yf^\ ^ 7 C^^l'^cZt^f^^^-1 ^^2 Classification of Lines ^^'^ Equation when the Origin is in the Curve 163 CONTENTS. Xi BOOK VI. SPACE — POINT AND LINE — PLANE — SUBJACES. Space defined 165 Co-ordinate Planes — Axes — Angles 165-171 Distance between two Points 171 Line and Co-ordinate Axes 172 Equations of a Straight Line in Space 173-176 Interpretation of the Equations 175-177 Equations of a Line passing through two Points 177-179 Lines, Intersecting and Parallel 179-lbl Angle between two Lines 181-185 Examples in Construction 185-186 OF THE PLANK. Equation of a Plane defined 186 Equation of a Plane 186-188 Traces of a Plane 188-190 Line Perpendicular to a Plane 190-191 SURFACES OF THE SECOND ORDER. Equation of a Surface defined 192-194 Surfaces of Revolution 194 Equation of Surfaces of Revolution 196 Sphere — EUipsoid — Paraboloid — Hyperboloid 196-198 Surfaces of Single Curvature 198 Equation of the Surface of the Cylinder 199 Equation of the Surface of the Cone 199-200 Intersectionofa Conic Surface by a Plane 201 Circle— Ellipse— Parabola— Hyperbola 202-204 ANALYTICAL GEOMETRL INTKODUCTION. 1. There are four kinds of Geometrical Magnitude : viz. Lines, Surfaces, Volumes, and Angles. In Geometry these magnitudes are presented to the mind by a pictorial language, and their properties are deduced by constant reference to the magnitudes themselves. Instead, however, of keeping the magnitudes constantly before the mind, we may, if we please, denote them by Algebraic symbols. Having done this, we operate on these symbols by the known methods of Algebra. The results obtained, will be equally true for the geometrical magnitudes and the symbols by which they are represented. We next interpret these results ; and this leads to the development of the properties of Geometrical Magnitudes. GEOMETRICAL CONSTRUCTIONS. ' 2. The coNSTBUcnoN of an Algebraic expression, is the operation of finding a geometrical magnitude equivalent to it. Construction of expressions of the first degree. 3. Construct the expression, a ■+■ b. 14 ANALYTICAL GEOilETEY. Draw an indefinite right line AB. From any point as A^ lay \ • \—B off a distance AG equal to «, and then from (7, a distance CD equal to J, and AD will be equivalent to a -\- h. 4. Construct the expression, a — h. Draw an indefinite right line AB. From any pomt as ^, lay ^ t; ~t7~'^ off a distance AD equal to a, and then from jP, a distance DC, in the direction toicards A, equal to 5 ; AC will then express the difference between a and J, and hence, is equivalent to a — h. If 5 is greater than a, a — h wdll be essentially negative : A C i \ 1— ^ will then be negative, which is shown by the point C falling at the left of A. We have, in this example, the geometrical interpretation of the negative sign: viz. If distances in one direction are regarded as 2^ositivey those in a contrary/ direction must be regarded as nega- tive* 5. Construct the expression, a — b ■}- c — d. Draw an indefinite line AB, From any point, as ^4, lay off Jr j^ "^ ^ ^i ^ the distance AC equal to a, and then the distance CD^ in the opposite direction, equal to * Bourdon, Art. 89. University, Art. 96. Note. — All the references are to Dayies' Bourdon, Davies' University Algebra, and Davies' Legendre, Geometry, and Analytical Trigonometry. INTRODUCTION. 15 b. From Z>, lay off BE^ to the right, equal to o, and fthen from jEJ lay off JEF^ to the left, equal to d. The line AF will be equivalent to the algebraic expression. It will be negative, because the sum of the negative terras, in the algebraic expression, exceeds the sum of the posi- tive terms, and this is indicated by the direction of the line AF From the above examples, we conclude that, Every algebraic expression of the first degree will repre- sent a line; whence, it is called, linear, 6. Construct the expression, — Draw two indefinite right lines AJB^ AF, making any angle with each other. From A, lay off a distance AG—Cj also the distance AJ3 = a. Then from A, lay off AD = b; join C and Z>, and through J] draw JBF parallel to CD; then will AF be equivalent to the given expression. For, we have by similar triangles,* AC that is, therefore, AB : : AD : AF; a :: b : AF; AF = ab ^ Construction of expressions of the second degree. 7, Construct the expression, ab. The degree of a term is the number of its literal factors.f Hence, ab is of the second degree. * Legendre, Bu. IV. Pi op. 15. j bourdou, Art. 25. Univ. Aj-t. 12. 16 ANALYTICAL GEOMETKY B B Draw the indefinite straight line AJB. Lay off, from A to Z>, as many units of length as there are units in a. At X>, draw DC perpendicular to AB^ and make it equal to as many units of length as there are units in h. Then, the rect- angle ADCE^ will contain as many units of surface as there are units in the expression, a X 5. Hence, we con- clude that, Every algebraic expression of the second degree repre- sents a surface:^ I Construction of an expression of the third degree. 