-a), or of ^^g^^^^ ig q. Hence the first term of the development is is/ a. Differentiating then F{x)y we have lir^IJx^^^^'^^ ^°^ ^^-^)-^^-<^^ w^ch, for a=a, '^^-j^- Consequently the true development of F{x-\-h) agrees with Taylor's form, as far, at least, as the first two terms. Differentiating again, we have d?F(x) 1 . ^ =—7 — -. — |-21og(iK— a)+3, which, for a;=a, is — «. Hence the true development departs from Taylor's form at the third term : the first three terms are as follows : — and we know that log 7t, which enters the third term, does not admit of development, according to the positive mtegral powers of h (p. 524). ASYMPTOTES TO CURVES. 555 575. Failure of Maclaurin's Theorem. — The failing cases of Maclaurin's theorem are not at all analogous to the failing cases of Taylor's. Tlie latter theorem becomes inapplicable in any case only for a particular value of the variable : the former, when inapplicable at all, is inapplicable generally for the function, whatever value be put for the variable : for in Maclaurin's theorem, the coefficients, for the same func- tion, are fixed in value, and are unaffected, therefore, by changes of value in the variable : these coefficients are determined by putting for the variable in the general values of them. If this substitution of for x cause F{a)}, or any of the coefficients derived from it, to become infinite, the theorem is inapplicable, no matter what finite value be put for x. It is inapplicable, moreover, when this substitution causes F{x) and all its derived functions, to vanish, as in F{x)-=e~^, 57(5. Asymptotes to Curves.— A rectilinear asymptote to a curve is a straight line which the curve approaches continually nearer and nearer to contact with, but which it actually touches only at an infinite distance. It may thus be regarded as the extreme or ultimate position of the tangent ; so that the determination of the asymptote, whenever such line exists, is the determination of the tangent, on the hypothesis that the co-ordinates of the point of contact are one or both infinite, while at the same time the intercepts of the axes, between the origin and the line thus determined, are one or both finite : if this latter condition have not place, it is plain that the entire line itself, as well as the point of contact, must be infinitely distant, and therefore no part of it can have existence within finite bounds. The equation of the tangent at any point (a/, i/) of a plane curve is (5J28) and, therefore, by putting in this 2/=0, and x=0, successively, we have for the intercepts of the axes of x, and y, between the origin and the tangent, the expressions ■(^-^'> ^ndy'--x'^ [1]. If for a/ =00 , both these are finite, they will determine two points, one on each axis, through which an asymptote passes. If for a/=ao , the first expression is finite, and the second infinite, the first will determine a point on the axis of x, and the second will show that a parallel to the axis of y, through this point, will be an asymptote to the curve. But if, on the contrary, the second expression is finite, and the first infinite, the asymptote will pass through the point in the axis of y determined by the finite value, and will be parallel to the axis of x. In like manner, if for 2/'=co , one or both of the expressions [1] are finite, the curve will have an asymptote determinable in a similar way. Should both expressions be infinite, the curve cannot have an asymptote : — no line can be parallel to both axes. When the expressions [1] are both for a;'=oo , the asymptote will pass through the origin : it is, of course, unnecessary to say that if one of the intercepts [1] be 0, the other must be 0. 556 ASYMPTOTES TO CURVES. Ex. 1. Let the curve be the common parabola, its equation being Here — =( — V, .«. y—x --^=2(ma;)^— (mx)*=(mx) , whicli is oo for x=.co. dx \x/ dx Also since -r^ is for a;=oo , the first of [1] is also oo : dx hence the parabola cannot have an asymptote. (2) Let the curve be the hyperbola, its equation being y=- {x^^—o?)^. dy h , ^ ,,-i / dy\ a;2-a2 a* Here ~-=- xix^-a") ^, .'. x-( y^-j- ) =x =-, dx a ^ \^ dx/ X SB which is when ic=oo : hence the hyperbola has an asymptote, and it passes through the origin. To find its equation, which must be of the form dy' dy , 2/=- - ic, we have only to put a;=oo in the foregoing expression for — , that is, ^-x{^-a^)~'^J~ in - x(a^—a^) *=-——-——, and we find the equation to be b 2^=+- ; so that there are tvx) asymptotes. 577. In many cases asymptotes to curves may be more readily deter- mined, without the aid of the differential calculus, from the consideration that to the same abscissa the difference between the ordinate to the curve, and the ordinate to the asymptote, becomes less and less as a; increases, and vanishes when cc=cd . Or else the difference of the abscissas to the same ordinate becomes less and less as y increases, and vanishes when y=cD . If neither of these results have place, there is no asymptote ; if both have place, there are at least two asymptotes: an example will illustrate this method. Ix Let the equation of the curve be xy—ay—lx=0 : then y= . X — (t Performing the division, y=6H 1 — --^ X x^ Now as X increases, the difference between this and h becomes less and less, and vanishes altogether when a=(X): hence the curve has an asymptote y=b. Again : from the equation of the curve, x= — ^=a^ [-—:+..., y — l, y yi therefore the curve has another asymptote a?=a. We thus see that the object is to develope one of the variables accord- ing to the descending powers of the other, and then to reject all the terms involving the variable in a denominator ; those retained, equated with the other variable, if of only the first degree, will represent an asymptote : if they are of a higher degree, there will be no rectilinear asymptote. ASYMPTOTES TO CURVES. 557 When, as in the example above, the equation to the asymptote is simply y=A, or x=B, showing that that asymptote is parallel to an axis, then there may be an asymptote parallel to the other axis, as above, so that it will be necessary to ascertain whether or i^ot such be the case. The de- velopment spoken of is to be effected either by common division, or ex- traction, the binomial theorem, Maclaurin's theorem, after putting - for ar, or for y, or else by indeterminate coefficients, as in the example following. (2) Let the equation of the curve be y^-{-a!^^axy=0. Assume y=Ax-\-B -\-Cx-^-^... J .'. f=A^ci^-\-dAWx^+..., and axy=:Aasi^+Bax-{-..., .-. y'^+st?-axy={A^-^l)x'-{-{SAW-Aa)c(P-^...=0, .-. ^3-fl=0, ZA^Ji-Aa=zO, ... ^=-1, B=-l. a . Hence the line y=—x—- la an asymptote. o Applying this method to the two examples previously given, we have, in the case of the parabola, y^=imx, ora;=— 2/H0?/+0H f- , 4m y 80 that the curve has no asymptote. In the case of the hyperbola, y=±- {aP—a^)=z±,( - x—- — !-••• )» •'• y=±- « denote the two asymptotes, ft \fl Ax/ ft Note. — Whenever asymptotes exist which are parallel to either of the co-ordinate axes, as in the case of the first of the preceding examples, they may be readily ascertained to exist by noticing that for some finite value of one of the variables the other will become infinite : thus, taking the example referred to, namely, {x—ob)y—hx-=.^, we see that for ^=a, we have x?/— a6=0, which can be possible only when 2/=ao : the equa- tion x-=a is therefore that; of an asymptote. The proposed equation is also {y—b)x—ay=0, which for y=h, is Oxx—ab—0, and this requires that a:=co : the equation y=b is therefore that of an asymptote. Examples for Exercise. (1) Prove that the curve y^=ax^—oc^ has the asymptote y=—x+-z. o (2) Prove that the axis of x is an asymptote to the curve y{a^-\-x^)—a\a—x)=.0. (3) Show that the axis of x is an asymptote to the logarithmic curve y=a''. (4) Prove that the Cissoid of Diodes, namely, the curve y'^=--^ , has no asymp- AO/ — X tote. (5) Prove that the curve y*(a;— 2a) — a;'-|-a^=0 has the asymptotes fl;=2a, y= ±(a5+a). (6) Prove that x=a, and y='X-\ — — , are asymptotes to the curve 3 558 SPIRAL CDBVES. 578. Spiral Curves. — If, while a straight line revolves about a fixed point S, a point P move along it, this point will trace out a curve called a spiral. From the mode of its generation, it is pretty- clear that a spiral curve is analytically repre- sented, in the most convenient manner, by means of polar co-ordinates : the object of the present article is to show, by employing polar co-ordinates, how the subtangents, asymptotes, &c., of spirals are to be determined. Let the curve be any whatever ; and let the point S be taken for pole, SX being the fixed axis. Put SP=r, and the angle PSX=Q : then if PR be a tangent at any point P, and PN the normal, and if RSN be perp. to the radius vector SP, the part SB will be the polar subtangent, and the part SN the polar subnormal. If the curve were originally referred to rectangular axes, SX, and a perpendicular to SX, we should have a;=rcos^, y=rsin^, and a^-\-y-=r^ ; and the equation of the curve in polar co-ordinates would be of the form r=F{ff), or F(r, e)=:0. Now SR=SP tan SPR=r tan {fi—a)=r tan 0— tan l+tan^tan a.' But tan 6=-, and tan a=-r-» X dx the axes being rectangular : hence the subtangent SR is All But the differential coefficient -j , which implies that x is the principal variable, must be replaced by the following expression when the variable is changed for 0, namely (572), dy dj dx . gj^/dxdy..dxd_l\ dx de 'da'" ^^-"^Vde ''d^J^y' d^^dif)' . , . dx dr And since — =-— cos ^- dd d0 dy dt ■r sin 6, and —z=. ■ ^ sin 6-\-r cos^, di d6 we have by substitution, dr SP^ dr Subtangent T,=^i2=r2^- .-. Subnormal iV,=;SiV=-—=—. d^ HR d6 Also for the angle P, between the tangent and the radius vector, we have , ^ SR , dr tan P= — =r-i- — ,. r d6 When for r=co the subtangent is finite, then, and only then, the curve has an asymptote, as is evident. (1) The logarithmic spiral. — If the point P so move along the revolving line SP that the arcual measure of the angle between the radius vector SPIRAL CURVES. 559 SP and the fixed axis SX, has a constant ratio to the logarithm of the radius vector, the path of P is the logarithmic spiral : lo"" V its equation, therefore, is— |-=a, or, wliich is the same, r=a<^. This justifies the name given to the curve ; for if a be the base of any system of logs, the log of the radius vector r will be the number ex- pressing the arcual measure of the exponent 0. Differentiating this equation, we have d^ d6 log a log a If a=e, the base of the Napierian system of logs, the subtangent at any point will always be equal to the radius vector of that point. dr 1 For the angle P we have tan P=:r-i-— •=, , d6 log a which being constant, it follows that this spiral cuts all its radii vectores at the same angle : it is in consequence of this peculiarity that it is fre- quently called the equiangular spiral: — if a=e, the constant angle is 45°. When r=oo , T^ is also oo , hence this spiral has no rectilinear asymp- tote. (2) The spiral of Archimedes. — The equation of this spiral is r=aQ, SO that here the subtangent is equal to the length of the circular arc to radius r, which is described while the revolving line generates the angle 6 ; hence when 0=27r, the subtangent is equal to the whole circumference, so that if this spiral could be accurately constructed, we might draw a straight line accurately, equal to the circumference of a circle. Since T^ is infinite when r is, there is no asymptote to this curve. The equation r=a9'* comprehends an endless variety of spirals. If n= — 1, it is the hyperbolic, or reciprocal spiral. If n=|^, it is the parabolic spiral: if n=— ^, the spiral is called the lituus. (3) The hyperbolic spiral. — The equation being rQ=a, we have dr a dr a? a — _^ ^ length. Since r=.-^ when ^=0, r=QO : hence this spiral has a rectilinear asymptote parallel to the fixed axis, and at the distance —a from the pole. The radius vector, in its initial or first position, is therefore infinite, the generating angle being positive. (4) In a similar manner it may be shown that in the parabolic spiral the subtangent is ^1=2 — : and that in the lituus it is T,=z1 — . In this spiral the fixed axis is an asymptote. [Note. — For negative angles there is always another spiral.] 660 CTRCULAH ASYMPTOTES. Although the formulaB in the present article have been applied exclu- sively to spirals, yet they are equally applicable to every curve related to polar co-ordinates : thus, take the hyperbola, the polar equation of which, when the centre is the pole, is — ^ — , .'. when 6^*0082^=1, »'=oo 1 — (^ ccis^O 1— e^cos"^ hence the asymptotes are so situated as that cos2^=-=-— -— , .-. — — =taii2 0+l= — -, h h .'. tan 6=z +-, .*. the asymptotes are 2/=+- x. a a, A curve which is referred to polar co-ordinates is usually called a polar curve. 579. Circular Asymptotes. — Hitherto we have considered only rectilinear asymptotes, but if, in any spiral curve, r should continue finite, though increase without limit, it will follow that the spiral must make an infinite number of convolutions about its pole before it can reach the circumference of a circle, described from that point as centre, and of which the radius is the finite value of r corresponding to 6 = 00 . In such a case, therefore, the spiral has a circular asymptote. Moreover, if the value of r for 6=00 be greater than the value of r for every other value of 0, the spiral will be wholly included within its circular asymptote ; but otherwise it will be wholly without it. For example : — In the spiral whose equation is (r^—ar)6'^:=l, or 6=.——— -, 0=oo , when r=.a ; v(H — ar) and for a less value of r, S is imaginary, .'. the spiral can never approach so near to the pole as r=a during any finite number of revolutions : hence the asymptotic circle, whose radius is a, is within the spiral. If, however, the equation had been 0= »J{ar—r^) then, although as before, 6=00 when r=a, yet, since for all other real values of 0, r is less than a, the spiral must be entirely enclosed within the asymptotic circle of radius a. Perpendicular upon the tangent from the pole. — An expression for this perpendicular {SY) is often necessary, especially in astronomical in- quiries. CaU it 2> : then p^=r^ sin^ SPR=r'' -. *^''' ^^^ l+tan^^'Pii' H^^4^-— ) w But (578), tan SPE=r^% /. cof' SPR=\(^\ . dff r^\d6/ 1 du Idr ^ __ 1 „ /du\^ Put w=-, then — = — r— : hence [1], — =?t2+( — ) . For other expressions see (583). SECTORIAL AREA : POLAR CO-ORDINATES. 561 580. Rectification of Polar Curves.— It was shown at (544) that s being the arc of any curve related to rectangular co-ordinates, {tJ =^+ W • ^°" ^'^'^' Te=d-.d&^ ^"' ^'''^' ■d.-dTd^^ .•.C-y=('^y+('^y. Butsincea;=rcos^, aiidy=rsin^, — =cos 6—-r sin 0, and ^=sin <; — +r cos tf, This integral, taken between the limits 8=6, and ^=^^, will give the length of curve included between the two corresponding values of r, that is, between Sr^ and Sr^. 581. Sectorial Area: Polar Co-ordinates.— Let SAB be a sectorial area, and BSB' an increment of it = A Area. Then, if from centre S, and with radii SB, SB' arcs of circles Bb, B'b' be described, terminating in the radii vectores to B, B' the in- crement A Area, if taken sufficiently small, will always lie between the circular sectors, that is, between - r^A^, and - (r-f Ar)2A0 : hence 2 2 T— lies between - r^, and - (r-f A?')'*, .*. in the limit, when A0, 1 rfArea 1 „ and consequently A Area, becomes 0, we have — — — =- r^, ,'.Axea.=:-fr^dQ...\l\ taken between assigned limits of 6. Otheru^ise. Conceive the proposed sectorial area to be cut up into com- ponent sectors, by drawing radii vectores so that the angles (A6) between every two consecutive pair may be equal : if the extremities of these radii be joined, we shall have a sectorial polygon, which will approach nearer and nearer to the curvilinear area as the angles spoken of diminish. The area of any one of the component plane triangles will be generally ex- pressed by - r?"* sin Ad : but when A6 diminishes down to d^, and therefore r and r' become equal, the sum of the infinite number of these zero triangles, that is, the sectorial area, is - fr^dfi...[2], the difTerential of it being - r^d^.* This same differential, that is, the zero-triangle, is also - pcfe, 'p being the perp, {8Y) on the tangent at the extremity of s. (See fig. p. 558). * The following is good advice : — "We would recommend to the student in pursuing any problem in the Integral Calculus, never for one moment to lose sight of the manner in which he would do it, if a rough solution, for practical purposes only, were required." — De Morgam, {Elementa/ry IllvMrations, &c.). O O 562 SECTORIAL area: rectangular co-ordinates. 582. Sectorial Area : Rectangular Co-ordinates.— It is sometimes necessary to have an expression for a sectorial area when the co-ordinates are rectangular, and there is a very easy way of arriving at such an expression. In the above diagram SA, SC, being rectangular axes, {x, y) being any point B in the curve, we have for the area SABG the expression J xdy, making y the independent variable (546). Consequently, tlie sectorial area SAB is fxdy—- xy, because - xy is the triangle SBC. By diflferentiating tHs, we have d sectorial Area= » — • ^ence (last art.), , ^ ds xdy—ydx 23_?-— — _ — . From the first of these we have ds T^ /^T —=— , .*. s=. I — cZ^...[l], which is another form for the length of a polar curve. We shall now apply the above formulae to an example or two. (1) Required the length of a circular arc subtending the angle 6 at the centre. Taking the expression [1] of the present article, observing that in the circle j3=r, we have Cq rd6=.r6. Let ^= ) ^'' Subtangent SR=r tan Pz^—-——— =rr2— . d/r , _,^ r r' ds Normal PN=^—-=~=~. Bin P p dff Subnormal SN=-^=.^-s/{i^-p^ tanP p dr "di' oir . n 5^ rxdy-ydx-1'- PY=v cos P= v^(r2-^2)_y 584. Contact of Curves : Osculation.— Let y=f{x), Y=F{x) be the equations of two plane curves, in the former of which the constants a, b, c, are known, and therefore the curve determinate. In the latter, however, the constants A, B, C, &c., will be regarded as unknown or arbitrary, and therefore open to any conditions we may choose to subject them to. The first is therefore an individual curve completely given : the second is any one whatever of a particular class, species, or family of curves. The arbitrary constants which enter into the equation of a curve are usually called the parameters. 00 2 564 CONTACT OF CURVES : OSCULATION. Let a; take the increment h, and let the corresponding ordinates y, Y' be developed by Taylor's theorem ; then ^-^^dx ^+rfar 1.2+ c/ar* 1.2.3+ ^2]. The parameters which enter [2] being arbitrary, they may be de- termined in terms of x (whatever particular value we may choose to give to this variable), and of the known constants a, 6, c, &c., so as to satisfy as many of the conditions ^ dy dY dhi d'^Y ^ , ^ as there are arbitrary constants, or parameters in Y=F{x). The values of A, B, C, &c., determined in this way, when substituted in [2], will cause so many of the leading terms in both series to become identical, whatever particular value we put for x. Now the greater the number of leading terms in the two developments which are thus rendered identical, the nearer will the developments themselves approach to identity, provided h may be taken as small as we please : — a proviso, indeed, always to be understood in reasoning from Taylor's theorem. When the first of the conditions [3] exist, the curves have a common point, inasmuch as, for the same abscissa, they have a common ordinate : if this be the 07ily condition, we infer, therefore, that the curves in- tersect. When the second condition also exists, we infer (528) that the curves have a common tangent at the point : if these two be the only conditions, the curves have simple contact, or contact of the first order. When the third condition also exists, the approach of the two curves in the immediate vicinity of this contact will be closer, that is, the dif- ference between y' and Y will be less ; and they will touch with contact of the second order, as it is called. And so on. Hence of all curves Y=F{x) of a given species, that will touch any fixed curve y=f(x), at a proposed point, with the closest contact, whose parameters are all deter- mined agreeably to the conditions [3]. No other curve implied in the equation Y=F{x) — whatever values we give to the constants — can ap- proach so nearly to coincidence with the given curve y=f{x), in the immediate vicinity of the point of contact, as this ; so that no other curve of the same species can pass between it and the fixed curve. A touching curve thus determined is said to be the osculating curve, of that species, to the fixed curve, at the proposed point : its contact is of the ?uh order, if it have n-f 1 parameters, the values of which have all been fixed by their being made to satisfy the n + l conditions [3]. If the values thus determined happen to be such that they spontaneously, as it were, satisfy also the n-f 1th of the conditions [3], the contact at the point is, of course, still more intimate. If the contact of two curves be of an odd order, they do not intersect at the point of contact : — if of an even order, they do intersect. For in the former case the difference y'—Y\ of the series [1], [2], has an even power of h in its leading term ; and in the latter case, an odd power of h ; so that this difference y'—T, for sufficiently small values of A, has th© RADIUS OF curvature: eectangular co-ordinates. 565 same sign whether /i be + or — , in the one case, and opposite signs in the other case. This shows that in the immediate vicinity of the equal and coincident ordinates y, Y, the ordinate y' is greater or less than tiie ordinate Y', on both sides of the ordinate common to both curves, when- ever the contact is of an odd order ; but that whenever it is of an even order, y' is greater than Y' on one side the common point, and less on the other. In the diagram at p. 329, the linear tangent at P, which has contact with the ellipse, of an odd order (the first order) is wholly outside the curve in the vicinity of P, but the touching circle, in the vicinity of P, is inside the curve to the left, and outside to the right : its contact with the ellipse being of the second order, since there are three parameters, or arbitrary constants, in the general equation of the circle. For reasons stated at p. 327, the osculating circle is that which, of all curves, is preferred to test the curvature of another curve at any proposed point of the latter, and we shall, therefore, proceed to show how the radius of the osculating circle — the radius of curvature — is to be de- termined, be the given curve whatever plane curve it may. It is as well, however, just to remind the student, that if the curve be such that, for the point chosen in it, the coefficients [3] any of them be- come infinite, thus implying that for the assigned value of x, Taylor's theorem is no longer applicable, the preceding theory fails for that point. Should the failure commence at the second, or at a more advanced co- efficient, the curve will have such a peculiar form at the point, as not to admit of orders of contact, as we shall hereafter see : but if it be the Jirst dififerential coefficient -—=——, that becomes infinite, we know, indepen- duc dx dently of Taylor's theorem, that simple contact, at least, will be implied, since the two curves will have a common linear tangent at the point, this tangent being parallel to the axis of y. 585. Radius of Curvature: Rectangular Co-ordinates. — Let y=fix) be the equation of any given plane curve : it is required to affix such values to the arbitrary constants or parameters a, /9, p, in the general equation {x—uf-{-{Y'-^f=p\ of a circle, that the particular circle thus fixed may osculate the proposed curve at an assigned point (x, y). By [3] the parameters are to be determined from the three con- ditions ^ dx~~ dx^ dx~~ dst?' Differentiating (a;-a)2-}-(r-/5)2=p2, we have (a;-a)-|-^(r-/3)=0, ^ dx Hence, introducing the above conditions, putting y for Y, &c., we have, for determining «, ^, p, so soon as the point {x, y) on the curve y=f{x) is fixed, the three equations (a-«P-|-(y-A)2=p2 [1], X—a ^d^T 1 (a:-a)2 r^ r-/5" ■ dx^- r-/3 (r-/3)3- (r-/3)3- 566 BADius OF curvature: rectangular co-ordinates. (x-a)+£{y-(i)=0, or (a?-«)+y(y-^)=0 [2], g(y_^)3=_p2,or2>"(y-^)'=-p2 [3]. From [2], (x-aY=p'^y-fi)^ .'. [1], (p'Hl)(y-^)'=p2, .'. [3], y-^=- p'^+l which, substituted in [2], gives a;— a= . And from these we get for the m'(»'2+1) p'^+1 co-ordinates of the centre, a=x ;; , (i=y-\ ;;— ...[4], and for the radius, P=(4±2)!=-{(|y-e-i.»}=-(^)%y(p. 498)...[5]. SO that the point {cc, y) in the proposed curve being assigned, we have only to put its given co-ordinates in the expressions [4] and [5], in order to find the centre and radius of the circle which will touch the curve at that point, and be in closer contact with it there, than any other circle what- ever : the symbols p\ y" stand for the first and second differential co- efficients derived from the equation of the given curve. From [2], which has place even when the contact is but of the first order, we learn that the centre (a, ^) of every circle touching a curve is always on the normal at the point of contact : for [2] is the same as /S— 2/=— —- (a— a;), which shows («, /S) to be a point on the normal at (x, y), (p. 498). ay dx If this normal be taken for the axis of y, the equation of the circle will be a!2+(y-A)^=/»2, the point of contact (0, y), and ^ orp'=0: the expression for the radius of curvature is then p= -, 586. The sign of p" will always make known the direction of curvar ture : for if a linear tangent be drawn at the proposed point (a?, y\ we shall have for the ordinate of that tangent corresponding to the abscissa x-^h, the expression y-\-pli, and for the ordinate of the curve to the same abscissa, the expression y+p'Ti-\\p"h?-\- , and it is obvious that if p" be plus, this latter ordinate, for a small value of h, must exceed the former ; and thus the tangent will proceed from the point in both directions, between the curve and the axis of x, that is, the curve, at that point, will be convex to the axis of x. If, on the contrary, p" is minus, then the ordinate of the curve will be less than that of the tangent, so that the curve will proceed in both directions from the point between the tangent and the axis of x, that is, it will be concave to the axis. If y be at the point, p will be oo , whether p'=.0 or not, so that the osculating circle degenerates into a straight line. As this straight line has contact of the second order, the parts of the curve in the immediate EADIUS OF CURVATURE : RECTANGULAR CO-ORDINATES. 567 vicinity of the point, in opposite directions from the point, will be on con- trary sides of the tangent, as in the annexed diagram, seeing that there must then be intersection as well as contact (p. 564), provided, that is, that f'" is not also 0: but if /''=0, and the next following coefficient finite, the contact will be unaccompanied by intersection : — the curve will be convex at the point, if the first finite differential coefficient, being of an even order, be +, and it will be concave if the sign be — . A point P, at which the tangent intersects the curve, or at which the curve changes from convex to concave, is called a point of contrary flexure, or simply a point of in- flexion. At this point, the curve being neither con- cave nor convex, the curvature is : — the reciprocal of the radius of curvature, which, as we have seen, is 00 ; and generally, the degree of curvature of a curve at any point is always measured by ] -j-rad. of curvature at that point. For exercises in finding the rad. of curvature by means of the formulae [5], the student may take the examples worked by a different method in the Ana- lytical Geometry (pp. 327, 350) : we shall merely add here the following problem. Problem. — To determine those points in a given curve, if any such exist, at which the osculating circle shall have contact of the third order. In other words, to find for what points in the curve the values of a, ^, p, determined as above from the first three conditions of [3] p. 564, satisfy also the fourth condition. The differential coefficient y, as derived from equation [3] at p. 564, is here to be equal to the corresponding coefficient derived from the equa- tion of the given curve ; so that the values which satisfy the three con- ditions at p. 564, are also to satisfy the fourth condition and this they can do only for abscissae, or values of iv, which are real roots of the equation : if, in any particular case, the roots should be all imaginary, there will not exist any point in the curve at which the contact shall be of the third order. But when the abscissae are real, the points indicated will be those of maximum or minimum curvature. For since _ (p^^+l)i . dp_ -S(p'^-\-l)ipyi-{.{p'i-\-i)lp" ' ^^ -p" '" dx ^"2 ' which, in the case of a maximum or a minimum value of p, must be 0, .*. the numerator, or —Zp'p"^-\-{p"^-\-V)p"'=zO ; or, dividing by^", and putting y—^ for its equal— (page 566), we have P (y— A)i>"'+3yy=0, the equation to be satisfied above. (See also art. 594.) Note. — In the figure at p. 329, the contact exhibited is of the third order; and at the principal vertices of the curves discussed in the Analy- tical Geometry, the contact of the osculating circle is always of the third order. 587. In the preceding investigations x has been taken for the inde- pendent variable : but if any other quantity be chosen, the foregoing ex- 668 EADIUS OF CURVATURE : POLAR CO-ORDINATES. pression for p will require modification. To give complete generality, therefore, to the formula for the radius of curvature, we shall now suppose the independent variable to be left quite arbitrary, a; and y being functions of it. Under this hypothesis p' and p'^ must be replaced by the expres- sions at (572), that is, by (dy) {cPy){dx)- {d}x){dy) the vincula implying that the independent variable in reference to which the differentials of x and y are taken, is arbitrary : when chosen, the dif- ferential of this independent variable, and its proper powers, are to be intro- duced as denominators of the above differentials. Making these substitu- tions, the expression for p becomes ^^ {{dyYMdxff^ _ {dsf ^ {d^y){dx)-{d'xKdy)- {d'y){dx)-{d'x){di,y ^^ ^^**^ W' which is of the utmost generality, and will furnish the correct formula whatever be the hypothesis as to the independent variable : if ^ be chosen, in which case we shall have {dx)=l, and .•. (d-x)={),ihe formula will become that at (585). If s be chosen, then we shall have (afe)=l, .-. {dsf=(dx)^-[-{dyf=l. Differentiating this, {d^){dy)-\-{dPx){dz)=0 [2]» and by substituting in the denominator of [1], the expression for {d^y\ deduced from this equation, and remembering that the numerator of [1] is 1, we shall have _ {dy) _dy Jl?x ^ (cT^a;) ds ' ds^' If the left-hand member of [2] be added to the square of the denom. of [1], the denom., which must remain unaltered in value, will be {{dyf^-{dxf}{{d:'yf-^{d^xf}={djh,f+{d;^x)\ .-. p=i.^y|(^^^y+(^gyj. 588. Radius of Curvature: Polar Co-ordinates.— In rectangular co-ordinates, let t denote the angle which the linear tangent at P makes with the axis of a?, then, leaving the independent variable arbitrary, .-. [1] above, -^=-, .-. P=^y..\r\. Now the angle r=e-\-SPT (see fig. p. 568), that ^ .=.+sia-.?, ... «,)=(,,)+^JJ^?g. Bnt (583), W=;;^^, also W=;^^ ■■ hence /7 V {dp) (ds) (dr) * Any first differential coefficient, in which the independent variable is assigned, may always be replaced by another, in which the independent variable is arbitrary {p. 653). CHORD OF CURVATURE. 569 By making the substitutions for r and p at the bottom of p. 560, this expression, after some reductions, may be converted into the following, namely. ^~u' u{d6f-\-{dd){d^u)-{dv){d^6)'' •'• '' u .,1^ ' i{;+ 1+^^^ , 1 the radius of curvature at the point (r, G), u being put for -. [A useful formula for the ellipse is investigated at art. 590.] 589. Chord of Curvature. — If from C, the centre of curvature, a perp. CK be drawn to the radius vector SP, the angle included between this perp. and the radius of curva- ture CP, will be equal to the angle SPY, since the lines including this are perp. to those including the former angle. The perp. thus drawn from the centre of the circle to the chord of it from P passing through S, must necessarily bisect that chord PD — called the polar chord of curvature : hence, since PK=.CP sin C=p sin SPY, we have Polar chord of Curvature=2p mn /SPr=2p -...[4]. Note. — It may be worth while to notice, in reference to [1] above, that if As be an increment of the arc s of the curve, and normals be drawn from its extremities, these, being perpendicular to the tangents at the same extremities, will include the same angle as those tangents, namely, the angle At. If the arc As actually coincided with the arc of the equi- curve circle, the two normals, up to the point where they intersect, would be equal, and that intersection would be the centre of curvature : and it is plain that this centre is approached nearer and nearer as At, and con- sequently As, diminishes ; As so that the ratio — - approaches nearer and nearer to ^, without limit d8 to the degree of nearness, as Ar diminishes: hence, as before — =p. dr 590. Of all curves the ellipse is that in reference to which (for the purposes of Astronomy and Geodesy) the radius and chord of curvature are the most important : we shall return for a moment to that curve. By the formula at p. 329 1 6'2 /h'\^h^ V r 1 S2 P= sm^SPY a' Now referring to the figure at p. 334, if x be put for the angle PNF, we have 570 earth's ellipticity. OF sin OP' F sinPFP' ^^ ^ . c cos SPY ^_„ -~— = z= , that IS, -= — : =e, .*. cosSPY=esinX, OP' sin OFF' sinA. ' ' a smX ' ' .'. sin SPY=y/{l—e^ sin2 A), .-. by substitution, putting also a(l— c°) for — , we have a(l-c2) P— J' (l-e2sin2x)i a very useful expression for the radius of curvature of the elliptic meridian of the earth, at a place of which the latitude is x. It may be otherwise investigated, without a reference to [4], thus. By p. 328, and equations 3, 4, p. 316, _6^_ a3&3 ^1__ "^ ab{a^ sinV+62 cosV)i (a^ sinV+&2 cos^a')* Now if A be the latitude of P, that is, the angle PN£=pOB, then since a=90°+pOBf we have sinV=cos'X, and cosV=sin% ^2 a %2 _ a{l-e^ {a^cos^X-\-IP&m^k)i {a'^-{a^-b^sm^X}-2 (l-eSsin^x)! 591. In order to show the practical application of this formula, let D, D', be the measured lengths in feet of two degrees of the earth's meridian, the middle points of those degrees being in the latitudes X, x\ respectively. Then these measured arcs, being very small in comparison with the entire circumference of the osculating circles at the middle points, may be con- sidered each as coincident with a degree of the corresponding circle, without any practical error. Now the chief object of measuring degrees of the meridian is to de- termine the figure of the earth, or the ratio of its polar and equatorial diameters, or the amount of a — b , called the earth's ellipticity, and which we know, from independent considerations, to be but a small fraction, - difiFering but little from unity. a We thus know that e- can be but a small fraction, and therefore e- sin- x a fraction still smaller ; so small that the powers of it may be disregarded in a practical matter of this kind : hence, neglecting these powers, p=a{l-e^){l-e^sm^xf^=a{l -e2)(l+? ^ sin^x). Also length of l°=p j^=i>, •-. i>=^ a(l-c2)(l+2 e^ smSx), and ^ -^a(l-e')(l+2 e'sinV), .-. by actual division, ^=l+^e'ismPx-sm^x'), ^ 2 D-D' 2 D-D' 3 D'{sm"X—sin:^X') 3 D' sin {X-\-x') sin {X—X')* which determines e^ or 1 — 5, and thence -, and 1 — , the ellipticity. The equatorial radius a may be found by measuring a degree on the EVOLUTES AND INVOLUTES. 571 equator, and thence b the polar radius becomes known. By measurements of this kind, these two radii are found to be 3962*8 miles and 3949*6 miles respectively, the ellipticity being — - (See Airy on the Figure of the Earth : Ency. Met). 592. Evolutes and Involutes. — If to every point of a given curve an osculating circle were to be applied, the centres of these circles would trace out a second curve called the evolute of the former: the original curve in relation to this is called the involute. Denoting, as before, the centre of any one of the osculating circles indifferently by («, /3), it is plain that to find the locus of this point, or the evolute of the curve y=F(^), we shall have merely to eliminate x, y, from the three equations at (385), namely, the resulting equation in a, /3, will be that of the evolute. For example, (1) Let the proposed curve be the common parabola y-= 4 wio;. Here /=-, .'. P"=— ^, •• ^'+^=i;r+l=-+l» Substituting these values of x and y, in the given equation, we have 4 fi^=—— {a—2m)% the equation of the evolute. This curve is called the semi-cubical parabola : by removing the origin to the point on the axis whose abscissa is 2w, the equation takes 4 the simpler form /3'=— — «'. Because /S=0, when «=0, the curve passes through the new origin, that is, the focus of the original parabola is midway between its vertex A and the vertex V of the evolute : moreover, since for every positive value of a, /S has two values, equal and opposite, and increasing as a. increases, while for every negative value of a, /3 is imaginary, it follows that the evolute consists of two equal branches of infinite extent, wholly situated to the right of the origin, one above the axis of X, and the other below it, and each receding from that axis more and more as in the figure : the abrupt termination of the two branches touching AX at V is called a cusp. That VX is a tangent may be easily proved from the equation of the curve giving -—=0, at the point V : thus -r=^zz — r, or (7, representing d» ^ da, 27wi /3' ' ^ '=' dB ct^ ^ a constant factor, -r=-C -5=C«^=0, when «=0. 572 EVOLUTES AND INVOLUTES. The following are added as exercises for the student. (2) Prove that in the circle the centre itself is the evolute. (3) Show that the equation of the evolute of an ellipse is and that it has the form in the annexed diagram, the points a, h, c, d, being cusps. [See the end of the Answers.] (4) The involute of the rectangular hyperbola whose equation in reference to the asymptotes is a:y=ar, has for equation (i8+«)^-(^-«)^=(4a)*. 593. The following theorems show the geometrical connection between the involute and the evolute. Theorem 1. Normals to the curve are tangents to its evolute. Let y=F{x), and ^=f[a), be the equations of the curve and its evolute. The tangent to the latter at any point (a, /5), is 2/-^=^(x-«) [1], where x, y are the current co-ordinates of the line ; also the normal to the former at any point {x'f y) of tlie curve is y—y'= — - {x—sd) [2], and this normal passes through the point (a, iS), the centre of curvature of {a/, 2/'). As, therefore, the straight lines [1], [2] both pass through the same point (a, /3), they will be identical, provided their tangents of in- clination be the same, that is, provided d^ 1 «'2+l — = — 7. Now, if for brevity we put „ ■=;, we have (585), 10 I da dq ,9 , dq ,/ / , dq\ d&_ dq ^ dp ^ da_dP_ dx doS " dx ' dx da hence the normal and tangent coincide. Theorem 2. The difference between any two radii of curvature is equal to the arc of the evolute comprehended between them. Regarding a as the independent variable, we have by differentiating the equation (y-^)^+(^-«)2=p2 [3], But [2] p. 566, remembering that «'=$^=$^-i-—, ly-p) ^+(a;_a) — =0, dx da da da da CONSECUTIVE LINES AND CURVES. 573 Dividing the second of these by the square root of the first, we have v{(D +^14:> ^^ ^^ (^^*)' 4:=l- - — '+^- Also for any other arc s' terminating at a point where the radius is p', —s'=:p'-\-C,.:s-s'=p—p, that is, the difference of the two radii is equal to the arc of the evolute comprehended between them. From these theorems it follows that if n, string be fastened to one extremity of the arc of the evolute, and be wrapped round it, and then stretched in the direction of the tangent, and continued up to the involute or original curve, this tangential portion of the string CP (fig. at p. 571) will be the radius of curvature of the point F of the involute which it meets ; and if the string be now unwound, the point P will trace out an arc of the involute. It is on account of this property that the names involute and evolute are given to the curves thus connected ; but it is right to notice that although every plane curve has but a single evolute, it has an infinite number of involutes, since the tangential portion of the thread wound round it may terminate at any point whatever. 594. Consecutive Lines and Curves.— When the equation of a curve contains a parameter (584), that is, a constant, the value of which is unassigned or arbitrary, it is plain that by giving a series of different values to this parameter the equation will furnish a corresponding series of different curves, all, however, belonging to the same family of curves. Let F{x, y, a/)=:0 represent any such family of curves, x' being a parameter, and whatever value of of fixes one of these curves, let any other, intersecting it, arise from changing x' into x' -\-h'. this latter curve will, of course, continuously approach the former as h continuously diminishes from li=zh down to ^=0, when the two curves will coalesce, and the points of intersection will settle into fixed positions : the object, at present, is to find these ultimate positions : — the intersections of the consecutive curves, as curves brought into coincidence in this manner are called. Now, by Taylor's theorem We equate this to 0, because, by the hypothesis, both F{x, y, a/)=0 [1], zsidi F{x, y, x'+h)=zO [2], dF(x V a/) and consequently, dividing the development by ^, — -^r — ^=0 [3]. ax Hence, when 7i by continuously diminishing becomes 0, and, therefore, the curves consecutive, their intersections, which would be generally given by the simultaneous equations [1], [2], are, in these special circumstances, given by [Ij and [3] : in other words, the points {x, y), determined by the solution of the two simultaneous equations ■n, ,. ^ du dF(x, y, x') „ n=F(., y, .0=0, _=^-i^=0 [4], are the points of intersection of the consecutive curves. 