8. Construct the expression ahc. Draw an indefinite line, and lay off AB equal to the number of units in a. Draw, in the plane of the paper, jB(7, perpendicular to AB^ and make it equal to h. At (7, suppose CD to be drawn perpendicular to the plane A a B of the paper, and made equal to c. Then, having drawn the other lines of the figure, ABC-D will be a rectangular parallelopipedon equivalent to the expression ahc. Hence, Every algebraic expression of the third degree will repre- sent a volume.] Construction of expressions of the zero degree. a 9. Construct the expression. b * Logendre, Bk. IV. Prop. 4. Sch. 1. f Bk. VII. Prop. 13. Cor. INTRODUCTION. IT Since there is one literal factor in the numerator, and one in the denominator, the quotient is an abstract number; and hence, con- tains no literal factor: therefore, the degree of the term is 0. Draw AH^ and make it equal to h. At J9^ draw JZ5, perpendicular to AH^ and make it equal to a, and join A and B. Then, - = tan A to the radius 1.* o If a were made the hT]^othenn«!o, - would denote the cosine (7, or sin i? a to the radius 1. Hence, Every algebraic expression of the zero degree represents the s,ine^ cosine^ tangent^ cfic, of an arc or angle^ to the radius 1. It follows, therefore, that every abstract number has a geometrical interpretation; for, it will always denote some function of an arc described with the radius 1. Homogeneity of terms. 10. We have seen, that there are four kinds of alge- braic terms, wliich may represent geometrical magnitudes;-. viz. terms of the 1st degree, which represent lines; terms of the 2d degree, which represent surfaces; terms of the third degree, which represent volumes ; and terms of the zero degree, which represent the functions of angles to the radius 1. Since no other magnitudes occur in geometry, no alge- * Legendre, Trig. Art. 37. 18 ANALYTICAL GEOMETRY. braic term of a higher degree than the third, can have a geometrical equivalent. K such a term occur, we can only find its geometrical equivalent by regarding all the factors but three, as numerical. Since like quantities, only, can be added or subtracted, it follows, that if two or more terms are connected to- gether by the signs + or — , they must be homogeneous,* If they are not so, in form, it is because the geometrical unit of length, generally denoted by 1, has been omitted in the algebraic expressions, wherever it has occurred as a factor or a divisor; and this must be restored before finding the geometrical equivalent. Thus, if we have, ah + c, the first term is of the second degree, and the second, of the first degree. The degree of the second term is changed (without altering its numerical value), by intro- ducing the linear unit 1, as a factor, and we then have, ah + \ X c^ vhich is a homogeneous expression. If we have the expression, a + he — dfg, it may be made homogeneous by introducing the factors of 1 ; we then have, Ixlxa+lxJc — dfg\ which is homogeneous. • Bourdon, Art. 26. University, Art. IS*, INTRODUCTION. 19 EXAMPLES IN CONSTEUCTION. 1. Construct the expression, a^ + li^. Draw an indefinite right line, and lay off -4 (7 = 5. At (7, draw CB perpendicular to AG, and make it equal to a, and draw AB : 2 then, AB will be the equivalent of 2. Construct the expression, c^ — h"^. Draw an indefinite line, and on it lay off AB = c; and on AB^ as a diameter, describe a semicir- cumference ACB, With -4 as a centre, and a radius equal to b, describe an arc, intersecting the circumference at C Then draw AG and GB. 2 2 Now, since A GB is a right-angled triangle, AB —AG — 2 — 2 is equivalent to GB ;f hence, GB is the equivalent of c2 - 52. 3. Construct the expression, -y/ab, and y^. Draw an indefinite right lino ABGj and from any point, as Aj make AB = a, and then BG = b. On ^ (7, as a diameter, describe a eemi-circumference, and from J?, draw ^Z> perpendicular to AG, intersecting the circumfer- ence at 1) : then, BB will be the equivalent magnitude. ' • Leg., Bk. IV. Prop. 11. f Leg., Bk. IV. Prop. 11, Cor. 1. 20 AXALTTICAL GEOilETET. For, BD^ = AB X BC" = a x b; hence, by extracting the square root of both members, J3D = yoJ. If we have -)/«, we have simply to introduce, under the radical, the factor 1, thus making the expression of the second degree. "We then have, yo" = -/a x 1. Making AB = a, and BG = 1^ we have the same con- struction as before. 4. Construct the roots of the equation of the first form, £c2 -f 2px = q.\ After making the second member of the equation homo- geneous, and placing it equal to ^^, we have, x^ 4- ^jyx = I X q = b\ This equation can be put under the form, x{x +■ 2/?) = 52 . from which we see, that 5 is a mean proportional between X and X + 2p. To construct these values of X, draw AB, and make it equal to b. At B, erect the perpendicular BC, and make it equal to j9, and join -4 and C. With C as a centre, and CB as a radius, describe * Leg., Bk. lY. Prop. 23, Cor. 2. f Bour., Art 117. Univ., Art U7. INTRODUCTION. 