574 CONSECUTIVE LINES AND CURVES. Suppose, for example, it were required to determine the point of inter- section of consecutive normals to any plane curve y'=F{x'). The normal at any point \x\ y') of the curve has for equation y-y'= — 7 (x-x), or (y-y')p+x-x=^0 [5]. This corresponds to the first of equations [4], a/ being the parameter. Differentiating it with respect to a/ of which / is a function given by the equation of the curve, we have 80 that consecutive normals intersect at the centre of curvature as might have been anticipated. And we see that the position of tliis centre is de- termined simply by combining the equation [5], of the normal, with its differential, those of the proposed curve being regarded as the only variables. This suggests a ready extension of the present theory : from a curve consecutive to a fixed one we may pass to the curve consecutive to this latter, then to a third consecutive curve, and so on : if, for instance, three such consecutive curves have the same intersections, these will be determined by the simultaneous equations ^, ,v ^ dF(x, y, x) ^ cPFix, ?/, x') ^ ^^- from which the two quantities x, y, may be eliminated, and thus an equation of condition determining a/ obtained. Take the case of conse- cutive normals considered above : differentiating [G] with respect to x\ we have then the three simultaneous equations i3f-^)p+x-x=0, (y-y>''-.|>'2-l=0, (y-2/>'"-3i>Y=0. And eliminating x, y^ from these, we get (l)'2+l)_p"'-3jp>"2=0, as at page 667, the condition which determines the points («', yf) of the curve yiz=^F{af) of greatest and least curvature. A geometrical significance is given to these analytical indications as follows : conceive a normal drawn from an assigned point in a curve : we may so select a point on this normal that a circle described from it as centre shall pass through the assigned point, and shall also cut the curve once again as near to it as we please, but not in general more than once as near as we please, seeing that the centre must not be taUen out of a fixed straight line : when the centre by con- tinuously moving towards, at length reaches the centre of curvature, the two points of the curve coalesce, and the circle becomes that of ordinary curvature : — the centre of it is the point of intersection of the fixed normals, with its consecutive normal. Again : we may conceive two neighbouring normals to be such as to in- tersect on an intermediate third normal, and as they approach, to preserve their intersection on this normal up to coincidence with it : these neigh- bouring normals must have the peculiarity that their three equations have place simultaneously for the point of intersection common to all: but, when this point takes up the position due to the three normals, when they all coalesce, or become consecutive, their general equations become re- placed by the simultaneous conditions [7], which fixes the point {x\ y") of ENVELOPING CURVES. 575 the curve ; -which point is such that the circle described as above passes through it and two neighbouring points of the curve, all three points coalescing when the centre is that of curvature. 595. Enveloping Curves. — The equations [4] after the differen- tiation implied in the second is performed, determine the point (or points) (x, y) in which the curve represented by the first equation is intersected by its consecutive curve, the parameter x' being any assigned constant whatever : hence, if from those equations we were to eliminate this arbitrary constant, the resulting equation must equally apply to every point of intersection between each curve of the family F{x, y, a;') = 0, and its consecutive curve. This resulting equation, therefore, must represent the locus of all the intersections, which locus will always be a curve touching every individual curve of the family at the points of their conse- cutive intersections. The evolute is a particular instance : this curve is the locus of the intersections of consecutive normals to the involute, and we have seen that this locus touches all the individual normals at those intersections : but to prove the proposition generally, take any one of the family of curves [4], a' being constant for that one, and differentiate its equation : we thus have (clu^_du du d^ .^ \dx) dx^dy dx ^ ^' Now the equation thus differentiated becomes that of the locus, when the value of x\ as given by the second condition [4], and which is .'. a func- tion of X and y, is substituted in it. Differentiating then this equation of the locus, a/ being a function of ic, we have (rfw) du du dy du dx! ^ but by the condition just referred to, -r>=0> at each point of consecutive dy intersection, that is, at every point of the locus : hence — in [1], as deduced from the equation of the fixed curve for the point (a;, y), is the same as — in [2], dx as deduced from the equation of the locus, for the same point [x, y)\ consequently both curves touch at that point. The following example will illustrate this theory of envelopes. Ex. Between the sides of a given angle are drawn an infinite number of straight lines, each cutting off a triangle of constant surface s : required the curve to which all these lines are tangents, that is, their envelope. The equation of any one of the lines indifferently is y=ux-i-0, the sides of the given angle being taken for axes : the intercepts or portions Q cut off by the line are /3, and , so that the constant area is ft 8=-2;sm^, .-.«=-- sine, .: y=--xsme-^0 [1]. This then is the general equation comprehending all the lines, /3 being 576 SINGULAR POINTS OF CURVES. a variable parameter, and occupying the place of xf in the above theory. Differentiating [1] with respect to /3, we have agreeably to [4], p. 573, -?:. sin 0+1=0, .-.^^J^,,.-. W' 2'=-2Tir^+^e' •••"^=dr^' the variable parameter being eliminated. The envelope is, therefore, an hyperbola, having the sides of the given angle for asymptotes. Examples for Exercise. (1) Prove that the envelope of all the straight lines comprehended in the equation y=.ttx-\ — , where m is fixed, and a a variable parameter, is a parahola. a (2) Prove that the envelope of all the straight lines indicated by the equation yz=.itx-\-{(i?a-\-Wy , where a is a parameter, is an ellipse. (3) Show that the envelope of all parabolas, comprehended in the equation y=.itx — a:^, is another parabola. 2/) (4) A straight line of given length c is placed in all possible positions with its two ex- tremities on two rectangular axes : show that the equation of the curve to which the Iff varying line is always a tangent is x +?/ =c . 596. Singular Points of Curves,— Any point at which a curve presents such peculiarities as distinguish it from the neighbouring points, without regard to the position of the axes of reference, is called a singular point: for instance, points of inflexion, and cusps (pp. 567, 571), are singular points. Multiple Points. — If two or more branches of a curve unite in a single point, either by crossing each other there, or by simply touching, the point is called a multiple point, its degree of multiplicity being indicated by the number of branches issuing from it, or passing through it. The analytical characteristic of such a point is evidently ~ =- for the co-ordinates of the point ; and the multiple values of this are readily found, without the calculus, by the rule given at (564), the examples to which might all have been introduced under the present head : we shall here add one or two others. (1) Required the nature of the point at the origin of the curve By the rule referred to Hence two branches of the curve pass through the origin, touching the axis of X on opposite sides of it. (2) Required the nature of the point at the origin of the curve By the rule, (■^Y=<«, .-. C?!)=^'=+0. SINGULAR POINTS OF CURVES. 577 Hence two branches, to each of which the axis of ;c is a tangent, spring from the origin, one on each side of that axis : they both proceed towards the right, for if x vanish negatively, the result is (-)=±0v/— 1, so that the two branches form a ciisp : the curve is the semi-cubical parabola figured at p. 571, the origin being at V. The equation of the curve itself shows that no part of it exists for negative values of x. (3) Required the nature of the point (a, 0) of the curve i/^=x{x^af. Removing the origin to the point, we must replace x hy x-]-a, so that then the equation will be f={x+a)aP, or 2/2=a;3_,_ax2, /. (^y=x-\-a, .-. £=±^a. Hence two branches of the curve cross the axis of x at the point, the in- clinations to it on opposite sides being equal. Conjugate Points. — It sometimes happens that the equation of a curve is such as to be satisfied not only for the co-ordinates of every point in the curve, but also for the co-ordinates of an additional isolated point, entirely detached from the curve, and thus having no place in the continuous series of points belonging to the curve : such a point is called a conjugate point. For instance, the curve figured in the margin has for its equation a(y—hf=x^—ca?, the origin of the rectangular axes being at 0. This equa- tion is satisfied for ;c=0, since the corresponding value of y is the real value y=^h : these co-ordinates, therefore, mark out a point P on the axis of y : but between the values a?=0, and x=c{=OC), there are evidently no real values for y ; nor are there any corresponding to negative values of ;» : P is therefore a point entirely isolated from all real points in the plane of the curve : — it is a conjugate point. The differential calculus, to be applicable to any particular point, always requires that that point be one of a continuous series of points : the point in question may terminate that series, or mark the limit or boundary of a continuous curve, but in the absence of all continuous connection with neighbouring points, the calculus cannot deal with it. In examining whether a singular point is a conjugate point or not, the legitimate way of proceeding is this : — Observe whether the values x, y, when they reach the singular case, difi'er either of them from a real value, only by the imaginary zero y/ —n, in whichever direction that zero be arrived at : if so, the real values, the imaginary symbol being suppressed, will be the co-ordinates of a conjugate point. But if the zero is real when reached in one direction, and imaginary when reached in the opposite direction, the point is a limit or a cusp. For example (1) Required the nature of the point (—a, 0) of the curve y''=:[x-\-afx. Here, for x=^a, we have y=0^/— a, whether x=—a be reached by decreasing or increasing values of x, showing that on each side of the point the ordinate is imaginary : the point is therefore a conjugate point. (2) Required the nature of the point, whose abscissa is a, of the curve (c^y— a^)'=(a;— 6)*(x— a) , a<6, or h—a=k. P P 578 SINGULAR POINTS OF CURVES. Solving this for y, we have ch/=i{x—hf {x—ay-\-x^, which for x=a, gives y=-^-^0^—7fi. and since the zero remains imaginary, whether a=a be arrived at from a succeeding or a preceding value of jc, we conclude that the co-ordinates =a, y="i> Ibelong to a conjugate point. In each of these examples y passes through a continuous series of imaginary values, between the extremes of which there is one value in which the symbol of impossibility is of the form 0^/ — n, which, being in value nothing, may be suppressed: the real value of?/ which then remains equally (in conjunction with the corresponding value of x) satisfies the equation of the curve, and therefore indicates a real isolated point. Conceiving imaginary values of y thus to generate an imaginary curve —as the imaginary ordinates of an ellipse belong to the imaginary hyper- bola having the same axes — we see that the real conjugate point is passed through by the imaginary curve : — the ordinate of the imaginary point being t/=w-HO\/—ri, and that of the real point y=m, the latter being inferred from the former, inasmuch as 0^/ —n may be suppressed without loss to the value of y. What are here called imaginary curves in the plane of the co-ordinates, by a peculiar geometrical interpretation of the imaginary symbol n/ — i, may be replaced by real curves out of that plane. On this subject the student may consult Vol. ii. of Peacock's Algebra. What by the French analysts are called courbes pointillees, or punctuated curves — curves traced by isolated points or dots — may be inferred to be an assemblage of real points in the co-ordinate plane, each (or rather an imaginary point super- imposed upon it) belonging to an imaginary curve in that plane, and for which the ordinate is y=m-|-Ov/ — n, or belonging to a real curve out of that plane having this ordinate. To continuous values, whether real or imaginary, the calculus is, of course, always applicable. [Note. — Many errors have crept into analysis from neglecting to take account of the signs with which variable quantities vanish in the extreme cases of a formula. Some of these were pointed out by the author in a Paper read before the British Association at the Cambridge Meeting of 1845. It may be instructive to mention an instance here. No less a man than Abel has stated, in reference to the series, 1 11 - ^=sin ^— - sin 20-1-- sin 3^— (p. 218), that "lorsque 0=9r, ou — tt, la serie se reduit a zero, comme on voit aise- ment." Now up to these values of 9, the series is complete, without the correction introduced at p. 220 ; and when becomes tt, sin 9 vanishes positively, sin 29 negatively, sin 39 positively, and so on : and when 9 be- comes — 27r, the signs are the opposites of these, so that in the extreme cases we have :^ SINGULAR POINTS OF CURVES. 579 from which we infer that - «'-7-0=oo =1 +--f -+ , 2 A o which accords with what we otherwise know to he true. See art. 207.] C'm'p^ are also singular points. They may be detected as explained in ex. 1, p. 571, and in ex. 3, p. 572. Points of inflexion have already been briefly noticed at p. 567, but their analytical indications off'er themselves more readily from examining those by which convexity is distinguished from concavity : we shall, therefore, investigate the latter indications here. Convexity and ^Concavity. — In order to ascertain whether at any pro- posed point P a curve is convex or concave to- wards the axis of x, let t be the angle which a vi linear tangent at the point makes with that axis, I ^^^^J"' then \p?^ y-p dy . „ dr d?y ^^ ^ tan r=^~r. .'. d tan T=secV -— =.-r^„ dx dx dx* so that ■—, must necessarily have the same sign as -7-. dx^ dx Now if , be positive, we shall know that the angle r increases as x ax increases, and diminishes as x diminishes, which can happen only when the curve proceeds from P {y being positive) upwards from the linear d'^v tangent — in both directions, as in the first diagram : hence — ^ dx indicates convexity. dr But if — be negative, then we shall be apprized CLX that T diminishes as x increases, and vice versa, and, therefore, that the curve proceeds from P {y being positive) downwards from the linear tangent, in both directions, as in the second diagram : hence -r-?^ nega- ax live indicates concavity. Points of Inflexion. — But if, in approaching P from a neighbouring d\ point, and upon passing through it to a point on the other side of it, r^ should change from -f to — , or from — to +, the inference will, of course, be that the curve is convex on one side of the point, and concave on the other, so that P must then be a point of inflexion. As, however, a continuous quantity cannot pass from + to — , or from — to -f, without becoming or 00 at the point of passage, it follows that a point of in- d^y flexion is indicated by — =0, or =00 , and by this coefficient changing sign at the point or points determined by these conditions. If the change be from -f to — , as ar increases, the curve is convex up to the point, and then becomes concave, as in the first diagram ; and if the change be from — to -f, the curve passes from concave to convex, as in the second diagram. Note. — In the preceding examination, the point P has been considered p p 2 680 SINGULAR POfNTS OF CUEVES. as above the axis of x, or to have a positive ordinate ; but if the point be on the opposite side of the axis, y being negative, convexity and concavity will be respectively indicated by signs the opposite to those above, so that generally a curve is convex or concave to the axis of x, according as y and ~ have like or unlike signs. In the application of these tests it is necessary to ascertain that the point which appears to satisfy them is not at an infinite distance : if either the x or the y of it be infinite, then, of course, the tests go for nothing. The first of the following examples will show the necessity of attending to this. Ex. (1). Required the peculiarities of the curve (a;— 2)i/=(a?— 1X«— 3), ^~ (x-2) ' •*• dx~ {x-2)^ ' •*• dv^" (a;-2)3* Now from a;=0 up to a;=2, this coefiScient is positive, but beyond this value of X it is negative: hence, the point whose abscissa is x=Q,, would be a point of inflexion, were it not that for a;=2, y=.±>: we infer, there- fore, that a parallel to the axis of y, at the distance a?=2, is an asymptote. At all points within the limits a;=], a;=2, y is positive, and the sign of the second differential coefficient, within this extent, being also positive, we infer that throughout this range the curve is convex to the axis of x. Again: within the limits a;=2, a?=3, y is negative, and so likewise is the sign of the second diff. coef. : throughout this range, therefore, the curve is also convex to the axis of x: but beyond 07=3, where it crosses the axis, it is concave, since y and — have unlike signs. Lastly, for values of x less than 1, ?/ is negative and - ^^ positive, as also for all negative values of x: hence from £c=l, where it crosses the axis of x, the curve proceeds to the left below the axis, without termination, and is throughout concave to it. The curve whose peculiarities we have been discussing is, in fact, an hyperbola, the proposed equation being a:i/— a:"^— 2i/ + 4x— 3=0 (see p. 354). (2) Required the peculiarities of the curve «=(1 -^-x^yy. _ X ^ dy_ 1-x^ ^ d^_ 2x{x^-S) ^ 1+x*' • * dx {1-^xY * * daP'~ (l+a;2)3 • This becomes for x=0, and for a?=±v/3, for all which values y is finite. For values of x immediately preceding x=0, that is, for small nega- tive values of x, the second diff. coef. is positive : hence it is positive throughout the interval x=0, x= — s/^y this latter being the only negative root of Saj"^— 6aj=0 : for the same interval y is negative, .-. the curve to the left of the origin, and up to a;= — v^3, is concave to the axis INTEGRATION OF RATIONAL FRACTIONS. 581 of a*. In like manner is it shown to be concave to the axis from x=0 to x=-\- v/3, throughout which interval the second diff. coef. is negative, and y positive : hence the origin is a point of inflexion. Moreover, beyond x= — \/3, in the negative direction, and x=-\- -s/3, in the positive direc- tion, the curvature again changes, there being inflexions at these points (P, P), beyond which the curve proceeds indefinitely right and left, and is convex to the axis. Since y is =0, not only for i»=0, but also for a!=co , it follows that the curve, after crossing the axis at the origin, and becoming concave towards it in each direction, as far as the two other points of in- flexion, as in the figure, again bends, and continually approaches the axis, which is thus an asymptote. The tangent to the curve at the origin has for inclination -;^=1 ; it is, therefore, inclined to the axis of x at an angle ax ° of 45°. The direction of this tangent is equally determined, without differentiation, by the rule at (564), from the equation of the curve y-\-x^y=x, wMcli gives ( -^-\-xy=l, and this, for x=0, y=0, is 1. These two examples must suffice as specimens of the way in which curves are traced from their equations, and their geometrical peculiarities dis- covered. For further examples on this and the other parts of the differential calculus, the student may refer to the collection of " Examples in the Differential Calculus," by Mr. Haddon, in Weale's Series. 597. Integration. — In the articles already given under this head the integrals determined are, with one or two exceptions, what are called the fundamental integrals : — the elementary forms into which, when prac- ticable, other integrals are to be decomposed, or reduced. The student will be prepared to expect that this reduction is to be effected, not by help of any extension of the principles of integration, but solely by our availing ourselves of the resources of common algebra : a chapter devoted to the integration of differential expressions, not included in the forms just adverted to, is in fact only a chapter on algebraic expedients, and algebraic transformations, in reference to a specific purpose. When an object is to be accomplished solely by these means, the successful mode of procedure, in every individual case, cannot be taught by precepts : — much must be left to the skill and penetration of the algebraist. There are, however, a few general methods of operation which, to certain classes of differentials, may always be applied with success : these are 1. Integration by decomposing Eational Fractions; 2, by Kationaliza- tion ; 3, by Formulae of Keduction ; 4, by Parts. These four methods we shall here briefly explain. 