21 a semi-circumference cutting AC ia ^, and AG produced, in J) I then will A£J be equal to x. For,* AJE:{AB+ 2EG) = Xe' = b^; or, x{x -f 2p) = b\ Finding the roots of the given equation, we have, jc' = —p + V&M^S and jb" = —p — y/b^lTj^, Having constructed the triangle AB C, as before, A will represent the radical part of the values of x. For the first value of x, the radical is positive, and is laid off from A towards C: then — ^ is laid off from C to JBIy leaving AJEJ positive, as it should be, since it is estimated from A towards C. For the second value of a?, we begin at D, and lay off DC equal to — j9 ; we then lay off the minus radi- cal from G to A^ giving — DA, for the second value of X. Let us now see if this value of x will satisfy the equa- tion, — x{- X + 2p) = ^.2, or, - AD(- AE) = l\ or, AD X AE = AB . Hence, the two values of x, are + AE, estimated from '^A towards D, and — DA, estimated from D towards A. 5. Construct the roots of the equation of the second form, a^ _ 2px = q = 1 X q = b^, * Legendre, Bk. IV. Prop. 30. 22 ANALYTICAL GEOMETEY. Finding the roots of tliis equation, we have, and x' = JO — V^^ + J»^. Having constructed the x' = 'p-\- ^/IP- + x>\ To construct these values of x. figure, as in the last exam- ple, the first value of x will be the line AD^ estimated from A to D. The second value will be 4- EC — CA, the first esti- mated from E to (7, and the latter from C to ^ : this leaves, for the reduced value — EA, estimated from E to A. The positive root, in the construction for the first form, corresponds to the negative root in the construction for the second; and the negative root in the first, to the positive root in the second. This is as it should be, since either of the forms changes to the other, by substituting — X for X. 6. Construct the roots of the equation of- the third fornL aj2 + <2px = — q = — 1X2'= "-^^' Solving the equation, we have, x' z= —p + vi>2 _ 52^ and x" z= — p — ^/^-^, To construct these values, draw an indefinite right line FA^ and from any point, as A^ lay off" a distance AD = — jo, and, since p is negative, we lay off its value to the left. At J9, draw DC perpendicular to INTEODUCTION, 23 "FA and make it . equal to h. With (7 as a centre, and CB — p as a radius, describe the arc of a circle cutting FA^ in B and E. Now, the value of the radical quantity will be BD or DE. The first value of x ^vill be - AB plus DE^ equal to — AE. The second, will be — AB + ( — BB^ equal to — AB : so that both of the roots, being negative, are estimated in the same direction fi-om -4, to the left. Therefore, the two roots are — AE^ and — AB. *I, Construct the roots of the equation of the fourth form, a;2 — 2jtxc = — q = — 1 x q = — b\ Solving the equation, we have. p + V^2 _ j2^ and x" = p — Vp2 b^. A ^ --....D_^,'-B To construct these values of x. Construct the radical part of the values of x, as in the last case. Then, since p is ^ positive, we lay off its value AB^ from A towards the right. To AB, we add BB, which gives AB, for the first value of X, K from AB, we subtract BE, the remainder, AE, is the second value of x. Both values are positive, and are estimated in the same direction, from A to the right. In the last two forms, if p and h are equal, the two ralues of x become equal to each other.* The geometrical construction conforms to this result. * Bourdon, Art. 11*3-117. University, Art. 146. 24 ANALYTICAL GEOMETRY. For, when p = b, the arc of the circle described with the centre C\ will be tangent to AJ^, at JD'j and the two points ^ and B will unite, and each root will become equal to AD. If P is greater than p^, the value of jc, in the last two forms, will be imaginary.* The geometrical construction also indicates this result. For, if b exceeds />, the . circle described with the centre (7, and radius equal to />, will not cut the line AB, Hence, T/ie imaginary roots of an equation give rise to con- ditions in the construction which cannot be fulfilled/ and t/iis should be so, since the imaginary roots can never appear, unless the conditions of the equation are incon- sistent with each other. Bourdon, Art. 116-117. University, Art U6. ANALYTICAL GEOMETRY. BOOK I. POINT AND STRAIGHT LINE PROBLEMS TRANSFORMATION OE CO-ORDINATES POLAR CO-ORDINATES. Definitions. 1. Analytical Geometry is that branch of Mathematics which has for its object the determination of the forms and properties of the Geometrical Magnitudes, by means of Analysis. 2. In Analytical Geometry, the quantities considered may be divided into two classes : 1st. Constant quantities^ which preserve the same values in the same investigation; and, 2d. Variable quantities, which assume all possible values that will satisfy the equation which expresses the relation between them. The constants are denoted by the first letters of the "alphabet, a, b, c, &c.; and the variables, by the final let- ters, X, y, z, &c. 3. The terms, straight line, and plane, are used in their most extensive signification. That is, the straight line is supposed to be indefinitely 2 26 ANALYTICAL GEOMETRY. [bOOK I. prolonged, in both directions ; and the plane, to be indefi- nitely extended. Points in the same plane. 