598. I. Integration of Rational Fractions. — An algebraic fraction is rational when numerator and denominator involve only positive integral powers of the variable. Every such fraction may be decomposed into others X^, X^, &c., such that X^dx, X.jix, &c., may always be inte- grated by the rules for the fundamental 'forms. Whenever the highest exponent of x in the numerator is greater than the highest exponent of x in the denominator, actual division of the former by the latter will convert the fraction into an equivalent expression, of which a jpart only is fractional, the part prefixed being integral. The 582 INTEGRATION OF RATIONAL FRACTIONS. fractional part will have the highest exponent of x in the numerator less than the highest in the denominator, because rational integral terms con- tinue to occur in the quotient till this is brought about. Every rational fraction is thus reducible to one in these latter circumstances, united or not to non-fractional terms involving only positive integer powers of the variable. Terms of this kind, connected with dx, being always integrable by the rule for powers, we have only to inquire how the fractional part of the differential is to be integrated : the mode of decomposition explained at pp. 156-7, suggests the course to be adopted, thus : 2 1 2 log a;— 2 log (05+ 1) + — -- ar+1 2(a;+l)2 These two examples show the advantage of the decomposition taught in the Algebra, but there is another, and, in most cases, an easier way of proceeding. 1. When the roots of F{x)=0 are all unequal. — In this case the com- ponents of the rational fraction "^r- may be found as follows. Let the roots be x—a^ x—a^ x—a^ ^~^n Now since for x=aj^ each numerator F(x) vanishes, it follows that for this value of x the above expression is reduced to the vanishing fraction which its first term then becomes, so that (561) we must have • • ^■=j;w' ^^ ^''"'^' ^''=Wi' ^'=:w' *"• Hence the roots of the equation F(^)=0 having been found, if these are all real and unequal, the several numerators of the partial fractions [1] into which ^r^ is to be decomposed, may be got thus : — Divide ^a?) by the first diff. coef. furnished by F{x), and in the quotient put for x the roots a^, ajj, &c., successively. -g — dx be proposed. Equating the den. to zero, the roots axe found to be ai=l, 0^=2, and a^—Z, .*. for the numerators of the three partial f(x) aP—Z fractions we have what '~rz=—— — - becomes when 1, 2, and —3 are successively F{x) oar — 7 113 put for X : that is, we get A^=-, - ^2= — 3> ^3= — 3> •*• / dx=z— I ■ / dx. The latter is — / ■:; . >» . ^ ^^t Which, putting 3 for x+2, IS -y P+"r2 J TTf Restoring, therefore, the value of z, we have J*-^-^J-^ log (^-l)-i log (a^+«:+l)+v^3. ton-i?^|. The following are left for the student to integrate.* 5x-\-l , 6a^+l , 2a ^ Sx-5 , 2aa; , rK2+a;-2 ' a;2-3a:+2 ' a'^-ar^ ' rg2_Qx-{-% ' (a+a)^ 599. II. Integration of Irrational Functions.—- Many functions involving radicals may be rationalized by certain obvious sub- stitutions, as in the following example. Irrational Monomials. — Let it be required to integrate x^-a jdx. Then, since the common den., given by the reduction of the fractional ex- ponents, is 6, put 2? for «, by which substitution the form is rationalized ; for we have X^—a, ^—O'aJSJ ^~^a^j, which, by actual division, gives a mixed quantity, the fractional part of which, as well as the integral part, is rational : the integration may there- fore readily be performed, and then the value of z restored. By the method here exemplified it is obvious that, whatever be the fractional ex- ponents {jp, q, &c.), every expression may be rendered rational, if it be in- cluded in the general form axP-\-l3fl-\-k(i. , a'xy-\-b'xfi'-\-hc. ' where the irrational quantities are all monomial. When they are binomial, the general methods of proceeding are as follows : — Irrational Binomials. — Integration of the form x"'{a-\-hx''ydx. Cases coming under this general form — where the exponents are whole or fractional, positive or negative — are of very frequent occurrence. The * As this work has already exceeded the prescribed limits, space cannot be afforded for many examples for exercise in this concluding portion of the volume : this is of less consequence to the student, as the shilling volume of "Examples on the Integral Calculus," in Weale's series, furnishes an abundance suitable for the purpose. 586 INTEGRATION OF IRRATIONAL FUNCTIONS. form may always be rendered rational whenever the exponents satisfy one or other of two conditions called criteria of integrability : these are as follow : — First Criterion of Integrability. — The condition for this case is, that whatever be the individual exponents wi, n, p, we must always have {m-{-l)-^n=. a, positive integer=N...[l]. Substitute la for a-\-bx", then for x we have 1 Wl ^ fit a:=Q:Z^y, ... ar=Q^y, .'. x'^-'dx^-^iz-ay'^dz. Mult, by aj, .-. [1], x'^dx=i-rj^(z-a)^-'^dz, .'. x'^{a^W)Pdx=-^{z-a)^-hPdz. The original is thus converted into a form which may always be inte- grated : this is plain, for TV"— 1 is either a positive integer or 0, and in the former case the binomial may be developed into a finite series of monomials. Note. — Should iV— 1 be a negative integer, and p also an integer, the coefficient of dz would evidently be a rational fraction. And to a rational fraction the expression may always be reduced in this case, though jp be a fraction : for let - be the fraction |); then putting?/* for z z, we have ti/~^dy for dz, so that, writing —k for the negative exponent iV— 1, we shall have to integrate {r^^a)-^ty*-^dy=tj^-^^^dy, which, the exponents being whole numbers, is a rational fraction, whether q be positive or negative. Second Criterion of Integrability. — The condition for this case is that |-p= a negative integer =— iV...[2]. Multiply the first factor by a;"^, and divide the second by the same, .-. x'"{a+hx^)Pdx=x'>'+''P{ax-''+h)Pdx. This is an expression still of the same form, so that if we substitute x for ax~"-\-b, the transformed expression can differ from that arrived at in the former case, only in this, namely, that a, b, will be interchanged, that —n will appear instead of n, and that m-{-np will replace m. Hence the transformed expression here will be x'>'{a+lx-^)Pdx= -{z- a) '^-hPdz. Ex. 1. Required the integral of a;\a -\-bar)^ dx. Here m=3, w=2, p=^i and (m+l)-i-w=2=i^. 2 Hence the first criterion is satisfied, and we shall now work the example by the process, and not by mere substitution in the formula. INTEGRATION BY SUCCESSIVE REDUCTION. 587 Put a+i^=., .: x=(^")^ .-. ^=(?=?)Li. (,_«)?, ^ .*. xHx:^ — -(z— a) tfz, .*. a?dx=i—{z—d)dzy ~" 62 V 5 3/~ 6* * 15 (2) Required the integral of ■ ^ ^ — ^ dx, or x-'\a?—x''Ydx. 00 1 <»»j_L.1 Herem=— 6, to=2, 2?=-, and \-p=—2z=—N. A 71 Hence the second criterion is satisfied: we shall solve the ex. at length. Mtdt. and dir. by x, .'. x-\a?-x^ldx=iX-\d?x~^-r)^dx. Substitute 2 for a?ar^-\, .'. x-^J"^^ , .-. x-'J^^^, .-. x-^dx^-^^^ dz. Divide by X-', .*. x-^dxz=——4zy .*. x-\a?x-'^^dx=:—-—-^z^dZf 15a* 15a''x* And in a similar way may the student integrate the following : — x{a-\-lxfdx, v?{a?-\'aFf^dx, a?{a?-\-x^)-^y a;-2(a+a:3)""W 600. III. Integration by Successive Reduction.—When neither of the criteria of integrability is satisfied by a differential of the preceding form, it cannot be immediately converted into an elementary diff'erential, as in the former case. The method of proceeding is then, by successive transformations, to reduce the differential to a form which is integrable by means of the fundamental rules : these transformations are effected by repeated applications of the formula for integrating by imrts : that is, by the repeated application of fudv^m—fvdu to fx'"{a-\-hx'')Pdx, Put (a-\-'bx'*)P=u, and x"*dx=dv, then v=. — — -, 771 4-1 .-. fx^{a->rlx-)Pdxz={a-\-hx»)p^^-I^fxrn+i{a-\-lx^)P-^x^-'^dx. m+1 m+1 Or, as in last article, putting z for a-^-hx"", m+1 m+1 688 INTEGRATION BY SUCCESSIVE REDUCTION. Again: since zP=:zP-^(a-{-hx^)=azP~^-\-hzP-^x'", it follows that fx'^zPdx=afx"'zP~^dx-{-J)fx'"+"sP~^dx...[2.'] And subtracting this from [1], and transposing, there results, afx^zP-Hx^zP — - — ^-^ \-^^-^J x'^+^zP-^dXy m+1 m+1 whatever be p. Put ^ + 1 for p, then this becomes, after dividing by a, fx'^zPdx^zP-^^- — — - — ^^ , ; ,, V a;"'+"2Pc;a;...[3.] a(m+l) a(w+l) The formulae marked [1], [3], may obviously be useful when m is nega- tive and n positive : they are not applicable, however, when m-|-l=0, as then the denominators vanish; but in this case formulae of reduction are not wanted, because, since =0, the first criterion of integrability is Kh satisfied. By transposing the integrals in [31, we have x-^^nzPdx=zP^^- -— ^^ '- fx'^zPdx^ o{pn-\-n-\-m-\-l) o{j>n-^n-\-m-\-\) whatever be m. Put w— n for w, then this is changed into o(^»+m-fl) b{pn-\-m-\-l) If, instead of subtracting [2] from [I], we had multiplied it by J^^ and then added, we should have got \ m+1/ m+1 m+1 ' so that, dividing by the first coefficient, we have fx^zPdx=zP- ^7"^' +. "''^'^ ^ ^ fx^zP-Hx..S5.-\ jpTi+m+l ^rt+m+1 ■■ -^ And multiplying by the denominator, and transposing, A'".p-.c?^=_..^!::::!+^!!±^±i/:,.,p^^, which, putting 2^ + 1 for p, gives a{p-\-l)n a(p+l)?i ^ -^ Finally, dividing [1] by the coefficient of the last term, and transposing the integrals, we have fxr-^-^P-Hx=Zv'^-'^fx-^ZPdx, which, by putting m—n for m, and p+l for |), becomes J x^zPdx=zP+^-—— ,, ,Z /x"'-»gP+'dt;...[7.] We thus get the following collection of formula. INTEGRATION BY SUCCESSIVE REDUCTION. 689 Formula for the Eeduction of fx'^{a+hx''ydx, or faf'z^da. I. fx'»zrdx= ^"' . gPH ^^^. J x^zP-Hx. II. „ = gp+. + ^^^ ^ ^ J x^'zv^'dx. a(m+l) a(m+l) " w+1 m+1 Should the particular values of m, n, p, in any example, be such as to cause the denominator in any of these formulae to vanish, that formula will, of course, be inapplicable: but in these circumstances, we should always have either (m+l^-7-w=0, that is, iV— 1=— 1, or else \-p=^, that is, iV— 1= — 1; n ■ and therefore, as shown in the Note, p. 586, the integral is then always reducible to that of a rational fraction. To exemplify the use of the above formulae, let x'{a+bx')-^da! be proposed for integration. Then, since »i=5, n=2, and p=— |, we have by III., fx^{a+lx^)-hx=fot^z-Ux=:~ z^-^fxh-Ux. 1 x^ 1 2cs 1 Again : by III., m being now =3, fQi?z—Mx=— z^——fxz— Mx. Now fxz~^dx=f{a-\-bx^)~'^xdx=- (a+lx^)^ : hence 156 \ h ^ V'J' The formula III. was chosen because it was easy to foresee that two reductions by it would bring the proposed to an integrable form, but V. or VI. would have done the same thing. The following integrals, reducible by III. and IV., m being a positive integer, are in frequent request. /x'^dx __a'"-^ a(m— 1) r* x^-^dx 590 INTEGRATION BY PARTS. / dx _ s/ja+hx^ b{2-m) /^ dx x'"y/{a-\-bx^)~a{l-m)x«'-'^ a{l-m)J x"'-^ ^ (a+bx^) / dx _ >y(a-bx^) b{2-m) p dx X ^{a-bx')~a{l-m)x»'-^'^a{l-m)J x"'-'^{a- bx^y We here see that at every repetition of the process m is reduced by Q ; hence, when m is even, it is ultimately reduced to 0, and when odd, to 1 ; and these ultimate forms may be directly integrated by the fundamental rules. To these we shall add the integral of the trinomial form {a + bx+cx^y dXy to which many others may be reduced. By the rule for solving a quadratic equation, we know that a-\-bx-^cx'^z=ci(x-{--\ 4^^}=^(i'^+^' •'• dx=zdy, And / = — / 2 — — sin-i _£_ — — gm-i 1 J ^{a+bx-cx') s/cJ s/{A-y^) ^/c s/A s/c {iac+by 601. IV. Integration by Parts.—This method has already been employed in obtaining the preceding formulae of reduction : we shall here show its application to certain logarithmic, exponential, and trigono- metrical forms. 1. To integrate XXo^xdx, where X is a function of x. In the formula fvdv=uv—fvdu, put dv=Xdx, and w=log''ar, then fXdx .log»x=log»xfXdaf-/ (fXdx . n log«->a: — \ or, putting, for brevity, yXc?a5=X„ fX log^'xdx^X^ log«a;-7i / —1 log«-'.r(Z«...[l.] Now, if n be a positive integer, the repeated application of this formula will finally reduce the integration to that of — - dx, so that the proposed form can always be integrated, provided we can integrate in succession the functions Xdx=dX,, ?^dx=dX,,~'dxz=dX,, ...,^dx. XX X As an example, let it be required to integrate {a^ -i-x")-^ alog xdx. Herew=l, X=:{a^-\.x'^)-ix, .-. Xdx=dX^=:{a^+x^)-ixdx, .'. Z,=(aHa;-)i r{a'^+x^)i />_a2+x2_ /> a'^dx r* xdx INTEGRATION BY PARTS. 691 The first of these is -a log ^^ ^ ^ ' -, and the second ^/ia^+x^ (pp. 508, 604), X V ViJ^^r ^VK+^-)-N/(«'+^-)+a log . One of the most useful cases of [1] is that where X=x'^. This form has already been integrated at p. 509. If, however, rn—O, and n= — 1, the form is /■ , which, simple as it J\ogx ^ is in appearance, has never yet been integrated, except by series, that is, by developing (log^c)"^ (see art. 602). ' 2. To integrate x"'a'dx. Putting in the formula for parts, u=x'", dv=a''dx, and .*. v=z , we have log a Jx'"a'dx= , .- J a;"*-'a*c?r, log a log a a formula which, by repeated applications, gives fx"'a'dx= \ x"^-- a;'«-»+ ' x-^-'^-..,±—-— — [, log a\ log a log^a log»'a j the upper being the sign of the last term when m is even ; the lower, when m is odd. When m is negative, the terms within the brackets do not come to an end ; in this case the substitutions in the formula for parts must be w=a', (fv=x-"'f?x, and .'. v=:- rr— — r; whence ' (m— l)a;'«-> /a''dx_ a* log a f a^dx x'" ~~(m— l)x'»-' m^i J ic'"-'' by repeated application of which, the final integral becomes / — -, *y CD which, however, like the simple integral above, can be found only by series, that is, by developing a*', and then integrating term after term (see art. 602). 3. To integrate s\n"'xcos'^xdx...[A']. First, let w be a positive and odd integer : then replacing it by 2p-f J, and putting (1— cos^a;)^ sin x for its equal, sin^/'+'a;, we have /cos»a;(l— C08%)P sin a;f?a5=— /cos"a;(l— cos2ic)Pc^ cos x=—fz''{\—z'^)Pdz, and since ;? is a positive integer, this may be converted by the binomial theorem into a series of monomials, all integrable by the rule for powers, whatever be the exponent n. Next, let n be a positive and odd integer=2^ + l, then we have /sin'»a;(l— sin2a;)P cos xdx=fdn'^xO.—&v[i"x)Pd sin x=fz'^l^—z'^)Pdz, which, as before, is integrable, whatever be the exponent m. Lastly, let m + w be an even negative integer= — 2p, then putting sin'^ic sin'^a; co3''a;= cos"'+"a3=tan'"a5 cos~'^^ic=tan»'fl5 sec-^'aj, cos'^a; 593 INTEGRATION BY PARTS. we have/sin'»a; cos'»ar{l-^z')P-'dz, which may be integrated as in the former cases. But if the exponents do not satisfy either of the above conditions, then putting the differential under the form sin"'~'a; cos"a3 sin xdx, and assuming sin'"~'d?=M, cos"a; sin xdx=dVf we have du=.(m—l) sin'"-^ cos xdx, and v= ; — , and the formula for parts, fudv=:.VkV— fvdu, gives /, sin"*"* a; 008**+ 'as wi— 1 /• . „ .„ , y sin'^x cos"a:cZx= --; —rJ sin'"-^^; cos^+^arcte, n-\-\ m+1 which is useful when n is negative and m positive. It may be put in another form thus. In the last integral put cos"a;(l— sin^x) for cos^+'^a;, .*. ysinw'-'a; cos'*+2a^a?=ysin'"-2 cos^ax^x— ysin'^a; cos"a5a: cos"-*^ m-\-nJ sin"*+2a; cos"a; _ -1 1 w+1 /"* dx " m+w sin"*-ia- cos"+*iC m-\-nJ sin'"a5 cos"+2x ' And putting w— 2 for m in the first of these, and w— 2 for n in the second, and then transposing, &c., as at p. 592, we have / dx _ -1 1^ m+w-2 r sin^'x- cos'*aj ~m— 1 sm»»-^a; cos^-'a; to— 1 ,/ £ dx 1 1 m+?i — 2 /* dx W— 1 ^ £ ** n— 1 sm'"~'ic cos"~'a3 w— 1 ^ sin*"a5 cos""^^' ' which ultimately reduce to the same integrals, as in the former cases, or else to /'.-^-=rf ?^+^V= log — = 1«8 *»" « (P- 592). %J Sin a: cos a; ^ \cos jc sin a;/ cos a; Hence the form [^] can always be integrated without series, provided m and n, whether positive or negative, are integral. We shall merely add, in conclusion, that when m and n are positive whole numbers, the forms ^vcH^xdx^ and cos^icdo;, may be immediately integrated by means of the following relations (259-60) :~ 8in2x=-- cos 2a;+-, 1 3 8in3a;=— - sin 3a;+- sin aj, 4 4 1 .1 « . 3 sm*x= - cos 4»— - cos 2x+- , o 2 o &c. &c. cos2x=- c o 2 o &c. &c. The following are a few examples for exercise on the preceding articles : — si.v?xdxy sin'x coo^xdxy sin^a; cos^xdx, sin^oj cos*a;da;, dec co&*xdx, m^xdxy sin'x cos''ajdx, — t-t r— • sin"*ic cos'^u; The formula for parts will apply successfully to many other differential forms besides those considered above ; and in particular examples, algebraic artifice will often conduct more readily to the desired result than any general rule : but since the Integral Tables of Meyer Hirsch give at once the integrals of nearly all the forms which have been integrated — ^just as a table of logs gives the log corresponding to a number, thus saving the labour of computation — we refer to that very useful work for ample practical information of this kind.* * These Tables (the English reprint) are now becoming rather scarce : they may, however, be had of Mr. Maynard, mathematical bookseller, Earl's Court, Leicester Square. Q Q 594 SERIES OF JOHN BERNOULLI, 609. Integration by Series. — In the preceding articles we have given all the methods of integration in finite terms, to which general pro- cesses have as yet been applied. Considering the endless variety of forms which algebraic and transcendental expressions may assume, these general rules are but very inadequate for all the demands of analysis. But, as previously remarked, algebraic artifice and ingenuity will often transform an individual differential into a shape which will bring it under the control of established methods ; though, for the successful treatment of it in this way, no formal directions can be given. As a last resource, we must develops the proposed expression in an infinite series ; integrate its terms separately, and thus be content with an approximation to the value of the required integral. As examples of this mode of proceeding we shall take the two forms left unintegrated at p. 591. (1) To integrate by series. Put loga;=«, or x=e^ : then, since log £C . r dx _ rdx{ \ ,1 >g^ (tog^ ) ■J i^x-J 7\''+i^x+^+-r+-2:r+-i = log (iog.)+ iog.+i.22|^%l.V... (2) io integrate by series. Here developing a*, we have /a'^dx Cdxi^ . , (log a)2 „ (log a) 3 ) = log .+ log a ..+i . fc|2>-'^+i. '^W.... In a similar way we may find /dx n 111 ^-^=y (1 -icH^* -««+.. .)dx=x-^^-^-t^i^ , . . =taji-'ar. See also p. 512, 603. Series of John Bernoulli.— For exhibiting the general value of an integral m an infinite series, the following, known as John Bernoulli s series, is that usually adopted: it is obtained thus :— Let X represent any function of ^, and put SXdx=F(x)', then, by Taylor's theorem, we have » j j F{x-h)=.fxdx-Xh^^l^l_ t^T^A. dx 2 dx^%Z w ^ril ^bitrary, put h=:x: then F(x-K)=F(0\ which is therefore what fXdx becomes when x=Q : to mark this state, write it IfXdxl which is the desired series, and in which the constant annexed to it, and SERIES OF JOHN BERNOULLI. 595 written within brackets, — being what the integral becomes when a;=0, — is the arbitrary constant. It is plain that for a definite integral, where x takes assigned values, the series is useless, unless for those values it be convergent. But it is difficult to ascertain when this is the case, seeing that the series does not proceed according to either the ascending or descending powers of x, inasmuch as this variable enters also the several coefficients. This inconvenience is avoided in the following analogous series. Applying to the proposed integral the theorem of Maclaurin, in- stead of that of Taylor, we have /x..=c/x^H[^+[f]|+[g£ 5+' the brackets implying that the enclosed functions are to be taken in the state of x=0 : the term here placed first is the arbitrary constant. The best way of applying this series is to proceed as follows : first develope X by Maclaurin 's theorem ; we thus get then multiply the several terms by x, ^x, ^x, &c., and annex the arbitrary constant : actual integration is thus dispensed with. We can afford room for but one example. Required the integral of a'^dx-i-^l^x). By division, l-i-(l_a:)=l-(-^+a^-f ... Also a^=l-^log a.x-i-^^^^x^-{-... The product of these is l + (l+log a>+ A+log «+■ J^ \x^-\-.., ''J J— ^=x+(l+log a) -+(^ l+loga-fi-^^-+.... This process of course applies only when X is developable, from the commencement, according to the ascending powers of x. 604. Definite Integration by Series. — In order to find fXdx=F{x), be- tween the limits x=a, and x=h, in a series, put ?;— a=/i, .-. h—a-{-h ; and by Taylor's theorem, putting [X]^ for what the enclosed function becomes when x=a, we have ^(.)-PW=A'x<^=m^+[g]|+[f2 2.3^ If hy the interval between the limits, be sufficiently small to render this convergent — as it is here assumed to be — the definite integral may be approximated to by summing the leading terms : but if the interval h be too great for this, it must be subdivided into a sufficient number of smaller intervals, and it is best to make these all equal ; we shall then have to substitute for x, in the coefficients, a, a-\-hf, a + ^h\ &c., up to h—hf — each series proceeding according to the powers of h' — one of the equal component intervals : the aggregate of these series, dismissing those advanced powers of h' that may be disregarded on account of their small- ness, will be of the form flXdx=A Jt+A ,h?-\-A^^-\- .... But a better way will be to approximate to the value of the definite in- tegral by the method of equidistant ordinates, explained at (312). QQ 2 596 SUCCESSIVE INTEGRATION. [For fuller information on the subject of Definite Integrals, see De Morgan's Calculus, and Vol. II. of Price's Infinitesimal Calculus : also a very ingenious Paper by Mr. Eawson, in the " Manchester Memoirs," Vol. VII., New Series.] 605. Successive Integration. — In all the preceding examples of integration it is the first differential coefficient that is given to deter- mine the primitive function from which it has been derived. But if, instead of the first, it be the nth diff. coef. which is given, then, by a first integration, we shall arrive at the preceding, or n — 1th diff. coef. ; by a second integration, at the n— 2th coef., and so on, till w^e reach the original function. Since at each integration a constant is introduced, it follows that the complete primitive ought to contain as many arbitrary constants as there have been integrations to arrive at it. Thus, let and after n integrations, we shall in this way get an expression involving n arbitrary constants. Hence, if we multiply X, and the successive results, by dx, and inte- grate at each step — omitting the arbitrary constants — we shall at length get Xn, to which the remaining terms, involving the several constants, may be appended, conformably to this expression. For example : let f^ sin xdjiy^ be proposed ; then we have y sin xdx=—cos x, — ycos xdx=^—s,m x, — fsin xdx=: cos x=X^ r x^ .*. J 3 sin ax£x^=cos x-f-Ci —-\-C^-\-Cy Again: Required the curve whose equation is — 4=(7„ or — =0. dx? dx^ Omitting constants, ^^=0, .'. y=Ci^-^-\-C^-{-C^-\-C.. Hence the curve is a parabolic curve of the third order. We shall add one example more : an example in which an infinite series is necessary. Let the integral be p[\ -\- 3(r)-^dx^ : then by the binomial theorem. Now, instead of proceeding here as in the former examples, integrating step by step, we shall get the final integral X^ at once, if we replace a}\ x\ x\ x'\ &c., by 05* ^ X^ xS^ 1.2.3.4' 3.4.5.6' 5.6.7.8' 7.8.9.10' ^^'^ as a little reflection on the effect of successive integration will show : THE CENTRE OF GRAVITY. 597 / dx^ V(i+^-') '2.3.4" afi :+ 1.3x8 1.3.5a;W 2.3.4.5.6 ' 2.4.5.6.7.8 2.4.6.7.8.9.10 + C,f^+C,j+C^+C,. This method of deriving the final integral at once, when X is develop- able in a series according to Maclaurin's theorem, was first given, in a general rule for the purpose, in the author's separate treatise on the Integral Calculus, p. 91 (1831). The plan is to replace a;'\ ^\ a;^ x\ &c., in the development of X, by g-n ajn+J ajw+a aj"+3 _^___ — — ^— > ^(» • 2.3...%' 2.3. ..(/i+l)' 3.4. ..(/t+2)' 4.5. ..(/t+S)' '' and then, annexing the terms containing the arbitrary constants, we shall have the complete integral fXdx''. X. APPLICATIONS OF THE CALCULUS TO MECHANICS. We shall terminate this treatise by giving a compendious view of the way in which the principles of the calculus are applied to statical and dynamical inquiries. 606. The Centre of Gravity. — Let ABC represent either a plane surface or a solid body : — if the former, let it be regarded as situated in the horizontal plane (of x, y). The co-ordinates X, Y, of the centre of gravity G, or of the point where the vertical line from G pierces the horizontal plane — the plane of re, y — are exhibited at p. 384. Now let CN be an increment of the body or plane, comprised between two planes, CM, DN, perp. to the axis of x: then the corresponding increment of the numerator of X (p. 384) will be the sum of all the particles in the slice CN, each mul- tiplied by its abscissa. If we call the increment mn, h, it is plain that however small we take h, that is, however slender the slice CN may be, the sum just alluded to will always be comprised between these two, namely, the sum of the same particles when each is multiplied by Orn=x, and the sum when each is multiplied by On=a!-\-h; that is, putting S for the numerator of X, and AS, AB, for the corresponding increments of this and of the body, AS will always be intermediate between a^AB and (x+h)AB ; but the ratio of these is ^^"^ ^ — =l-f -=1 in the limit, or when 7t becomes ; xAB X hence the ratio of the intermediate quantity AS to either must, in the limit, be 1 ; that is, A^ . _^ xAB~ ' •'' xdB where S is the sum of the infinite number of elements constituting the numerator of X : hence in the limit, -=^=1, /. -^=1, .'• dS=xdB, .'. S=fxdB, ^ fxdB , . ,., „ fy^'^ rr T Z= — — -, and m like manner, Y— — ^...L^'J* 598 THE CENTRE OF GRAVITY. where it is to be observed that the integrals are to be taken between the assigned limits, and where B, though called above body, may stand for either line, surface, or volume. If its figure be such that it cannot be analytically represented, then the co-ordinates of the centre of gravity can be expressed only thus, namely : — _ 2{xm) ^_ Siym) where m is any particle of the body, l{m) being the sum of the particles, or^. The foregoing reasoning merely amounts to this. A thin vertical slice of the body is conceived to be taken : the particles in it are regarded, by way of a first approximation, as all at the same horizontal distance from the origin : this supposition approaches nearer and nearer to the truth as the thickness of the slice is diminished, and actually reaches the truth when the slice, by continuous thinning, becomes a mere section : — a plane or line of particles; or, when 5 is a line, a single particle (^dB) : see p. 561, foot-note. We shall give an example or two of the application of the preceding formulae. (1) To find the centre of gravity of the parabola, y=2(ax)K For a plane curve, dB=ydx, or B=J ydx (p. 511), therefore ^ fyxdx la'i-fx^dx 3 , . , - ^ ^ • ^ Z=— ? = — ; ; — =-ic, which, from a;=0 to x=x^ is -ou. Note. — It should not be passed by unnoticed — as in most books it is — that, agreeably to the general expression for X, we ought to have used 2i/ instead of y ; for here B is only half the parabola : but as the 2 would occur equally as a factor in numerator and denominator, it is suppressed. In like manner, tt is suppressed in num. and den. of ex. 3. (2) To find the centre of gravity of a circular segment, y=i{%'x—x^)^. fyxdx f {Irx—x^^xdx "" fydx \ area of sag. Now, by adding and subtracting the first of the terms following, we have f{2rx-x'^ixdx=f{2rx-x^)h-dx-f{2rx—a?)^{rdx-'Xdx) =rX- area of seg. — -(2ra;— a;^)i=rX- area of seg.— — , Z o 2 o 2^/3 2r3 ir .*. X=zr—- • . For the semicircle (v=0, y=zr),X=z- =t~- 3 area of seg. ^^ ' "^ '' 3 area Sa- (3) To find the centre of gravity of a segment of a spheroid. The equation of the generating ellipse being y^=—{2ax—x^), _fy^xdx_f(2ax—x^}xdx_%ax^—^x*_8ax—Sx^ "~ fyHx ~ f{2ax-x^dx ~ a^^-W ~ 12a-4jj' For the hemispheroid (a;=0, £C=a), X=— . 8 Since the expression for X is independent of 5, it remains the same THE the(5kems of gdldinus. 599 when h=a : hence, if a sphere be described on either axis of a spheroid, any two segments — one of each — cut off by a plane perp. to this axis, will have the same centre of gravity. The following are added for exercise. (1) The distance of the centre of gravity of a semi-ellipse from the base, or minor axis, is Z=4a-i-3ir. (2) The dist. of the centre of gravity of a paraboloid, of alt. a, from the vertex, is ^ 2 3 (3) The dist. of the centre of figure from the centre of gravity of a circular arc, is JSr=r chord -f- arc. 2 (4) The dist. of centre from centre of gravity of a circular sector is -r chd. -j- arc. o (5) The centre of gravity of the surface of a spheric segment is at the middle of its 607. In all the preceding examples, the bodies are symmetrical as respects the axis of x, on which axis, therefore, the centre of gravity is situated. But when such is not the case, and this centre is out of the axis of X, then the Y of it also must be found by the second of the formulae [I.]. And here it must be remembered that the dB is not the same as in the expression for X : — in this the slice is parallel to the axis of 2/, in the other it is parallel to the axis oi x: in the one case x is sup- posed to vary uniformly, and in the other, t/ to vary uniformly. But the expression for Y may be applied to the same slice thus : — Let B be regarded as a surface ; then dB=ydx is a line {CM) of particles : dif- ferentiating this variable line, y alone varying, we have dydx; which symbolizes a single particle, and .*. ydydx is the general expression for any particle in the line multiplied by its distance from the axis oi x: it is therefore the integral of this that forms the numerator in the expression for r. Integrating then in reference to y, we get \y''dx: this refers to a single entire line of particles, and .•. \fy^dx to all these lines, or to the whole surface, when taken between the limits of it. Hence for a surface, r=i4^. For a curve, ¥=^...[11.]. 608. The Theorems of Guldinus. — The expressions just de- duced furnish two remarkable theorems for determining surfaces and volumes of revolution ; for they immediately give For a curve, 2^Ys^=2^fyds. For a surface, 1*Yfydx=:.9rfy^dx. Now SwF is the circumference of the circle of which Y is the radius : it expresses, therefore, the circumference that would be described by the centre of gravity of the line s, if that line were to revolve round the axis of X. But ^Trfyds expresses the area of the surface which would be actually generated by this revolution (p. 517) : hence, 1. The surface generated by the revolution of a curve round an axis is equal to the length of that curve multiplied by the circumference described by its centre of gravity. 600 MOMENT OF INERTIA: RADIUS OF GYRATION. Again : SttF, in the second of the expressions referred to, is equal to the circumference which would he described by the centre of gravity of the plane surface, if it were to revolve round the axis of x ; and irfy'-dx is the expression for the volume actually generated (543) : hence, 2. The volume generated by the revolution of a plane surface round an axis is equal to the area of that surface multiplied by the circumference described by its centre of gravity. The application of these theorems to the determination of surfaces and volumes of revolution constitutes what is called the Centrobaryc Method. But instead of determining surfaces and volumes in this way, the theorems may be conveniently employed, when these surfaces and volumes are already known, to find the centres of gravity of their generating lines and planes : thus, let it be required to find the centre of gravity of a semicircle, and also that of a semicircumference. Then, since a semicircle, revolving about its diameter, generates a sphere whose volume is —5—, and o a surface whose area is 47rr'', we have only to divide the former expression by Stt times the area of the semicircle, which is -—, and the latter ex- pression by ^tt times the arc of the semicircle, which is 'rrr, to get the distances of the respective centres of gravity from the fixed diameter; which distances are, therefore, 4/* 2 J* For the semicircle, 7= — . For the arc of it, 7= — . 609. Moment of Inertia: Radius of Gyration.— The moment of inertia of a body turning about an axis is expressed by the sum of the products of all the particles of the body into the squares of their respective distances (r) from the axis of rotation (480). Putting then M for the mass of the l3ody, or the sum of all its particles, let dM denote a line of these particles parallel to (or equidistant from) the fixed axis, if the body be a plane ; a surface of them, if the body be a solid ; and merely a single particle, if it be a line : then the moment of inertia of the entire mass will be expressed by the integral J'r^dM, taken between the limits of the body. If the mass* could be condensed into a single point, at a distance k from the axis, such that Tc^M=.f7^dM, and .♦. such that lc=^y — ^7— , Tc would be the radius of gyration of the rotating mass (480). NoTE.^ — If the figure of the mass be such that it cannot be analytically represented, then the moment of inertia can be expressed only in the form k^M^lLir^.m), m being any particle of it. The evaluation of this expression, when m is a mere particle, is in general impracticable; but when M consists of several distinct masses, all rotating about a common axis, each mass may be regarded as condensed into a single particle at its centre of gyration ; and the moment of inertia of the whole, putting K for the radius of gyration, will then be K^M='%{7^.m)==.l?.m-\-Tc^.m,^-^Jc^.m^-\-..., the no. of terms being finite. 610. If the body were free, its resistance to progressive motion would MOMENT OF INERTIA: RADIUS OF GVRATTON. 601 be merely the mass moved ; but if it be constrained to turn about an axis, its resistance to angular motion is the above expression. In reference to motion about an axis, therefore, k'M always supplies the place of the mass moved, as if M were condensed in a point distant k from the axis. Just as the mass M is conceived to be concentrated in a point at the distance k from the axis, if M, be another mass condensed into a point at the distance a, the resistance of this to angular motion would be a^M^ : if this resistance be the same as the former, then k'^M=a^M^, so that, for such to be the case, the new mass M^, condensed into the assigned point, krM must be such that M, = — r ; and such a substitution of M. for M will ^ a- ^ not modify the resistance to angular motion. And here we must make a remark in anticipation of a perplexity which the student might otherwise feel in reference to the centres of gyration and of oscillation, into one or other of which points the mass of the rotating body is regarded as concentrated in the subsequent investigations. It is to be noticed that, as respects the first point, we have regard solely to the mass, dismissing all considerations as to gravity, or any other accele- rating influence, — no such influence being supposed to act ; in which cir- cumstances, a body once put in motion, would move uniformly for ever. But in speaking of the centre of oscillation, we take matter as we find it, obeying the force of gravity, since without some such force there could be no oscillation ; so that acceleration and time then become elements of consideration. (1) A slender rod of length L, revolving about its extremity. Putting r for any distance on the line from its fixed extremity, we have fj^dr 1^ . 1 Jc^= — - — =7r-=, which for the whole line, r=Z, is -Z'; L oL 3 and this, multiplied by the mass of the rod, is the moment of inertia. If the rod revolve about a point in it, at distances a, b, from the ex- tremities, the preceding integral must be taken between the limits (j3_J_53 1 I r=~—hy r=:a, giving P=— — — -. If a=h=-L, then P=—IP. OL/ 2 12 (2) Let the rod vibrate lengthwise, about a fixed axis directly above its middle point. Then, calling the perpendicular distance of the point of suspension a, and any distance along the rod from its middle point, Xj the distance r of a particle at a from the point of suspension, the integral being taken from x=:—-L, tox=-\--L,.'. Js^=a^-\-—L'. 2 2 12 (3) A circle turning about an axis, perp. to its plane, through the centre. Putting R for the radius of the circle, its area will be -TrR^ ; and at any distance r the area is wr'-, .*. dM=^'7rrdr; this is a ring of particles round the axis, at distance r. 602 MOMENT OF INEETIA : RADIUS OP GYRATION. (4) Let the circle turn about a diameter: then '2y being a chord parallel to it, dM=2ydx, ,. ^^=-^^= ^ . The numerator here satisfies the second criterion of integrability (p. 586), and taken between the limits x=—E, x=R, the integral is k^=-RK (5) A volume of revolution. The most convenient way of finding k^, for a volume of revolution, is to regard the whole moment of inertia as generated by that of the generating circle of the solid : this generat- ing moment (k^.^Try-), by ex. 4, is -tt^"^, the fixed axis being that of a; : hence the moment of inertia generated is If s be the arc, the ^ of the surface generated is F= — . 611. When the axis about which a body turns passes through the centre of gravity, the moment of inertia is always less than for any other axis parallel to this. For conceive two axes, perp. to the plane of the paper, to pass through the points C, G, the latter being the centre of gravity of the body ; and let P be a particle of the body projected on that plane, which may be allowed, as its distance from the axis remains un- altered. Join PC{=r) and PG{=r'); draw also Pp, perp. to the line through C, G; then for this, and every point P, we have P . CP'=P{C(P+ GP^+ CO . GP). But (421), S(P . GP)=Q ; .-. S(P . CP')=:S{P . CG')+^{P . 6^P»), .-. f7^dM=M. CG^-{.fr"dM... [11 where it is to be observed that, in the first integral, dM is the continuous arc of particles, of which P is one, all equidistant from C ; and in the second, that it is the continuous arc of particles, of which P is one, all equidistant from G : the integral extending to all the particles, or to the entire body in both cases. Hence, if to the integral jV'dM, in reference to an axis through the centre of gravity G, we add the constant M.CG~, the result will be the integral fr'dM, in reference to an axis parallel to the former through C. The radius h'=^ — — — is called the principal radius of gyration. Putting then h for CG, we have from [1], ]c^=h^-^1c'^...[2]. As an example, take the case of a slender rod L, revolving about an axis at the distance h from its middle : then (ex. 1), k^z=i--L^-\-h^ : this will agree with the former determination by putting a-}-6 forZ, and a— -(a+&) for A. 2 CENTEE OF OSCIIXATION. 603 612. Centre of Oscillation. — When a heavy body oscillates about a horizontal axis by the force of gravity, it is re- garded as a compound pendulum ; and it is an important inquiry to seek what must be the length of a simple pen- dulum that would perform its oscillations in the same time : in other words, to find at what distance from the axis a weighty point must be suspended by a thread without weight, so that its vibrations may be identical with those of the body. Conceive the line CG^ in last diagram, by a displacement of the body M, in a direction perp. to the axis through C, to be drawn out of the vertical, as in the annexed figure. Put CG=h, then the perp. Gq=h sin S ; d and since the sum of the moments about C, of the several particles of the body, is the same as the moment of the entire mass M, supposed to be concentrated at G (415), it is Mghsin ; also the moment of inertia of the mass is M{h'^ -\-kf^). Hence the moving force of gravity on the body, to turn it about C, is Mgh sin 6 ; and the resistance to this angular motion, or the mass moved, is expressed by M(/i^H-&'^) : therefore the angular acceleration is Sfe=p+F^-'W- MakoCO=^^...[l], then, if the whole mass were concentrated in 0, and connected to C by a thread without weight, the moving force producing rotation would be MgCO sin ; and the resistance to this motion, that is, the moment of inertia, would be M. CO' : and the angular acceleration would .'.be McjCO sin ^ 1 . ^ , . , . h . . , , The point 0, whose distance I from C is thus found to be is called the centre of oscillation of the body, and lis .*. the length of the simple pendulum performing its oscillations in the same time. The length I— CO may be expressed otherwise thus : — Mgh sin ^ 2(m) ^,,,, . ■■„, may be written gh sm ff ^. „ ' . , so that we have And it may be noticed here that if k be the radius of gyration, since A;2=(/iHfe''), [1] shows that .'=^0' ••• '''^='"' 80 that the radius of gyration is a mean proportional between the distance of the centre of gravity, and the distance of the centre of oscillation, from the axis of suspension, .*. if two centres coincide all three coincide. For an example of the application of the preceding expression for CO, seep. 616. [The theory of the simple pendulum must be deferred till we have established the diflferential expressions for velocity and acceleration. See p. 614]. 604 THE CENTRE OF PERCUSSION. 613. Tlie centres of oscillation and suspension are interchangeable ; that is, the body will vibrate in the same time whether it be suspended from C or from O. For suppose it to be suspended from 0, and put OG—h\ so that /, the length of the simple pendulum corresponding to the centre of suspension C, is l=zh-\-h\ The lengths /, V, corresponding to the centres of suspension C, 0, are given by the formula deduced above : hence we have V2 7/2 IJi yi IJI '=*+*'- '=''+T' ''=*'+!