4. We shall first explain the manner of determining, by the algebraic symbols, the position of points in a given place. For this purpose, draw, in the plane, any two lines, as X'AJl, TAT\ intersecting at A, and making with each other a given angle, TAX. The line XX, is called the axis of abscissas, or the axis of X; and YY^, the axis of ordi?iates, or the axis of Y, The two taken together, are called the co-ordifiate axes ; and the point A, where they intersect, is called the origin of co- ordinates. 'The angle YAX is called, the first angle; YAX\ the second angle ; XA Y^ the third angle ; and Y'AX^ the fourth angle. First Angle. 5. Let P be any pomt in the given plane. Through P, draw PD, parallel to AY^ and PC, parallel to AX. Then, AD, or CP, is called the abscissa of /- — -, the point P ; PD, or A C, is called the ordinate of P; and the lines PP, PC, taken to- gether, are called the co-ordi- nates of the point P. J BOOK I.] POINT AND STRAIGHT LINE. 27 Hence, the abscissa of any point, is its distance from the axis of ordinates, measured on a line parallel to the axis of abscissas; and the ordinate of any point, is its distance from the axis of abscissas, measured on a line parallel to the axis of ordinates. The co-ordinates may also be meas- ured on the axes themselves. For, AD, AC, are equal to the co-ordinates of the point P. The co-ordinates of pojipts are designated by the letters corresponding to the co-ordinate axes ; that is, the abscissas are designated by the letter x, and the ordinates by the letter y. 1. K the co-ordinates of a point are known, the position of the point may be found. For, let us suppose that we know the co-ordinates of any point, as I*. Then, from the origin A, lay off, on the axis of abscissas, a distance AD, equal to the known abscissa ; and through D, draw a par- allel to the axis .of ordinates. Lay off, on the axis of ordi- nates, a distance AC, equal to the known ordinate, and through C, draw a parallel to the axis of abscissas ; the point P, in which it meets DD, will be the position of the point. 2. When the co-ordinates of a point are kno^vn, the point is said to be give?i ; and we have, X = a, and y — h ; these are called, the equations of the point. Hence, The equations of a poi7it are the equations which ex- press the distances of the point from the co-ordinate axes. 28 ANALYIICAL GEOMETRY. [bOOK I. Second Angle. 6. Let US consider the given point P', in the second angle, TAX'. Tlie abscissa of this point is y CP\ 6r AI)\ and the ordinate, pi I P'D\ or AC. Since distances es- / Jc timated at the right of Y, have „, / j been regarded as positive, those ^ M at the left, are nes^ative;* hence, / the equations of the point P', are, JB = — a, and y = J. Third Angle. 7. Let US consider the given point P", in the third angle. The abscissa of this point is CP", or AD\ and negative. The ordinate is D'P", or AC, Since distances above the axis of ~ -Zi have been regarded as posi- _ tive, those below it are negative ; ^"\ hence, the equations of the point P', are, JC = — a, and y =: — b. Fourth Angle. 8. Let us consider the given point P"', in the fourth angle. • Bourdon, Art. 89. Universitv, Art. 96. i BOOK I.] POINT AND STRAIGHT LINE. 29 The abscissa of this point is C'P'", or AD, and positive^ The ordinate is DJP"', or A0\ and negative ; hence, the equa- tions of the point jP"', are, X = a, and y = b. Therefore, the following are the equations of a point in each of the four angles : 1st angle, 2d angle, 3d angle, 4th angle. X = +a, X — — a, X = — a, X = +«, y = ■\-b. ,V = +b. y — -b, y — —b. We see, by examining these results, that the signs of the abscissas in the different angles, correspond to the algebraic signs of the cosines, in the different quadrants of the circle ; and that the signs of the ordinates, correspond to the alge- braic signs of the sines.* ^ EXAMPLES IN CONSTRUCTION. 1. Determine the point whose equations are, jc =r 3, and y = 2. Having drawn the co-ordinate axes, lay off, on the axis of -X^ a distance AB, to denote the unit of length. Then lay off, from A to Z>, three times the unit of length. From A, on the axis of X M * Legendre, Trigonometry, Art. 69. 30 * ANALYTICAL GKOMETEY. [bOOK I, y^ lay off a distance AJE^ equal to twice the unit of length. Through D and JE^ draw parallels to the axes, and their point of intersection, P, will be the required point. 2. Determine the point whose co-ordinates are, JB = — 5, and y = 4. 3. Determine the point whose co-ordinates are, a = — V, and y = — 8. 4. Determine the point whose co-ordinates are, ic = 4, and y = — 6. Co-ordinate Axes, and Origin. 9. Since the ordinate of a point is its distance from the axis of -Z", the ordinate of any point of that axis must be zero. Hence, the equations of a given pointy in the axis of JXy wiU be, X = ± a, and y = ; the" plus sign before a, being used when the point is at the right of the origin, and the minus sign, when it is at the left. If we attribute to «, all possible values between and -f 00, the equations will embrace all points of the axis of -Zj at the right of the origin; and if we give to a, all values between and — oo, they wiU embrace aU points of the axis of -Z", at the left of the origin. Both these coiv. ditions are expressed by the simple phrase, X indeterminate. Hence, for all points in the axis of -ZJ we have, X indeterminate, and y = 300K I.] POINT AND STBAIGHTLINE. 31 10. For a given point of the axis of Y^ we have, a; = 0, and y = ±h', the plus sign before J, being used when the point is above the axis of JT, and the minus sign when it is below. For all points in the axis of Y^ we have, a = 0, and y indeterminate. 11. Since the origin of co-ordinates is in the axis of Yt its abscissa is zero; and since it is in the axis of JT, its ordinate is zero. Hence, the equations of the origin are, ic = 0, and y = 0. Straight lines in the same plane. 12. T/ie equation of a line, is an equation which eocpresses the relation between the co-ordinates of every poi?U of the line, - Equation of a straight line. 13. Let A be the origin of co-ordinates, and AJC, AY, the co-ordinate axes. Through A, draw any straight line, as AP, making with the axis of ^ an angle denoted by a. Denote the angle YAJT, included by the co- ordinate axes, by /3. Take any point of the line, as P, and draw PD parallel to the axis of Y: then, PD will be the ordinate, and AD the abscissa, of the point P, 32 ANALYTICAL GEOMETEY. [nOOK L Since PB is parallel to the axis of ordinates, the angle APD is equal to PAY\ that is, equal to /3 — a. / P Since the sides of a triangle are to each other as the sines of their opposite angles,* we have PD : AB : : sm a : sm (/3 — a). But PB is to AB^ as y, the ordinate ot any point ot the line AP^ is to the corresponding abscissa x\ therefore sin a : sin (/3 — a), which gives, y = sm a sin {,3 — a) «; and this is the equation of the straight line AP, sinco it expresses the relation between the co-ordinates of every point of the line. 1. If we draw a Ime parallel to AP, cutting the axis of Y at a distance from the origin de- noted by 5, and produce the or- dinate of the pomt P to P\ it is plain that the ordinate of the point P' will exceed the ordi- nate of the point P by the constant quantity b ; hence, the equation of the parallel line will be, sin a sin (3 — a) X -{- b. * Legendre, Trig. Art. 43. BOOK I.] POIT^T AND STRAIGHT LINE. 33 2. If the parallel cuts the ax:^ Y below the origin of co-ordiiiates, the value of y, in the second line, will be less than the value of y in the line AP^ by the constant quantity b ; and in that case, b becomes nega- tive, and the general equation takes the form. y sm a sin (? -■ ri) 3. Since the line PD is par- allel to the axis of T^ the angle APD is equal to the angle PA Y\ hence, the coefficient of X is the sine of the angle ichich the line makes with the axis of X^ divided by the sine of the angle which it makes with the axis of Y. 4. Thus far, we have supposed the co-ordinate axes to make an oblique angle with each other. It is, however, generally, most convenient to refer points and lines to* co-ordinate axes which are at riorht anirles. If we suppose YAX to be a right angle, «• /3 - a = 90° - a, and, sin (^ — a) = cos a.* * Legendre, Trig. Art. 63. 34 ANALYTICAL GEOMETRY. [book 1. The equation of the straight line then takes the form. y Bin a X ±h\ cos a or, y = tang ax ± b, the tangent of a being calculated to the radius 1. If we denote the tangent of a by a, the equation becomes, y = ax ± b. r f PK A /KY\ [/ x^^ — !x pin ^ Interpretation of the equation. 14. The line AP^ passing throucrh the orisrin of co-ordi- nates, and whose equation is y =z ax, has been drawn in the first angle. But the equation is equally applicable to a line drawn in either of the other angles, if proper values and signs be attributed to the tan- gent, a. The angle, of which a is the tangent, is always estimated from the axis AX^^ around to the left. 1. If the line be dra^\^l in the first angle, the tangent a is positive,* and the co-ordinates x and y, are both positive. 2. If the line be drawn in the second angle, the angle XAP will fall in the second quadrant, and its tangent, a, will be negative.* But the abscissas of points in the second angle are also negative : hence, a and x are both nega- tive: their product is, therefore, positive; lieiice, y is i osi- * Legendre, Trig. Art. 59. BOOK I.] POINT AND STRAIGHT LINE. 35 tive, as it should be, since it represents the ordinates of points above the axis of abscissas. 