- •■•^'-T=«' ••• *-T' ••• ''=X+*='- which, as we shall presently see, is a property of considerable practical importance. 614. The Centre of Percussion.— We have already seen that the resistance to angular motion about a fixed axis, or the moment of inertia, is expressed by :S(r2.m), that is, by r,.r,.m,+r3.ra.m,-f r3.r3.m3+..., w denoting a particle of mass, and r its distance from the fixed axis, or its leverage. Now, if ^ be such a distance from the tixed axis that 2:(r-m) may also be equal to this quantity will equally represent the resistance to angular motion : it denotes, therefore, a moving force or pressure, which, if opposed directly to the resistance of inertia, will neutralize or destroy that resistance. Hence, if in the plane in which CD vibrates, and perpendicularly to CD at the distance q from C, the body strike an immovable obstacle, that obstacle will receive the full force of the shock, and none of it will be ex- pended in straining the axis of suspension : — the mass will not incline to either side, but will, for the instant, be in equilibrio. The point thus de- termined, and whose distance from the fixed axis is 2(r2,m)-j-2(r.w), is called the cmtre of percussion : it is the point in which the moving force of the body may be regarded «8 concentrated, and we see that it is at the same distance from the axis of suspension as the centre of oscillation, for that distance is (¥-{-k'^)M^hM=:S{7^.m)-r-:Bir.m), (612). The blow with which a body, having angular motion, strikes an obstacle, will be the hardest when it comes in contact with the resistance at the centre of percussion : thus, the hardest blow with a straight stick will be when the object is struck at | the length of the stick from the hand, or extremity about w^hich it turns — see ex. 1, p. 601, where (70=F-7-A=-i/^-7--X. 615. In the preceding articles the peculiar character of the several in- quiries renders it necessary that we should view body or mass as made up of particles, or molecules: these are sometimes called elements of the body, as well as the differential slices of it adverted to at (547). Now what in common language is called a particle, or a physical atom, is not in strict- ness that which analysis symbolizes : physical particles necessarily have dimensions, and therefore magnitude ; and a finite number of them will make up any finite body, however great. In analytical investigations the so-called particle is a poi7it. By Euclid, nothing is attributed to a point but position : he never in- ON THE ATTRACTION OF BODIES. 605 troduces it but in connection with magnitude ; — magnitude either in actual existence, or in prospect : it is the same in analysis, with this ad- ditional information, namely, the directions, which the dimensions of the present or prospective magnitude take, are symbolized also. Thus, if (x, y, z) denote the position of a point in space, in reference to three rectangular co-ordinates, the calculus supplies us with the notation dxdydz, as symbolical not only of the point itself, but also of the directions which the dimensions are to take when magnitude is to originate from it. The symbols dx, dy, dz, do not mean little lengths : — they do not imply any lengths at all, but only the directions which foreseen lengths must take : magnitude is always in view, and magnitude generated in a certain way, though not yet in actual existence. In most physical investigations the student will no doubt find this strict rigour departed from, and dxdydz employed to represent a physical particle of minute dimensions ; but he should regard this form of expression as adopted more for brevity and convenience, than from necessity : — a mathematical particle is a mathematical point, and analysis does not require that we should give dimensions to it. In the calculus, dx, or ds, is a zero-line ; dxdy, a zero-surface ; and dxdydz, a zero-volume; and these zeros are all as distinct from one another, and, to the analyst, as significant, as the finite geometrical quan- tities to which they relate. There is another matter too which must be considered in physical ap- plications. In the theory of curves (and the same may be said of the theory of surfaces) the analytical equations refer exclusively to the boundaries of the figure : (x, y) always denotes a point in the outline, and every point in the interior is excluded, in virtue of the condition which fixes the relationship of x and y, one of these quantities being an assigned function of the other : but for a point not in the boundary of the figure, no relation exists between the co-ordinates, x being the abscissa of every point in y : for such a point, therefore, the variations or differentials of x and y are quite independent. We have thought it well to ofifer these remarks here, because students frequently entertain the impression that, in applying the calculus to physical inquiries, strict geometrical rigour must of necessity be dispensed with, and approximation only to the truth tolerated ; which is a mistake. In order to illustrate and confirm what has just been said, we give the following short article on Attractions. 616. On the Attraction of Bodies. — Imagine any curve on the plane of the paper : — a circle, an ellipse, a parabola, &c. ; and that the solid body here to be considered is generated by the rotation of this plane curve about the diameter or axis which lies across the page. Our object is to find an expression for the attraction of the body — supposing it to exercise such influence — on an assigned point in the axis, or in the axis prolonged. Let P be any point (or particle) in the solid ; call the perp. from P to the fixed axis, y ; and the distance of the foot of y from the assigned point attracted, x: let also 6 be the angle of rotation through which P has passed from its original situation on the plane of the paper ; that is, let it be the angle which y makes with that plane. Then, having regard to the generation of the body, the symbolical representation of the point P will be ydMydx ; and as, in nature, the attraction on a point varies directly 606 ON THE ATTRACTION OF BODIES. as the mass (which, for uniform density, is as the magnitude) of the attracting particle, and inversely as the square of the distance, - , ydOdydx the attractive influence of P on the proposed point may be expressed by ^ . Consequently the component of it, in the direction of the axis, is found by multiplying this expression by the cosine of the angle which the line from the attracted point to P makes with the axis, X which cosine is ~ TT : hence. = the attraction of a particle P...[l]. Now this expression must be viewed in reference to attraction, just as the point ydMydx is viewed in reference to magnitude : actual value is in neither case implied — nothing but the proper preparation is made for what is to come into existence. In each case the quantities are analytically exhibited in their zero state ; but instead of the vague symbol 0, a character is impressed by the notation, upon this initial, or ultimate, or evanescent state of the quantity, which significantly marks its connection, by the bond of continuity, with what is to be, or with what has been. It is the business of the calculus thus to preserve the distinctive peculiarities of different quantities, even when by continuous diminution, according to prescribed laws, they have passed into their ultimate or zero state. If we integrate the above expression in reference to d^, from 8=0 to 0=27r, our point, or particle P, expands into a ring ; and we have - xydydx , 2sr -= the attraction of a ring of rad.=2/, at dist.=a: from the point. (2/2+x2)i If this be now integrated in reference to dy, the ring becomes converted into a circular section of the solid ; so that 2* fl r l£ZiC=the attraction of a circle of rad. =2/, at dist. =.x from the point, {y'+^)^\ the integration extending from t/=0 to y=zanything. A.nd finally, the integration of this, within the prescribed limits of x, of which 2/ is a function, will give the attraction of the entire mass, when multiplied by the density of it. It may be observed here, in reference to the above zero-expressions for a ring — a mere circumference — and for a circle without thickness, that if we eliminate the zero-factors, the attractions of rings and circles will respectively vary as the following quantities, where r is the rad. of the circle, and a the distance of the attracted point, on the central axis, from the plane of the circle : -,and2Jl "l—X >5 L (j-s+aSjaJ We shall give a practical illustration of the foregoing theory, but before doing so, in order that there may be no obscurity about the process, we shall briefly review what has been done, thus : — 1. Having a body of revolution in prospect, we introduce, in reference ON THE ATTRACTION OF BODIES. 60T' to its generation, the angular quantity 0, and symbolize the point P thus, d&dydx, which point is not as yet controlled by any conditions, all the symbols being arbitrary and independent. 2. We next integrate [IJ in regard to dQ, just as if the other symbols were constants ; and taking 6 from to Stt, we get the attraction for a ring of particles, the magnitude of this ring being altogether arbitrary, as well as the distance of it from the attracted point : x and y are there- fore still as arbitrary and independent as at first. 3. By a second integration, in reference to dy, we fill up, as it were, this ring, and get a circle of any radius y, and at any distance x from the point. Up to this step no controlling conditions, establishing a relation between a and y, have been impressed upon these symbols ; but it now becomes necessary, for the purpose of applying the general result thus obtained to the special case before us — to the exclusion of all points not belonging to the particular body of revolution with which we have to deal — to take into account the prescribed relationship between x and y, the origin being at the attracted point, and thus to take the final integral only between the limits of the proposed generating curve. The several steps are, therefore, these : — 1. but without actually performing the subordinate integrations, it is usual to symbolize the whole operation thus. Iff — — - — , and the variables being at first quite independent, it matters not in what order the integrations are executed. Problem 1 . A paraboloid of revolution, of uniform density, attracts a particle at the focus : required the attraction of a slice comprehended be- tween two planes perp. to the axis, one passing through the focus, and the other at the distance a from it. The equation of the generating parabola, when the focus is the origin, is y~=4?n(^+m), which value of if, substituted in the general expression (3) above, gives cZx=2«- /^"Sl %:Adx dx , , a-\-2m ■=4{rwi log aj+2m 2m Problem 2. To find the attraction of a spheroid of uniform density, and very small excentricity, on a particle at its pole. Putting U for the fixed semi-axis, y«=(l±e)2(2iJa;-«2), the particle being at the origin, the upper sign applying when the spheroid is oblate, and the lower when it is prolate. Now e being regarded — as in. the case of the earth — so small that its square and higher powers may be neglected, the preceding equation may be written 608 VELOCITY. Hence the expression (3) is 2{r / J 1 [ dx J i (2i6;±4ii!exqi2ea.-2)2J =2^x-2^f{{2R±iRe)T2ex}-^x^dx^ 4 / 4 \ which, between the limits x=:0, x=^2R, is-irRi l±^e ), the attraction of the spheroid. [The integral is obtained by the formula of reduction, p. 589.] 4 If e=0, that is, if the solid be a sphere, the attraction is -w-JS. o We leave the student to prove that if the attracted point be without the sphere, and at the distance a from its centre, 4 R^ the attraction will be -?r— r. 3 a^ But he must be apprized that, as these results are purely numerical, they merely express so many times the unit of attraction, whatever this may be assumed to be. Such conclusions, therefore, only show the relative proportions of the attractive forces of different bodies, or how these forces vary for the same body at different distances of the attracted point. If the attracted point be taken anywhere within the sphere, the result will show that the attraction towards the centre varies directly as the distance of the point therefrom. It is plain from the above, when the point is without, that the attractions of spheres of the same density vary directly as the cubes of their radii, and inversely as the squares of the distances of the point attracted from their centres. Instead of the cubes of the radii, we may obviously substitute the volumes of the spheres, or when their densities are different, their masses. Suppose m to be the mass of the earth of radius r, m^ that of a planet, M that of the sun, and Z> the distance between the planet and sun : then we have m M Mr^ ~2''jyz'''9 '• —^29'^^**^^*^°^ °^ ^'^^ ^^ *^® planet. In like manner, — -^=attraction of planet on the sun, iJf+m 1 r^ - . --25f=united attraction of both. This expresses the attractive force mutually exercised by the sun and planet in terms of the terrestrial attraction g. 617. Velocity.— If the velocity which a body has at t seconds con- tinue uniform for t^ seconds afterwards, then, calling the increment t^—t of the time, At, and the corresponding increment of the linear space «i— s, A*, As we have for the velocity v at the time t, v= — , ' At' ACCELERATION. 609 however small At, and consequently as, which depends on it, may be. But if the velocity during any finite interval of time be variable, then this expression is not the true velocity at the time t, but approaches to it nearer and nearer as the interval At diminishes. In fact, if the velocity continuously increase or continuously decrease during the interval A^, it is plain that the preceding expression for v will always be intermediate be- tween the true value of v at the time t, and the true value at the time t^ ; that is, it will be a velocity intermediate between the velocities at the be- ginning and end of the interval A*. Now when by continuously diminishing A^ this increment vanishes, the three velocities coalesce, and we therefore have for the variable velocity, at the time t, ds v=.—, whatever be the path s which the body describes. at As In a case of uniform velocity this is of course constant, and equal to — , whatever be At. At 618. Acceleration. — Like as a uniform velocity generates equal increments of space in equal times, so a uniform acceleration of velocity generates equal increments of velocity in equal times. Putting, as usual, / for the amount of this uniform acceleration, we therefore have /= — . But if the acceleration during the interval At •'At continuously increase or continuously decrease, then this expression will always be an acceleration intermediate between the accelerations at the beginning and end of At, and since, when by continuously diminishing At this increment vanishes, the three accelerations coalesce, we must have for the acceleration at the time t, f=z — . If the acceleration be uniform, then of course /=a constant. dt ds dv d^s Measuring a in the direction of the motion, we thus have v=y, /=:r~T3"l^-'^^ • dt dt dt remarkable expressions of the calculus, showing (1) that the first differential coefiftcient derived from the space in reference to the time of describing it is the velocity generated, and (2) that the second differential coefficient is the acceleration generated, or, as it is usually called, the accelerating force. From these the fundamental equations of motion are readily derived thus : — Fundamental Equations of Motion. — 1. When the acceleration is con- stant. From the second of [1], dv=fdt, .-. v=ft-\-C...[2'\. If t become when v does, that is, if t be measured from the beginning of the motion, the constant G vanishes, and we infer that the velocities acquired in any times from the commencement are proportional to the times themselves : putting therefore v=ft in the first of [1], we have ds=ftdt, .-. s=i/<'=^...[3], no constant being added because s vanishes with t. From [3] we infer that the spaces measured from the commencement of motion are as the K R 610 ACCELERATION. squares of the times of describing them. Eliminating t from [2], [3], disregarding the constant C, we have showing that the spaces described are as the squares of the velocities acquired. If the time be reckoned not from the commencement of motion, but from when the body has acquired a velocity v^, then the constant in [2] will not be but C=v^, since by hypothesis v^ is what v becomes when t=0. In this case, the foregoing equations will be 2. When the acceleration is variable. — From equations [1] we have -=—,.'. fds=vdv, .'. ffds=-v^, s= I -dv, also«= / — ...[6]. The general equations [1] from which the foregoing expressions have been deduced, give the velocity and the acceleration of velocity along the path s of the moving body. But referring this path to two rectangular axes in its plane, we have for the component velocities and accelerations in the co-ordinate directions (l'v fill (1 (f fj'u vel. along X—--, vel. along F=-— . Accel, along X=^-t-, along Y=.-r-r. at at at' at^ Take, for illustration, the case of a projectile : here the equations of motion are 3-7r=0, -—zzz—g, since y is measured in direction opposite to tliat in whicli g acts. at' dt' Multiplying by dt, and integrating, the component velocities are Multiplying again by dt, and integrating, we have xz=zCt^ y=.C't—^, or putting oCfor C", y=ax—^t:^, as at page 433, where tan K=a. The following problems will serve further to illustrate these formulae. Problem 1. — To determine the vertical motion of a heavy body towards the earth, the force of gravity varying inversely as the square of the dis- tance from the centre. Let r= the rad. of the earth, a the distance of the body from the centre at the commencement of motion, x its distance after the lapse of t seconds : then the acceleration / at the time t will be given by the pro- portion 11 r^ d'^x ^ : ^ : : S' : /, .-. /=-.9'=-^, by [1]. This is taken negatively because x is measured in direction opposite to that of the motion. Referring now to the third of [6], we have ACCELEEATION. 611 The constant C depends upon the initial velocity, that is, upon the velocity of projection, or that which the body has at the distance a, where gravity begins to act: if this initial velocity be 0, then (7= — 2r^^-f-», and thus the velocity the body has at any time i, that is, after having fallen from the distance a to the distance x, is completely determined : it is X a ax When the body arrives at the surface of the earth, that is, when a?=r, it will have acquired the velocity /2ro'(a— r) , . , .- . ,,«<,. ,, «— ^ ^ •y=*/ — — ^, which, if a=oo , is«>=/^(2rgf), since then =1. ^ Ot ft Hence, from however great a height the body may fall, it cannot arrive at the earth with so great a velocity as this, so that if it were possible to project a body Vertically upwards with such velocity, it would go on to infinity and never stop — supposing, of course, that there is no resisting medium or other disturbing force. Taking the radius of the earth at 3965 miles, the above expression for v would be v=6'9506 miles; so that if a body were to be projected vertically upwards with a velocity of about 7 miles a second, and were to experience no resistance, it would never return to the earth. In order to determine the time «, we have V rV \1ga—x/ rV <2.g ^{ax—a^) ' 1 ,a r* X , \ / a C ,, „^1 ,2a;— a^ •'• t=. 4,/t- . / —r, :r:dx:=-K/r-} \/(ax—x^-\--a cos-' K, rV 2g J V(««-« ) »'^ ^9\y ^ « J which integral needs no correction, because a—x and t become at the same time. When a;=r, that is, when the body reaches the earth's surface, <=-»/-—! x/Car— r^OH'-a cos"* I, which is qo when a=QO. rv 2(7 (. 2 a r If the force of attraction be assumed to reside in a single point, then X, upon the body falling to this centre of attraction, becomes 0, and (still putting g for the acceleration at a given distance r) the whole time of falling from the distance a will be 1 / a l-..-._l=l ' „l 1g 2 rl^lg In like manner, if the fall be from any other distance a^, the time t^ is a similar expression, .