3. If the line AF" be drawn in the third angle, the tangent a will be positive, since the angle falls in the third quadrant ;* and since x is negative, the second member will be negative ; hence, y will be negative, as it should be. 4. If the line AF'" be drawn in the fourth anffle, the tangent a will be negative, since the angle falls in the fourth quadrant ; and since x is positive, the second member will be negative, and therefore, y will be negative, as it should be. As the same reasoning is applicable to the general form, the equation, y = aa; + J, will represent every straight line w^hich can be drawn on the plane of the co-ordinate axes, if proper values and signs are attributed to a and h, 5. The values of a and h are constant for the same straight line, but take different values when we pass from one line to another. They are called arbitrary constants^ because values may be attributed to them at pleasure, when only the form of the equation is given. 6. If, in the equation of a straight line y = ax + b, any value be attributed to one of the variables, the other becomes determinate, and its value may be found from the equation. * Legendre, Trig. Art. 69. 36 AJTALYTICAL GEOMETKT. [bOOK I. IfJ for example, we make, jc = 1, we have, y = a + b. SB = 2, gives, y = 2a -^ b, a; = 3, gives, y = 3a -{- b, &c., &c., &c. Or, we may attribute values to y, and find the corres- ponding values of x. If we make. y = 1. we liare, y = 2, gives, y = 3, gives. 1 __ 5 X - a 2 i X — a , IIB will be twice CB or AF\ hence, any ordinate, as EF^ will be equal to twice the abscissa, * Legendre, Trig. Art. 37. 3S ANALYTICAL GEOMETRY. [book I, AJFy plus AC, which is 4. If the coefficient of x were negative, the equation would take the form, y = — 2x -\- 4; the point D would then fall at D', and the line would take the direction C^\ Every straight line passing through the origin of co-ordinates will lie in the first and third angles when the co- efficient of X is positive, and in the second and fourth, when it is negative. If 4 were negative, the point G, would fall on the axis of T", below the origin. \ "^ A ! /w ^ X , Second Method. 17. A point which is common to two straight lines wiU be at their intersection ; and the co-ordinates of this point will satisfy the equations of both lines. Conversely, if the equations of two straight lines be made simultaneous,* and combined, the results obtained will be the co-ordinates of the common point. 1. Construct the line whose equation is, 2/ = - 6a; -f 12 . . . . (1.) If this equation be combined ^vith the equation of the axis of -Z", which is, X indeterminate, and y = (Art. 9), we shall have, a; = 2, ♦Bourdon, Art. 82. University, Art 88. BOOK I.] PROBLEMS. 39 which is the abscissa of the point iu which the line inter- sects the axis of JT. If we combine Equation ( 1 ), with the equation of the axis of Y, which Ls, a; = 0, and y indeterminate (Art. lo), ■we shall have, y = 12, which is the ordinate of the point in w^hich the line inter- sects the axis of Y, Having drawn the co-ordinate axes, \J at right angles to each other, lay off, on the axis of JT, AJB equal to twice the unit of length ; and on Y, AC equal to 12 times the unit of length, and then draw the line BC\ ~~^\ ^V" this will be the line required, since two points determine the position of a straight line. The line makes an obtuse angle with the axis X, as it should do, since the coefficient of x is negative. 2. Construct the line whose equation is, y — ax + h. S. Construct the line whose equation is, 2/ = 2a; -f 5. 4. Construct the line whose equation is, y = — X — I, 0. Construct the line whose equation is, y =: _ 2ic -I- 6. 4:0 ANALYTICAL GEOMETRY. [bOOK I. Equation of the first degree between two variables. 18, The equation, Ay + Bx-{- C = 0, is the most general form of an equation of the first degree between two variables, since there is an absolute term (7, and since each of the variables, y and ic, has a coefficient. This equation may be written under the form, B C which becomes of the form already discussed, if we make, 7 = «, and 7 = J. A A Havinj? dra\\Ti the co-ordinate axes at ripjht ano:lea to each other, if we lay off on the axis of !Fi a distance equal (J to — -j-i ^Q • • • (*•> 1. The value of a, found in Equa- y jf tion ( 2 ), is easily verified. For, y" — y' is equal to iVP, and x" — x' is equal to J/P; hence, ^ _ y" - y' MP ~ x" - x' ' and, consequently, equal to the tangent of the angle NMP^ to the radius 1.* Hence, a line passing through either of the points M or iV, Equations (3) and (4), and making with the axis of JT, an angle whose tangent is NP -v- MP^ will also pass through the other point. 