-. i^ : t^ : : a3 : a^. That is— the squares of the times of falling from rest to the centre of attraction are as the cubes of the distances passed through. Pkoblem 2. A body revolves about a single centre of attraction, to find the relation between the trilinear spaces described by the radius vector and the times of describing them. When a body, acted upon by a single centre of force, moves in a curve line, we may consider such motion to arise from a primitive impulse given to the body, which impulse alone would have caused it to describe a straight line ; but being continuously acted upon by a force out of this RR 2 612 ACCELERATION. line, it is deflected from its wonted rectilinear course at the very com- mencement of motion, leaving that course a tangent to its actual path at the point of projection; and since nothing draws the body out of the plane in which the centre of force and the line of projection are situated, the path of the body must be a plane curve. Hence, placing the origin of the rectangular axes at the centre of force, the rectangular components (Z", Y) of the acceleration F, at any point of the curve, are X=i^cos«, Y=F COS fi r r where r is the radius vector, or line from the centre of attraction to the point {x, y) at which the body has arrived in any time t, from the com- mencement of motion. Hence the equations of motion are the negative signs being prefixed because, F being attractive, the com- ponents X, Y, are measured in directions opposite to those of x, y. To eliminate F, multiply the first of these by y, the second by x, and subtract the products : we thus have yd?x—xd?y _^ ydx-xdy ^? -"'•• ~~dt -^• Multiplying this by dty and integrating, we have (582), 2 Sectorial Area=(7f-f C'u which being when <=0, we have C,=0, .*. Sectorial Area=-(7<. Hence tho sectorial areas described are as the times of describing them, so that equal areas are described about S, the centre of attraction, in equal times, whatever be the law (F) of the attracting force. This is Newton's remarkable principle of equal areas ; in the particular case of the solar attraction on the planets, the principle was first announced by Kepler. Problem 3. To determine the motion of a heavy body P rolling down a smooth curve, situated in a vertical plane. The motion of the body P is due to that component of the force g of gravity which is in the direction of the curve, and which component is g multiplied by the secant of the angle which a tangent at P makes with the 2/ of it : it is .-. 5-^ (p. 515). This .'. is the acceleration / of P down the curve, .*. [1] p. 609, d?s dy (ds As « , ^ , . •. integrating, ■jr'= — 1gy-\- Q. [Note. » is measured op^osiU to the direction of motion. ] Let }i be the height above the axis of x from which the body begins to descend, then when y=^, ?;=0 : hence 0=:-2^A+C7, .'.v'=2g{h-y), 0TV=>^{2gih-y)}. Since this expression is independent of x, it follows that if a body THE CTCLOTDAL PENDULUM. 613 move without resistance from any assigned height h down any curves whatever to points all at the same vertical distance y from a horizontal plane, the acquired velocities will all be the same as that which would be acquired by a vertical fall through the same height, be the curves long or short ; and the same would obviously be true were the actuating vertical force other than g. Hence no modification of path can increase the velocity with which a body arrives at any point from any given point, pro- vided the motion be due to a force constant in intensity and direction. The time of descent will depend on the nature of the curve : it is found thus. By [1], v^=. — —, .'. dt=z , that is, in the case of gravity, at V —ds /* —ds Peoblem 4. The Cycloidal Pendulum. — The generation and properties of the curve called the cycloid will be dis- cussed at p. 619, where it is shown that the curve is such that the e volute of each half of it is a curve equal and similar to that half. Let BAD be the curve here spoken of, CB, CD, being the two branches of its evolute : these curves, uniting in a cusp at C, are each the same in everything but position as the two equal arcs AB, AD. If, therefore, a simple pendulum be suspended from the point (7, and be made to swing between the two equal cycloidal cheeks CB, CD, the bob P will describe an arc ^P of the cycloid AD equal and similar to the arc Cp about which the flexible thread wraps itself. If I be the length CA of the thread, and s that of the arc AF, it is shown at page 620 that s'''=^ly, the curve being referred to AX, AC, as rectangular axes. The present problem is to find the time in which such a pendulum will make a complete vibration, or pass from F at the vertical height h above A to an equal height above A on the other side of it. Since 8=y/2ly, .'. ds=sj— . dy, .: [A] <= / 1 /I /» —dy I /I .2 Hence from y=h to y=y, we have t=-y/- . ( *— versin-^Tj^Y which expresses the number of seconds in descending from the height h to the height y, therefore the time of descent to the lowest point A, where 2/=0, is t=-v/-, so that for a complete vibration the time is 7'=5rA /-. Since this time depends solely on the length I of the thread, and is in- dependent of the length FA of the semi-arc, it follows that the time will be the same whatever be the extent of the arc of vibration. Moreover, 6J4 THE SIMPLE PENDULUM. at the same place, or for the same value of ^, the times of vibration of different pendulums are as the square roots of their lengths, and for the same pendulum, at different places, the times are inversely as the square roots of the values of gravity. Since CA is the radius of curvature of the cycloid at ^, an arc, of which A is the middle point, may be taken so small as to differ insensibly from a circular arc described from C, so that a simple pendulum swinging freely from C — the cycloidal cheeks being removed — will perform a vibra- tion in the time T above, if the extent of arc (the amplitude) be very small : but we shall consider this matter independently. Pkoblem 5. The Simple Pendulum. — To determine the time of vibra- tion in a circular arc of radius r. Let the vertical and horizontal axes originate at the lowest point of the curve, and as it will be more convenient, take the former for the axis of a;, and the latter for that of y : then for the equation of the circular path we have y -2rx X,.. [^^J -^^x-x^' • . rfs- V Y-^\dx) Y^^^ ^ {2tx-x^)' Consequently, by [A], changing y into a-, we have /—d8 r r* —dx sj{1g{h-x)\-:j¥gj ^{{h~x){2rx-x^)\- This cannot be integrated in finite terms, but it may be integrated to any degree of approximation by putting the differential under the form .. 1 /»• dx /^ a;\-4 the proposed differential may be replaced by a series of others, all of the same integrable form, namely, all of the form —x^dx ^^ /»o dx Now xdx 1 , /^o xHx h y^o xdx 1 , /^o xHx 1.3 ,„ ""777 57=— t;*"* / ""77; r=— -— wA'', and so on: ^ ^{hx-x^) 2 'y^ s/ihx-x') 2.4 ' hence for 2f=T, the time of a whole vibration, we have -'v/H^-aya)HMy(^)v }■ We thus see that if the arc of vibration be so small that w*/- . — . and the higher powers may be disregarded, the time of a complete oscillation will be the same as that in the cycloidal pendulum. Hence, for a very small amplitude or arc of vibration, the length of a circular pendulum beating seconds is r=Z=^-=-9r^, which .'. varies directly as the force of gravity; and from the expression Tz=.ve^-y this force varies at different places inversely as the squares of the times of vibration of the same pendulum. IMF THE SIMPLE PENDULUM. 616 Of course, a more exact expression for the time is Tve^ - . 1 1+^ f • From careful experiments by Capt. Kater, the length of the seconds' pendulum in the latitude of London was found to be 39-13929 inches, but these and like experiments could not be made by ^ such a simple pendulum as here described, for a heavy particle without sensible magnitude, and a thread without weight, form ±j^ but an imaginary pendulum : the actual pendulum employed is a heavy mass, the centre of oscillation of which being found, the length of an ideal pendulum that would vibrate in the same time becomes known, and may be conceived to replace it. The valuable property (613) that the centres of oscillation and suspension of a heavy vibrating body are inter- changeable, leads to a convenient practical way of finding the exact length of the equivalent simple pendulum. " The process used by Captain Kater was the following. The pendulum consisted of a bar AB oi plate brass, about 1^ inch broad, and I inch thick. At A was a knife-edge of the hardest steel, its back bearing firmly against solid knees of brass ; and at B a similar knife-edge. When the pendulum was in use these edges rested on horizontal plates of agate. At C was a large flat bob : at F was an adjustible weight, and at G a smaller weight, which by means of a screw could be adjusted with very great nicety. The principle of the operation was to observe the number of vibrations per day made by the pendulum when suspended on the knife-edge A, and again when suspended on B. If these were not equal, the sliding weights F and G were moved till they became equal. Then as A, B were at diff'erent distances from the centre of gravity, it was certain that B was the centre of oscillation corresponding to the centre of suspension A, and therefore that the distance between A and B was the length of the simple pendulum vibrating in the same time." Airy : " Figure of the Earth," Ency. Met. Knowing the length of the seconds' pendulum, it will be easy to find the time of vibration of a pendulum of any other length, or the length of a pendulum to vibrate in any other time, the place being the same : thus, I being the length of the seconds' pendulum, and l^ the length of any other, the number of seconds this last will vibrate in will be ^=k/ j=ti. Let ?i=20 feet =240 in., then «j=w^-— — -=2'5 sees. Again, to find the length of a pendulum oscillating once in 10 sees., we have /i=39.13929x 102=39139.29 inches. To find the gain or loss in a day by shortening or lengthening a pendulum. Let I be the original length, n the number of vibrations it makes in 24 hours, Al the length by which the pendulum is shortened, and An the number of vibrations gained in consequence: then, since the square roots of the lengths are as the times of one vibration, and that these are in- versely as the number made in a day, we have ^ B 9 616 THE SIMPLE PENDULUM. Al hence, neglecting the subsequent terms because of the smallness of Al, Anzszur—. At Suppose the seconds' pendulum shortened by the 300th part of its length, then Aw=24.60.60 •==24,6=144= no. of vibrations (or seconds) gained in a day. 600t The foregoing general expression is the same when Aw is the number of vibrations lost, in consequence of an increase of A? in the length of the pendulum. Hence, conversely, to find how much the pendulum must be lengthened or shortened in order that it may keep true time, the rule is this : — Multiply twice the length of the pendulum in inches by the number of seconds gained or lost in a day, and divide the product by the number of seconds in a day : the quotient will be the number of inches, or parts of an inch, by which the pendulum is to be lengthened or shortened. 619. Besides the ordinary purposes to which it is applied, the pen- dulum is also employed for the purpose of finding the heights of moun- tains, and the depths of mines : thus, 1. Let r be the rad. of the earth, h the height of a mountain, n the no. of vibrations in 24 h., and An the number lost at the top of the mountain, on account of the diminution of the force of gravity. Then since this force varies inversely as the distance from the centre 3'-^ '-'Z^'' 7-TT72- Also(p. 614)- : — : : -7- : -7-7 : : r : r+A, r-\-h n ^ . An , ^ An .'. = =1H nearly, .*. A=r — . r n—An n n Hence the height varies as the number of vibrations, or seconds, lost. 2. Again : let h be the depth of a mine, n and An as before, An being still the number of vibrations lost, because below the surface the attraction varies directly as the distance from the centre : here, therefore, g :g : : r : r—h. Also - : — ■ : -y - -TZ, • • "T * ""TT n» n n—An ^Jg ^^ ^r ^{r—h) r-h /n-An\ An . 7. o ^»* .*. =( ) =1—2 — nearly, .*. h=2r — . r \ n y n *^ n Hence the depth varies as the number of seconds lost. (1) Suppose that at "the top of a mountain a seconds' pendulum loses at the rate of 10 seconds in 24 hours : required the height of the mountain regarding the radius of the earth as 4000 miles. Here r=4000, w=24.60.60, An=10, .'. hz= ^^^^'^^ =—=zl a mile nearly. » ' ' 24.60.60 54 ? «* "^« "^^J- (2) Again : suppose the pendulum loses 3 seconds at the bottom of a mine : then , 8000X3 6 ., ,^^^^ ^=24760:60=18 "^^^='''^^^^*- \ COMPOUND PENDULUM. 617 We shall terminate these inquiries with a problem which will show the application of the formula [2] at p. 603. It will exhibit to the student how such expressions as S(r.w) and 'E{r'^.m), apparently implying the summation of an infinite number of quantities, may be often turned to account without any such usually impracticable summation at all. Problem. — A pendulum consists of a long slender rod CA, and two balls P, Q, of which the larger P is fixed at A, and the smaller Q is moveable : given the time * of a vibration to find the variation of the place of Q corresponding to a small variation of t. Disregarding the weight of the rod, suppose the mass of each ball to be compressed into its centre. Let CQ=a;, CA=:a, then, by the formula referred to, the length I of the equivalent simple pendulum will be 1=- ^ I p ; and supposing the latitude of the place to be such that the length of the seeonds' pendulum is 39*2 inches, we have 39.9 . Sf!±^. .1 ..2- J_?^±^' ' qx-\-Pa ' ' ' ~39-2 Qjx+Pa ' g , Q'a:*(fa;+ IPQaxdx—PQaHx 6q " 89-2(Qa;+Pa)2 . 78-4^(Qx+Pa)' . . »^— Q2^2_^2PQaa;-PQa2 Now the variations A(, Aa;, of t and x being small, they may be re- garded in a matter of this kind as proportional to their differentials ; suppose for instance a=158 inches, a;=126'9, P=20 lbs., Q=l lb.; then f=2 sees.; let the change in the position of Q be such as to cause a loss of 15 sees. in 24 hours, or 86400 sees., then 86400 : 2 : : 15 : '000347 the loss in 2 sees. =M : hence Aa;=l*84 inches, the corresponding displacement of Q, Examples for Exercise. (1) Required the radius of gyration of a cylinder revolving about its axis, the radius of the circular base being r. (2) Required the rad. of gyration of a sphere of radius =r. (3) Required the rad. of gyration of a cone of altitude a, about its axis. (4) A sphere oscillates about an axis at the distance a from the centre : required the distance below the centre of oscillation. (5) A vibrating cone is suspended at its vertex : the alt. of the cone is a, and the rad. of its base na : required the length of the equivalent simple pendulum. (6) A clock gains three minutes a day : by what part of the length I must its pendu- lum be lengthened to make it beat seconds ? • The diameters of P, Q, are so small in comparison with the lengths CA, CQ, that the centres of gravity, and the centres of oscillation of P and Q, in reference to the point of suspension C, may be regarded as coincident. [See inference at bottom of p. 603. ] 618 THE BALLISTIC PENDULUM. (7) A clock loses 1 min. a day : how much must the pendulum be shortened to make it keep true time, the place being London ? (8) A seconds' pendulum loses at the rate of 21 "6 seconds when at the top of a moun- tain : required the height of the mountain, taking 4000 miles for the radius of the earth. (9) If a pendulum be 39-37 inches in length instead of 39-1393, how much will the clock lose in a day? (10) A seconds' pendulum at the bottom of a mine lost 2 sees, a day : required the depth of the mine. (11) A sphere, whose radius is two feet, is suspended from a point 10 feet above its centre : in what time would one vibration be made by this pendulum on the top of a mountain 2 miles high, the acceleration of gravity at the bottom being 32g ft. ? (12) A smaU weight W is suspended from the extremity of a long slender rod of length a : it is required to find at what distance x from the point of suspension another small weight w must be placed, so that the pendulum may perform its vibrations in the least time possible. (13) Of similar right cones each attracts a particle at its vertex : prove that the attrac- tion varies as the height of the cone. (14) An upright cylinder attracts a particle in the prolongation of its axis : prove that the attraction is proportional to the height of the cylinder plus the difierence of the dis- tances of the particle from its upper and lower edges. 620. The Ballistic Pendulum.— This is a heavy block of wood suspended by an iron rod to a strong iron axis : it was devised by Kobins, and employed by him, and afterwards by Hutton, for the purpose of de- termining the initial velocities of cannon-balls. The projectile is fired directly against the vertical face of the suspended block, and enters the wood, which, with the ball in it, is set oscillating by the force of the impact. Let B be the mass of the ball striking the pendulum with the velocity V, M the mass of the body struck, and M^ the mass which, placed at the point of impact (regarded as in the vertical through the centre of gravity), might be substituted for M without modifying the motion : then putting a for the distance of M^, or the point of impact, from the fixed axis, we have Jlfi=FiIf-i-a« (p. 601) ; and since the momenta are the same before and after impact (482), if v be the initial velocity of M^ and B united, after impact, we must have (B+'Jy=By, ... .=^1^= initial vel. „£ M,. Now I being the length of the equivalent simple pendulum, and the angle of a semi- vibration, the velocity of the extremity of /, when it descends to the lowest point, is the same as what would be acquired in falling down the versin G to radius I ; that is, it is As/ (2gl versin ^)=2 sin ->s/(gl), 2 which is ,♦. to the initial velocity of ilfj, as I is to a, that is, „ . ^ ,, ,, BVa^ BVal . 6 , Z : a : : 2 sin-v'(^Z) : -^^^j^, '• ^^;i:j:]^=2 sm -^{giy..[ll If h be the distance of the centre of gravity of the pendulum, including the ball, from the fixed axis, we shall have (612), THE CYCLOID. 619 .-. [1], BVal=2 ^m^{B+M)liW{gt), .'. F=2 sin^^^^^(gl). Suppose, after impact, that the pendulum makes n oscillations in a minute, then (p. 614), n ^ g ^ ^^ ' vm 2 B 9rna in which formula weights may he substituted for masses. Note. — The impact is directed as nearly as possible to the centre of percussion, in order that little or no force may be expended in producing shock to the fixed axis. (For the practical details of numerous experi- ments with the ballistic pendulum, see Hutton's Tracts.) 621. The Cycloid. — If a circle AmY, roll without sliding along a straight line BB, and continue always in the same plane, the curve BAFT) described by any point F in its circumference is called a cycloid. The following properties of it are obvious (see next figure). The semi-circle AmY= YD, arc PQ=QD, ,'. arc Am= YQ...[ll Fmn being drawn parallel to DB : also CQ is parallel to A Y. Now the arc ^m being = arc (7P, .*. arc mY= arc PQ, .: the chords PQ, to F are equal and parallel, .'. mP=FQ, .*. [1], arc Am:=mP...[2]. Put now nP=Xj -4%=y, also the angle nOm=.6, then AO—nO=iy=.r (1 — cos ^), and [2], arc -4m-|-wm=a:=r(^-f- sin S). Or since nm=»y{2ry—y^f and arc Am=r cos"* , .-I This latter is the difierential equation of the curve, — being the trig. ay tan. of the angle which a straight line touching the curve at P makes with the axis of y. Rectification of the curve, — Put s for the arc AF, then we have ^=y |l+(^y |tiy=v/(^)^y, .-. s=2{2rfy^=:^8ry=2^2ry; 80 that when y=^r, the arc AD is 4r=twice the diam. of the generating circle. Quadrature of the curve. — This is best effected as follows : — fxdy=xy—fydx=xy —f(2ry—y^)idy. The second term of this integral, taken from y=0, to y=^r, evidently expresses the area of the semicircle of radius r, that is, it is -irr^ ; and the first term xy, between these limits, is YDx2r=z2itt^, 2 620 THE CATENAEY. Hence the area of the semi-cycloid A YD is ^irr^, SO that the length of the whole cycloid is 4 times the diameter, and its area 3 times the area, of the generating circle. Properties of the curve. — These are very numerous, we can here notice but one or two of the more important. 1. The tangent to the curve at P is parallel to the chord ^wi, and .*. PQ, parallel to mF, is perp. to that tangent. For the expression for — ■ above is dx )^{An.nY) nm An z=—-z=. tan nAm. An Hence the chord CP parallel to Am is tangent to the curve at P. 2. The cycloidal arc AP is twice the chord Am oi the corresponding circular arc. For AP=i=2s/{A Y.An)=2Am. 3. The evolute of a semi-cycloid is an equal semi-cycloid. This property may be proved analytically by the method explained at (592), but the following geometrical proof is likely to be preferred by the student. Conceive a cycloid B'DD\ equal to the cycloid BAD, to be placed in reference to the latter, as in the figure, its vertex being at D. Draw PQ, QT, chords of the generating circles : then by (1) above, QT touches the lower cycloid at T. Also EB' =&rc ET, and FB'= arc ETQ, .: FE=:DQ= arc TQ= arc PQ: hence the angles TQE, FQC, are equal, and .*. PQT is a straight line, and, as just shown, it is a tangent to the curve at T, and, moreover, PT=2TQ= cycloidal arc TD by (2) above. Consequently the normal PT, at any point P of the semi-cycloid AD, is a tangent to the equal semi- cycloid DB', so that DB' is the evolute of DA, and a string, applied round the arc B^D, if unwrapped from D, and kept stretched, will by its extremity P trace out the semi-cycloid DA. The entire length of the string, or of the semi- cycloid, as shown above, is twice the diameter of the generating circle =4r : if this length be called I, then AP^=s^—^l2/. In the figure at p. 613, the semicycloids above are inverted, JB' being the point of suspension of the cycloidal pendulum. 