2. If, in Equation (2), we suppose y y' y'\ we shall have, a = X' X > = 0; M JV and this is as it should be, since under this supposition, the line becomes parallel to the axis of JT. 3. If we suppose a' = jb", in Equation (2), the ordinates y' and y" being unequal, we shall have. y y' A N J i AT tlierefore, a is infinite;! and hence, the line is perpendicular to the axis of -Z! J •Leg., Tr. Art. 37. f ^y ^rt. 71. Un., Art. 72. X Leg., Tr. Art 60. BOOK I.] POINT AND STRAIGHT LINE. 45 If we suppose y' = y'\ and at the same time, x' = ar"', the two points will coincide, and we shall have, Under these suppositions, a is indeterminate,* as it should be, since an infinite number of straight lines may be drawn through a single point. Equation of a line parallel to a given line. 22. Let y = ax -^ h be the equation of a given line (Art! 16). The equation of the required line will be of the form in which a* and 5', are undetermined. Two right lines are parallel, when they make the same angle with the axis of abscissas. Hence, if we make, a' — a, the second line will be parallel to the first ; and its equa- tion will be, y =z ax -\- h\ in which equation, V is undetermined, as it should be, since an infinite number of lines may be drawni parallel to a given line. 1. If it be required that the parallel shall pass through a given point, its position will be entirely determined. For, if the co-ordinates of the given point be denoted •Bourdon, Art. 71. University, Art. 72. 46 ANALYTICAL GEOMETRY. [book I. b7 a' and y', the equation of the parallel will take the form (Art. 21), 1/ -y' = a{x- x'), in which the quantities a, y\ x\ are known; hence, the position of the line is fixed. Angle included between two given lines. « 23. Let DC, BC.he the two given lines: y = ax + b, tiie equation of the 1st, y = a'x + b\ the equation of the 2d, in which cr, a', b, b\ are known. Denote the angles CDX and CBX, by a and a', and the angle DCB, by V, Then, since CBX = CDB + DCB* we have, F" = a' — a, and, tanff y = tangr (a.' — a) = 2. & — ^ "= ^ '' 1 + tang a' tang a to the radius l.f Substituting for tang a', and tang a, their values a' and a, we have, TT a' — a ^ 1 + a'a ••; • Legendre, Bk. I. Prop. 25. Cor. 6. f Trig. Art. 66. BOOK I.] POINT AND STRAIGHT LINE, 47 1. If the lines becomo parallel, the angle V will be 0, and hence. tanor y — a' — a = 0. 1 + aa Therefore, a' — a = 0; or, a' = «, a relation already proved (Art. 22). 2. If the lines are perpendicular to each other, "Fifill be equal to 90°, and its tangent infinite,! that is, a' — a hence, tang V = r— — - = 00 ; t 1 + a'a = 0. This last, is the equation of condition, when two ri^t lilies are at right angles to each other. If the tangent of one of the angles is known, the other can be found fi'om the equation of condition. Intersection of two lines. 24. Let y = ax -\- b, be the equation of the first; and y = a'x + b\ the equation of the second. The pomt in which two straight lines intersect each other, is found, at the same time, on both of the lines; hence, its co-ordinates will satisfy both equations. If, therefore, we suppose y • Trig. Art. 60. f Leg., Trig. Art. 60 X Bour., Art. 71. Univ. Art. 72. ot-^ / Ay-. -. «v 'U' 43 ANALYTICAL GEOMETliY. [bOOK L and a', in the equation of the first line to become equal to y and a*, in the equation of the second, the two equations will be simultaneous. Combining the equations, mider this suj^position, and designating the co-ordinates Df the point of intersection, by a' and y\ we find. a — a' and y' = ah' - a'h a a' 1. If, in the two last equations, we suppose a = a', the values of x' and y' will both become infinite. The supposition of a = a', is the condition when the two lines are parallel ; and therefore, under this supposition, their point of intersection ought to be at an infinite distance fi-om both the co-ordinate axes. If, at the same time, we suppose 5 = 5', the values of x' and y', will become equal to di\dded by 0, that is, indeterminate. The two suppositions will cause the lines to coincide; hence, their point of intersection ought to be indeterminate, since eveiy point of either line will satisfy both equations. A perpendicular from a given point to a given line. 25. Let y = a^ + 5, (1.) be the equation of a given line, and cc', y', the co-ordi. nates of a given point. BOOK I.] POINT AND STRAIGHT LINE. 49 It is required to draw from this point, a line perpen- dicular to the given line, and to find the length of the perpendicular. The equation of a line passing through a point, whose co-ordinates are x\ y\ (Art. 20), is y -y' = «'(^ - aj'), .... (2.) in which a' denotes the tangent of the angle which this second line makes vat\\ the axis of A". But since this line is to be perpendicular to the given line, we have (Art. 23), y * 1 + aa' = = 0; from which we have. • o' =z - 1 a Substituting this value for a', in Equation (2), the sec- ond line becomes perpendicular to the first, and we have,. y - y' = - -i^ - ^') ' . . (3.) It is now required to find the length of the perpen-- dicular. This is done, 1st, by finding the co-ordinates of the point in which the perpendicular intersects the given line; and 2d, by finding the differences between the co-ordinates of" this point and the co-ordinates of the given point ; 3d, by substituting these differences in Formula (Art. 19). Let us designate the co-ordinates of the point of inter- section, by ic", y'\ Then, since the pohit is on the given 4 50 ANALYTICAL GEOMBlTET. [bOOK