622. The Catenary.— This is the curve formed by a heavy flexible chain of uniform thickness and density when suspended from its extremi- THE CATENARY. 621 ties, and allowed to hang freely. Let C be the lowest point of the catenary, a= that length of chain of which the weight equals the tension at (7, <= that length of chain of which the weight equals the tension at P, «= the vertical line CM^ y=. the horizontal line, MP. s=CP. This portion CP of the chain is kept at rest by the tension at P, the tension at C, and its own weight, which is proportional to its length : these three forces are thus proportional to t, a, and s, respectively, and they act in directions parallel to TP, PM, and MT, .-.by the triangle of forces, - = S17=c°*^=T-- ••[!]' -=^l>=T----[2]- Also a PM dy "• -' a PM dy "■ " /ds\.,/dx\_s^^ dy_ a _ ads \Ty) -^'^Kd^/ -~^' ^"""^ •'• d^-::7(fT^y ''' ^~v(^h^' and since s=0 when t/=0, the integral is y=alog — ^^- -, .♦. e«=-^-^ — , and .-. e '^^^Ll — !. — i — a a a g^ y y dx 1 - ^ ...,=_(e«_g .)...[3].-.[l],-=-(e«-e «)=cotr, orcot^...[4], •••^=2/(^"' '^)dy=-^{e-+e -)-a;...[5]. ds 1 ^ ~y- a - * Differentiating [3], -=-(e-+e "), /. [2], <=-(e«+e «)...[6]. Now if iV^ be the number of which the Nap. log is -, then N=:e^, and the preceding equations give '=i(^-^)=""°'''°°*''4(^'-:^)'''=l(^+D-«='-''' y so that whatever value be assumed for - the corresponding value of N becomes known, and thence the proper values of s, fi, x, and t. These have been computed for y=lOO units of any kind, and a=a succession of equi-different values, as also for a =100, and y= a succes- sion of values 1, 2, 3, &c., And the results tabulated for use in reference to the construction of suspension bridges. For a specimen of these tables and of their practical use, the student is referred to Vol. iii. of the Woolwich Course. We must here bring these inquiries to a close ; and for further applica- tions of the Calculus, must refer the student to works specially devoted 622 NOTE ON INTEBPOLATION. to analytical Mechanics, and other physico-mathematical subjects. The writings of Professor Potter, of University College, London, are among the most recent and best that he can consult. End of the Calculus and its Applications. NOTE: ON INTERPOLATION (seep. 162). In order to be able to insert between the terms of a proposed series other terms, such that all may follow the same general law, it will be necessary to solve the following problem, namely, Peoblem. To determine the general relation which exists between two variables x and y, from knowing the particular independent cases. Let the general relation be then from the preceding known conditions, which we are to suppose to be as many in number as there are coefficients A, B, C, &c., to be deter- mined, we have /3 =:A + Ba-]-Car-+Dec^+ ^ fi,=A-{-Ba^-\-Cct^^-\-I)^t^^-\- ... &C. &c. J •[1], &c. &c. .-. ^ii:^=J5+C(a,+a)+i>(«iH«i«+«^) + =a ) [-[2], &c. &c. the values a, a^, &c., being known because the first member of each equation is known : and A has been eliminated. Proceeding in a similar way with the group of equations [2], B may be eliminated, and we shall have and so on. And by continuing this process we may thus eliminate the coefficients one after another, till we come to the last, the value of which may then be determined from the final simple equation. Suppose the general relation to be y=A+Bx + Ca!\ and that three conditions are given : the equations [2], [3J, become Substituting this value of Cin the first equation, we have NOTE ON INTERPOLATION. 623 B=a—b{ai-\-a), and since the assumed relation gires /3=^+^«+(7«', we get A=(i—aet-\-baeii, so that the relation sought is y=/3— aa+6aai+(a— &»!— 5a)a;+6x2, or y=li-\-a(x—a)-^h(x—a.){x—eti). If four independent conditions had been given, we should have got y=(i-\-a{x—ct)-^h{x—a){x—eti)-{-c{x—a){x—a^)(x—ei^...[4:], and so on. Now the general relation between x and y may evidently be such as not to agree with the form we have here assumed for it : ^ may not be a rational integral function of x : yet if we regard the equation be- tween ,v and y as that of some curve, the given independent conditions will denote so many points in that curve, and by proceeding as above, we shall determine a parabolic curve necessarily passing through all these points, and the closer they lie together the more nearly will this parabolic curve approach to coincidence with the assigned curve between the limits of the given co-ordinates. Between these limits, therefore, the general relation sought may be approximated to closer and closer — by replacing it, as above, by a relation of the assumed form — the more numerous the given conditions are, and the smaller the intervals between the successive values of x. Let these successive values a, a^, k^, &c., be in arithmetical progression, h being the common difference, 80 that ai^=a-\-h, a^=:et-\-27i, m^=a-{-Sh, &c., and let a;=«+Ai, .'. x—et=hi, x — ai:=hi — h, x—a^:=.h^ — 2^, &C. Put also /S,— /3=Aii3, /Sj— /3,=Ai|S„ ^^—^^■=z^^(i^, &c. : then [2], Ai/5 _Ai^i _Ai03 In like manner, putting \^^, A^^^, &c., for the second differences, A/Sj— A/3, A/Sj— A/3,, &c., we have by [3], A^ A^, ^=1^^^ ^^=i:2A-^' ^'' Putting also Agi?, &c., for the third differences AgiS^— AgjS, &c., we get, A^^ 1.2.3A3' Hence the formula [4] becomes &c. Or putting n for the number W-^Ji, whether this be whole or frac- tional, n(n—\) w(w— IVw— 2) y=/3+«A,/3+-^^A^+-l-_^__j!A3^+..., which is an extension of the formula at p. 160, to the case of n fractional. An example of its application is given at p. 163. End of the Elementary Course. ANSWEKS TO THE EXAMPLES FOE EXEKCISE. ALGEBRA. Definitions, p. 4. (1) 4. (2) 8. (3) 138. (4) 540. (5) 35. (6) 4. (7) 0. (8) 3. Addition, p. 5. (1) 24a. (2) 76y+9a. (3) 22^ax-7icz. (4) lllmx—ieny-lQ^. (5) l\axy-\-J}. (6) 10a«+4. (7) cz-16. '8) -Vj\axy-{-Q\hz. Case 11., p. 6. (4) 2a:c+2y4-41. (5) 2ic+3y+8a+&+20. (6) Zahx-\-10ay. (7) 6mx+lZny-\-Saz-\-20. (8) ll-+10^+3aa;+6?-3. ^ y X b (9) 15a&a;— 19aJy— 15ay+4a6z — 12ax+^. Subtraction, p. 7. (1) 2aa;+462/+7. (2) —Zabx—2z. (3) —5x2/2+8x2^+6. (4) —5mxy—7nyz-\-2a—8. (5) 15az-66y+9a;-l. (6) 3aa:-li6y-2z-6. (7) 7py—2Qaz—7n—m. (8) — 6ca;2— 9ey+4fc2+a+6— 4. Simple Equations, p. 10. (1) x=7. (2) ^=9. (3) x=q. (4) a:=4. (5) x=60. (6) a;=6. (7) x=-l^. (8) ;r= (9) x=-. (10) y=12. 6+m Questions, p. H. (6) x=210. (7) ar=40. (8) 8 J and 211. (9) £50, £30, £20. (10) 4. (11) 6f. (12) 3*- 20'"- past 12. (13) l74mUes. (14) 21^ days. (15) 120 gals. (16) 100 lbs. (17) 1st. 6 min. ; 2nd. 4 min. (18) 90 of each sort. (19) 11 half cr. 29shUgs.' (20) ^, £24; B, £36; C, £56. Brackets, p. 13. (1) [3(a+&)+l](x+y). (2) 5s/(x-a). (3) (3a+8)V(x-7/) + (a-3)V(a;+y). (4) S{2z-y^)-2x^+a-b. (5) (Za-7){x^y)-\-(a-b-i)xy-4z. (6) 9(x+y)-(a-hiXx^-y% (7) (4a-36)v'^+(6a-5J)v/y-J-<«. Multiplication, p. 14. (1) ISa^bxY (2) -28a^ly^xy7. (3) 15a363a;y. (4) -14a36ry. (5) 30a&3x/. Q (6) —-abK^yh. Case II., p. ] 4. (1) 12a362a;3_6a26xV. ANSWEES TO THE EXAMPLES FOR EXERCISE. 625 (2) -12a2Jic3/2_26a62a:2*4-10a6a:2 (3) —2a%cx-{-Sal^cxy—4:ab<>^xz. (i) ia^xhj+Sabx7/-\-12axy. (5) 3a6a^-7aV-2aa^2. 4 4 5 (6) -aPb^x^-h-xabcxf. Casein.,^. 16. (1) a;*-2a2aJ5+a^ (2) a^-l. (3) a*-a«. (4) a^-f|r'+?^+l. (5) 6a^~2Sx^-i-2Qx-8. (6) a^-y^. (7) 2a:*-44x3+5«2_4ia;+2. (8) 18aa:3-3(5a4-2)x2-(18a-6)a; +6. (9) aJa:2».^((i2_|_j2)a;»«+«+aJa;'"+». (10) i^+li^-?x+l. (11) a:«_?^3_^.i3^_l^. (12) a^x^-bY- (13) 9a2;c2_42a6a;yH-49iV. Division, p. 17. (1) -Zax^y. (2) -6a^xy^. (3) 9xY^z. (4) 26V a 7a^x^y 7a2a: (6) 36c • (b) 46y , , Qa^T*' (V) a"i«a;. (8)- >3 • Case //., p. 17. (1) a3-36a;+5«2. <2) 12x-dax^-\--. X (3) 4ax«-36y8+22*. (4) y-9. (5) 8a2a;+a-2ary. (6) 2a6' Case III., p. 18. (1) 2a:+3. (2) 2x^-^ix^-^Sx-]-16. (3) x'-\-x^7j-{-xy^+7/. (4) o*+a^a;+a'a;Hax3+jr*. (5) 6^:3+ 4^:24. 3a:+ 2. (7) l+^+^2_|_^^&c, (8) 3(^+{a-\-b)x-^a--\-ab+c (1) Fractions, p. 19. axy-^b x—2a (3)-?L. (4)(^W. ^ ^ a+6 ^ ^ a6 .^, a;3-3a;2+8a;-9 ^^> ^^2 • ,.. 2x2 2a^ a2+62-c^ ^^^-2^6-- Page 20. (1) a+26+^. (2) «^+2/+^ (3) a-S+^j. (4) 3X+2+ ^ 3ax (7) 3^+2^+2+l|i^ Addition— Subtraction, p. 21. 14 4a2 (^>^»- (^)ri^ (5) a—x a-{-x' Multiplication — Division, p. 21. 3x(2^-3) 4^ ^ ^ 20y ^ ^ 5y^ ^^ 12x (l+x2)(l-y2) ^^^^q:^- ^^^ xy ' <^^ ;,_y • ^^^'' 4(3x+4)' (11) ^^ a2) ^^^"'+^') ^ ^ 28V(«+x)' ^ ^ a«+a6+62- Simple Equations, p. 27. (1) x=i. (2) x=l. (3) x=:q. (4) a:=14|. (5) x=12. sp {6)x= ANSWERS TO THE EXAMPLES FOR EXERCISE a^—a (7) x=e. (8) a;=100. (9) x=l. (10) x=2i. (11) a:=l. (12) x=^ (15) x=a(b-l). (16) .=„(l_y^_i_). (")-=4^— (18) x4 (21) x=i. (22) a:=2. (23) a:=l. Simple Eqaations. Two Unhiotims, p. 30. (1) x=5, y=.7. (2) x=1, 2/=16. (3) a;=3, y=5. (4) a;=19, 2/=3. (5) a;=ll, 2/=4. (6) a;=8, 2/=2- (7) a;=12, 2^=6. (8) a;=13, y=3. (9) x=-, y=-, (10) a:=21, y=63. (11) a:=13,2/=7. (12) «= -02,^=2 -9. (13) x^-^1, y=-i^. (14) ^=(-^y, y=32. Simple Equations. Three Unknowns, p. 31. (1) a;=24, y=6, 2=23. (2) x=l, y=12, 2=60. (3) ar=2, y=-3, 2=1. (4) a;=12, 2/=20, 2=30. (5) x=d, y=2, 2=1. (6) a;=2, 2/=4, 2=7. (7) ^=10, y=2, 2=3. (8) a=2a, y=26, 2=2c. Simple Equations. Questions, p. 34. (1) ^, £50 ; 5, £30 ; C, £20. 3 (2) 14 rye, 36 barley. (3) (4) Man, 21^ days ; Woman, 50. (5) 32s. ; 14 poor. (6) 18, 22, 10, 40. (7) a=3, J=-l, c=-2. (8) 240. (9) 5JH days. (10) 221 lb. lighter ; 201^ heavier. (11) A, 105 min.; B, 210; C, 420. All, 60. (12) 420 oz. copper; 85 oz. tin. (13) Tin, 74 lb. ; lead, 46 lb. (14) 234. (15) A, 8oz. ; £, 5oz.; O, 3oz. (16) 160 oz.; 623SOVS. (17) Add 1 h. 5^ min. successively. (18) 8 gal, 6 gal., and 6 gal. (19) A, ^{22a-9b-8c) ; B,l^(21b -4c-6a); C, ~{20c-Zb-ia). (20) {a-hy (21) 62s. and 34«. (22) £81, 41, 21, 11, 6. Square Boot of a Poljmomial, p. 39, (1) 2x2-a;-l. (2) 3a;2-o. (3) 3a^-2x2+a:-4. (4) 2j^-i-3x^-x+l. (6) 2xy^-Bx^y+2a^. (6) -+^+1. y X Cube Root of a Polynomial, p. 40. (1) x'-2a;+l. (2) a;H2a;-4. (3) 2a;+3. (4) x^'—ax+a^ Sards. Reduction^ p. 41. (1) ivi4. (2) 2^/15. (3) 3^-6. (4) 2aa^f^3. (5) V^- (8) X V(«+M. (9) {a-\-h)^Zx. (10) ^{a^-x'). Surds. Addition — Svhlraction, p. 42. (1) 10^2. (2) ?s/10. (3) 4aV3.r. (4) 25^/2. (5) {a^x)^x. {&) as/ah. (7)^v^l5. (8) a^{a+h). (9) 2^-1. (10) 2(6^/2-f-v'6)+aV*. ANSWERS TO THE EXAMPLES FOR EXERCISE. ear Sards. Multiplication — Division, p. 42, (1) 6V432. (2) 7V15. (3) |>/3. (4) 32V2. (5) V3. (6) ^%/lSQO. o (7) n/6-|v108. (8) -Va^^- (9) a" (10) -evi2+v2). Binomial Surds, p. 43. (1) -^(2v/7+3V5). (2)i(3W3). {S)'-^^l x{a-^^x) 13,/10-42 37 (6) 3(V5-V3)(n/5+>/3) (7) 2^9+V6+V4). (3) -^---y;-^^^- (0)2^3. Binomial Snrds. Sqttare Root, p. 44. (1) ^(n/2+>/6). (2) ^(v'26+V6). (3) ^/6-2. (4) >/5-V2. (.5) V7+1. (6) 2^/7+^/14. (7) V(l + V6). Imaginary Qnantities, p. 45. (1) ^/21+2^/-7. (2) 6+v'10+(2V2-3V6)V-l. (3) 1-?^^/-!. (4) 6(1+V2)+(V2-4)V-1. (5) 3(v^5-V6)>/-l. (6) i(3V2+2V3)(l-2V-l). (7) -(117+44^/-!). (8) (a3-3a62)i:(3a26-&3)v'-l. Quadratic Equations, p. 49. (1) 4, -25. (2) 5, 21. (3) 4, -21J. (4) 3, -p (.5) 4i, -. (6) 25, 3. (7) 18, 3. (8) 1, 9. (9) 2±V5. (10) ±3, ^/-l. (11)9,-2,^^. (12) ^, £i^. (13)3,1,^11^^. (14) -l±^s/2. (15) 22, (-1)2. (16) 1, (-1)1 (17) 60, 235. (18) l,l(-3±^/-66). (19) 2,-3. (20)l(l±V3),l(l±V-7). (21) 1, -3±s/5 (22) -{_2±v'3±>/(3T4V3)}. Quadratics . Qibestions, p. 62. (1) ^, 9 m.; 5, 10 m. (2) £2, ^£3. (3) 80+10^/5, and - 10±10V6. (4) No. (5) Hyp. 20 ; Base 12 ; Perp. 16. (6) 4 and 7. (7) 8. (8) A, 30 h.; B, 20 h. (9) 16 and 24. Simultaneous Quadratics, p. 57. (1) (2) x=±-s/6r ±3 y=±lV6, ±1. a;=±3, ±->/2 y=±2, ±^^/2. (^) \^. .x=±6, T25V~330 W l2/=±5, ±30^/-330. <^ l;5 a;=8, 4 8. (6) y=-i, a;=l, 3, 2±5V-1 ^'' ly=3, 1, 2T5>/-1. S S 2 628 (8) ANSWERS TO THE EXAMPLES FOR EXERCISE. (2) 5465. (3) 49. (4) 2^\. (x=i, -2, l±v/-15 {y=2, -4, _1±^-15. cc=8, 4 <^) i::. (10) x=9, 6k y=i, 6i. (11) (12) 17±v/-283 =4, -1, ^=1, -4, •17+^-283 (13) 1,^1. (14) 1,=^. Miscellaneous Questions, p. 57. (1) 19, 9, 6. (2) ^(3±V6), ^(1±n/6). (3) 36. (4) 841 and 121. (5) 61 or 12. (6) Fore, 4 yds. ; hind, 5 yds. (7) iv^5andi(5+V5). 5 (8) Time, 4 h. and 6 h. Vel. - miles. „ (l±N/5)m (9) The roots of equa.z^ z (1±n/5K^0 "^ 2 * (10) 5, 12. (11) S=p^-2q. (12) S=p*-Ap\+2p^. (13) 3+n/(9+8w) (14) 2304, 1296. (16) No. (16) 6, 7, 8, 1 (17) 93. (18) ^^-^ 3W5 l±v^5 (19) — ^— ' 2 (20) ±1, ±3, ±5. 46 Arith. Prog., p. 62. (1) 31. (2) -. (3) 0. (4) -3. 5 17 2 3 (5) 140 or 280. (6) -, -, -» 3. 4- (7) -35. (8) 4. Geom. Prog. , p. 64. (1) This is an Arith. Prog.; 34. (5) ^;— ^. (6) ±3, 4^, ±6|. (8) 11. x-l (7) 6, 18, 54, 162, 486 Haxmonic Prog., p. 65. (1) 4, 6. (2) 3, 2% 12. (3) 3, 1. (4) 3, 4, 6. Prop, and Prog. Questions, p. 66. (1) 4 days. (2) 6 m. 6 fur. 20 yds. (5) 7 nights ; 63 yds. (6) £5825. 8s. S^d. (7) 9172 2293 33300' ^^ 8325' (8) 2 gal. from A, 12 from B. (10) 1, 3, 5. (11) 1, 2, 4. (12) 3, 9, 27, 81. (13) 2, 10. (14) 234. Piling of Shot, p. 69. (1) 364 and 650. (2) 392. (3) 3640. (4) 23405. (5) 6146. Binomial Theorem, p. 76. (6) a^-15a*b^-\-15a?¥-l^-\-{6a^b- 20a^b''+6al^)s/-'i.. (7) Z2a^x^-240a*x*y-\'720a^xY- 1080a2a;V+ 810aan/*-2433r^. ^„, 1 /, 2x , dx' 4x^ . \ (8) -/I +— „-+... ). (9) l-3a;2-f6x*-10x«+.... /..v 1/, . 6^ . 24a;2 80a:3 ^ (10) -^(i+— +— T 3 +...)• Binomial Theorem, Case II., p. 76. « ^+F6- 3u,-3 2.463 ' 2.4.66«^ ^ ^ +362 3.665"^ 3. 6. 96"" 3.5a;=» 3x2 "*"2.4a2+ 2.4.6a3 ...). ANSWERS TO THE EXAMPLES FOR EXERCISE. 629 ^ ^ Va^'* 2' 2V^ 2V *••/ <^^ ^+3^-3! (8) 2-^t:+ ,6.2* ' 3.6.9.27 2 2.5 3.2 ' 3.6.2^ 3.6.9.2* a^(DW...}. x^+ Interest, Annuities, &c., p. 88. (1) 4perc. (2) 12-04. (3) 22^ yrs. (4) £742. 6s. (5) £681. lis. 7d. (6) £551. 15s. 3d. (7) 25. (8) £822. 14s. 6d. (9) 7 yrs. nearly. (10) 24418270. Permutations, p. 90. (1) 362880. (2) 3628800. (3) 15120. (4) 1260. (6) 64. (6) 340. Combinations, p. 93. (1) 330. (2) 84. (3) 252. (4) 4, 70. (.5) 22. (6) 7. (7) 49639590. (8) 93600. Probabilities, p. 99. 10 31 27 (*>? <^)25- <^^-2- <^>loro- (8)ji^. (9)3Tr,7^. (10)1. W^' (12)? and I Theory of Equations, p. 104. (1) Q=3a;H4a-H13a;+19; i2=19. (2) Q=7x3+8xH28a-+90; 72=268. (3) Q=5a;3-37x24-218x-1302 ; 72=7799. (4) Q=7x^-dSx; R=-8U. (5) Q=xHll«^+47a;2+205a;+830 ; i2=3306. Trans, of Equations, p. 112. (1) x3-6x2-9x+54=0. (2) :t3-7a;+6=0. (3) 6a;3+51a;2+148a;+146=0. (4) 12x34-42x-2-25a;+3-5=0. (5) 19a?^+206ic3+793x2+1232.c+ 580=0. (6) a:^-7x-^7=0. (7) x2-i^,2^iV=0. (8) 2a;*-6x2+3a;-l=0. (9) a;=»-a;2-34a;-56=0. (10) a;3-6a:24-ila;-6=0. (11)2,-3. (12) 2, 3, 6. (14) 3a;«-6r2+2=0. (15)l,zli^3^Z±^^ (16) ±1, ±V-1. (17) a)«-3;c^+2x2_7^_5_o. (18) a^=-(^4+ai+a2+a3+a4). Solution of Equations, p. 134. (1) 3-2213261. (2) 2-4384783. (3) 1-0560897. (4) 2-209753. (5) 2-35833. (6) 2-3027756. (7) 1-284724. (8) 8765. (9) 2-080083823. (10) 9-886016. (11) -316604. (12) 1-283137. (13) -2-35833. (14) -5-018424. (15) -3-9073786. (16) -1-876837. (17) -1-584418. (18) -5-2467839. (19) 2-557351, 23-213112, 23-229537. (20) 2-8580833, -4432769, '6060183, -3-9073785. (21) All imaginary. (22) Two positive ; two imaginary. (23) All imaginary. (24) One positive ; four imaginary. Decomposition of Equations, p. 141. ^3 + ^(15-4^^3)2 2 (1) x=-l±- _, . V{2(9±V33} (2) x=2± 630 ANSWERS TO THE EXAMPLES FOR EXERCISE. (4) X=Z, 8±s/21 2_^_1±^{_1_V-1} (5) (6) 2+V-l±s/{-l+V-l} ^+{iw(^^+!)}^+^. Recurring Equations, p. 144. (1) «=!, 2 • 1 5±12v'-l (2)«:=2,- 13 (3) l±V-3 1 ^ 2. (4) -1, 14-N/-3 l-x/-3 ■1±n/-1 1±n/-1 (6) 1,-1,-1,+n/-1,-V-1. Vanishing Fractions, p. 145. (l)i (2) I (3)i (4)1». (6)t (6)^. maxima and Minima, p. 150. 1 2" -2^ (1) -a and -a. (2) No maximum the least possible are -a, -a. (3) ly/a and v^ a. (4) -. (5) Wliena?=-. (6) x=-. (7) No. (8) 1 and 2. (9) 8 and 4. (10) |. (n)x=i. (12) .=|.?. Indeterminate Equations, p. 153. 'a;=9, 20, 31, 42. Ja;=9, 20, 31, 42 ^■^^ (y=19, 14, 9, 4. y=9. <2) i::^^' (3) 4. (4) No. (5) 190. fa:=2, 4, 6, 8, &c. ^ ^ l2/=8, 11, 14, 17, &c. Indeterminate Equations, p, 154. (1) 10. (2) 24. (3) 1, 3, 6, or 2, 1, 7. (4) xz=5, y=l, 2=2. (5) 8, 10, 2 oz., or 10, 5, 5 oz. (a;=15, 50. (6)]y=82, 40. i 2=15, 30. Least Integers, p. 155. (1) 43. (2) 503. (3) 104. (4) 7691. (2) Indeterminate Coefficients, p. 158. (1) 4.r(x+l)-8a;+l. _2 ]_ a;+3 x+2' x-1 3(xHa;+l)* (4)?+-^-^-^. (1) Finite Series, p. 160. w(w+l) (2/1+1) (2)(^'y. (3) »-<2±ii(^^ (4) 430. (5) nK 2.3 (6) (7) w(w+l)(w+2)(%-f3) ?i(n+l)(?t+2)(w-f3) 2. 3. 4. (8) «2(2„,2_1). Infinite Series, p. 163. (1)| (2)1. (3)i. (4)" 18 (5)^. ml (7)1 ANSWERS TO THE EXAMPLES FOR EXERCISE. 631 (1) (3) (4) Recurring Series, p. 166. 1+iC ,^^ l+x (2) 1-a;- (l-xf l-\-x a-xf Convergency of Series, p. 169. (3) Divergent. (4) x=0, and ic=l. 1 (5) Limit= 192 PLANE TKIGONOMETKY. Bight-angled Triangles, p. 179. (1) 164-4, 296. (2) 60°9'. (3) 95-365, 120-097. (4) 384-05, 480-036. (5) 40°16', 72. (6) ^=54°36' 14", e=35° 23' 46", ^(7=92-23. (7) C=37° 3', ^^=138-24, i?(7=104-34. (8) 63°26'6", 26°33'54". (9) 66*78 ft. (10) 8000 miles. (11) 219 -8 ft. (12) 56 -649 ft. (13) Dist. 4100 '4 ft. Height, 289-125 ft. (14) Diam. 7916 miles. Dist. 138 miles. (15) 935-76 yards. Oblique Triangles, Caae /,, p. 184. (1) 22°37', 351,648. (2) a=309-595, 6=436-844. (3) C=61°13'47", a=101-617. (4) ^=45n3'55", or 134°46'5". f 41°28', 106°32', 72-36 ; or ^ ' ll38'=32', 9°28', 12-415. Oblique Triangles, Case II., p. 158. (1) ^=103°19'21" ; £=44°22'13" c=288-894. (2) ^=53°54'11" ; 5=99°53'33" ; c=9-729. (3) 5=26°52'42", {7=21°0'6", a=3387-974. (4) .4=62°15'25", B=ZrU'B5\ c=632-08. (5) ^=18°21'20", C7=33°34'40", 6=2400-036. (6) ^=98°9'16', i?=36=49'40", c=1194-7. Oblique Triangles, Case III., p. 189. (1) ^=54°33'24". : (2) 41°24'35", 55°46'16", 82°49'9". " (3) ^=40°33'12", i?=78°13'l", C=6ri3'47". (4) 4=37"59'53", B=52°Z7'i7'\ (7=89°22'20". Miscellaneous Problems, p. 189. (3) i>^=301-01, 2)(7=629-101, i»5=719-622. (4) 1-49 mile. (5) Height=210-44ft. Dist. =250 -79. (6)2)^=5m.,i>a=4-892m.,Z)5=7m. (7) 621ft. (8) 84 ft. (9) P^ =709 -33, PC=1042 -66, Pi?=934 yds. (1 1) A=63 -964 ft. (12) 44 -46 yds. Applications of Formulse, p. 206. (13) ^=16°. (14) Bin^=iv(-2±V5). (16) ^=15°. (16) a;=18'' or 64°. (17) x=i5\ (18) mP-n^=i^mn. Trigonometrical Series, &c., p. 224. (1) cos ^+»/— 1 sin ^. 2 2 (2) cos -^— v'— 1 sin -6. o 6 (4) sin{0+i(^-l)(«+^)}X sin - ri(«+ !r) -^ sin - (a + «•) . (5) ;S'=tanw^, (6) f. 1 ah' — a'b i( X 7 ). \ a— a / ah'— ah Problems on the Circle, p. 308. (1) A circle of half the diameter. (2) y-l=(3±2V2)(l-x). / am (6) A circle of centre ( : asjini > and radius=: m— 1" 634 ANSWERS TO THE EXAMPLES FOR EXERCISE. (8) 4(l + «i2)(V+&^+c)= (2ai6j4-aii+a)"'^. CONIC SECTIONS. Problems and Theorems, p. 325. (1) Equa. of asymptotes, x=.2, y= — Point, 3/=— o- (2) A parabola. (3) An hyperbola. (4) An hyperbola. (5) An ellipse. (6) An ellipse. STATICS. Examples, p. 371. (1) R=V191b. (2) R=135-651b. (3) P=10, 14-14, and 5-17 lb. (6) 2-354, and 5-18 ft. (6) 6'571b. (7) 30-62 lb. (8) 60-9 and 49 -02 lb. i(\\ X. A cos a (9) tan^= 2 —sin « (10) p-w s/{<^'^'+{a-Wdi=-«*- (3) ^=6-'+12^. (^>l=-- a^* ^ ^ dx~ { i = {l-x^l ^^^ dx'^'H^x x' ^ ' dx (3+a;2)V«* 636 ANSWERS TO THE EXAMPLES FOR EXERCISE. Log. and Exp. Functi (1) dy dx" =l+loga;. (2) dy dx~ " log a; \ lo (3) dy _ dx~ =loga.a<^*c. (4) dy_ dx' a -x^ia'+a^' (5) dy_ dx~ 1 (6) dy_ dx" >Jax ' {a-x)x (7) dy _«'«.? 'log a dx~ X (8) dy_ dx nxn-^e'. ^a+xs) wt= ' dx a;V(l+a^)' Trig. Functions, p. 502. /-.^ dy . ^^^ c??/ 1 (1) —=—asmax. (2)—= — sma;. (ix dx X (3) ^=(l+a:sec2^)etanx, "• dx (5) ii= ~ dx ^{l-a^x^)' (6) ^ = ^ . ^ ' dx x/(2-e-) 2 (8) ^ = ^ ' dx Xs/(^,n^x^-\) Integration, p. 505. (3) -(4^+3)^+(;. (4)-log!C(a'±<.)J.« (7) 27 . (8) 2^9 \ "3/ Maclaurin's Theorem, p. 525. (2) sin-..=x+|-;+ll^%... (3) cos""'a;= x ... (4) (tan«;)*=a:''+ — +— + ... (5) cos"a;4-*2'»cos"-^;j:sina;4- — - — ^co%^~^x%\x^x-\- . . . 2 (6) (a4-6a;+ac2+...)"=a«+^''~^i« x2 u n(.-l)(.-2) ^ 2.3 +7t(»i— l)a*'~"-6c a^'+... * This example belongs to the case of ai't. 536. ANSWERS TO THE EXAMPLES FOR EXERCISE. 637 Compoimd Functions, p. 533. ,„. (dii^ , xdy m i<^H- I ^ ^ Xdxi ^{x^-2axy (4) |^|=(coscc)«!n'X f , sin^a:;') ■J cosxlog cos a; }-. I cos iC j ^^^ XTxi-VTTx dx Implicit Functions, p. 534. ^'dx y' d;^^ y^ ^ ' dx y-x dx^ {y-xf ~ dy dx~ dy_ dx' y= a? (4) Zx-2h-a - ^{x-b) • y (6) VanisMng Fractions, p. 639. (1) 0. (2) -| (3) 0. (4) -1. (5) \ (6) 0. (7) 0. (8) 1. (9)-. (10) -1. (11) 1. (12) 1. VanisMng Fractions, p. 541. V •1, -2. Maxima and Minima, p. 547. (4) a=h. (5) x=e. (6) The equal conjugates. (7) Rad. of base =-V'2Xalt. (8) r=-. (9) 4^ in. (10) 109°28'. AT (11) 116°14'21". (12) x=% a min, ; i/=— 2: that is, for adjacent values of x, y will be a greater negative number than for a;=2. The Pendulum, &c., p. 616. (1) \sJ2. (2) r^l (3) V4 (^) "^ 5a (7) -055 in. (8) 1 mile. (9) 33-885 sees. (10) 978 ft. (11) 1-73 sees. (12) x=-{.y{^'^-\-ww)-w}. w Page 572. The co-ordinates of the four cusps are a=0, i8=± — r-, and ^=0, x=± b a^-h^ THE END. u^ nv GENERAL LIBRARY UNIVERSITY OF CALIFORNIA— BERKELEY RETURN TO DESK FROM WHICH BORROWED This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. APR 6 1954 LI 5i3 RECEIVED DEC b '66 -1PM LOAN DEPT [ULi&i967gB ' LD 21-100m-l,'54(lQ878l6)476 Young. Elem. msctb mialrlcff OCl 28191.iD^^..^.Ju(M W O 19 1^ rEe li 1920 ^v4 i h 845^^ UNIVERSITY OF CALIFORNIA LIBRARY