U line, its co-ordinates will satisfy the equation of that line, and we shall have, y" = ax" +h^ (4.) and since the point is also on the perpendicular, its co. ordmates ^ill also satisfy the equation of the pei-pendicular and give, y"-y' = -Ux"-x') . . . (5.) (J/ If we eliminate x'\ from these two equations, we shall have, ,, _ a^y' 4- Gx' + h ^ ~ 1 + a2 Subtracting y' from both members, we obtain, y' — ax' — h , . Substituting this value of y" — y\ in Equation { 5 ), we have, a{y' — ax'— h) 1 + a" X" -X' ^ ^ ^\ , . ^. . . . (7.) Let us designate the length of the perpendicular, by P. Since the distance between two points, whose co-ordinates are, x", y'\ x\ y\ (Art. 19), is we have, by substituting for x" — x', and y" — y', their values found in Equations (7) and (6), jp — y' - ^^' -f> . *1. If the given point should fall on the given line, BOOK I.] TRANSFORMATION OF CO-ORDINATES 51 its co-ordinates would satisfy the equation of the line, and give, y' = ax' + 5. This supposition would reduce the numerator of the value of jP, to 0, and consequently, P would be equal to 0. A ly TRANSFOEMATION OF CO-OEDrNATES. 26. The equations of a point determine its position with respect to the co-ordinate axes (Art. 5). The co- ordinate axes may be selected at pleasure, and any point may, at the same time, be referred to several sets of axes. Let A^ for example, be the origia of a system of co-ordinate axes, and A\ any point whose co-ordinates are a and b. Through A\ draw two new axes, respectively parallel to the first. The co-ordinates of any point, as P, referred to the primitive system, are AD^ PD ; and its co-ordinates referred to the new system, are A'D\ PD'. The point P is equally determined, to which ever system it be referred. 27. It is often necessary, for reasons that will be here- ^er explained, to change the reference of points from one system of co-ordinate axes to another. This is called, tlie transformation of co-ordinates. The axes to which the points are first referred, are called. Primitive Axes / *and the second axes, to which they are referred, are called, New Axes, 52 ANALYTICAL GEOMETRY. [bOOK I. In changing the reference of points from one system to another, all that is necessary, is to find the co-ordinates of the points referred to the primitive axes, in terms of the co-ordinates of the new origin and the co-ordinates of the points referred to the new axes. To pass from a system of co-ordinate axes, to a parallel system, 28. Let A be the origin of the Y y primitive system, and A' the origin of the new system. Suppose the co-ordinates of the origin A\ to be, AB = a, and JBA' = b; and let us designate the new axes, by JC' and Y\ and the co-ordinates of any point, referred to these axes, by x' and y'. Then, assuming any point, as JP, we shall have, AB = AB -f BB, and BB = BB' -f B'P, Now, since AB is the abscissa of P, and BB = A'B\ its abscissa referred to the new axes ; and since BB is the ordinate of P, and B'B its ordinate referred to the new axes, we have, X — a '\- x\ and y = h -\- y\ in which, the primitive co-ordinates of any point, are ex- pressed in terms of the co-ordinates of the new origin, and the new co-ordinates of the same point. 1. The new origin may be placed in either of the four angles of the primitive axes, by attributing proper signs to its co-ordinates, a and h. It is also to be observed, that BOOK I.] TRANSFORMATION OF CO-ORDINATES x' and y' have the same algebraic signs, in the different angles of the new system, as have been attributed to x and y, in the corresponding angles of the primitive system. ^ To pass from a rectangular to an oblique system. 29. Let A be the common origin, AX^ AY^ the primitive axes, and AX!^ AY\ the new axes ; and let ns designate, as before, the co-ordinates of points referred to the new axes, by x' and 2/'. Denote the angle which the new axis of X makes with the primitive axis of X, by a, and the angle which Y makes with AX, by a' ; and let P be any point in the plane of the axes. Through P, draw Pi? parallel to the axis of Y, and PF' parallel to the axis of Y' ; draw also P'P parallel to the axis of Y, and P'(7 parallel to the axis of X. Then, AB = AR -f PP, "will be the abscissa of P, referred to the prunitive axes ; and, FB = J3G + CP, will be its ordinate. Also, AP' will be the abscissa of P, referred to the new system; then PP^ will be its ordinate. But, AR = AP' cos a,* that is, AR = x' cos a. and, RB P'C = PP' cos a' y' cos a' ; Trig. Art. 87. 54 ANALYTICAL GEOMETBY. [bOOK I. hence, a = a' cos a, -{- y' cos a'. We also have, jP'H = CJ3 =z AP' sin a ; that is, €B = a' sin a, and, FC = Fr sin a' = y' sin a' ; hence, y = a' sin a -f y' sin a'. Hence, the formulas are, X = x^ cos a 4- 2/' COS a', y = a' sin a -{- 2/' sin a'. 1. If it were required, at the same time, to change the origin, to a point whose co-ordinates, referred to the prim- itive system, are a and 5, the formulas would become, X = a -{- x' cos a -f y' cos a', y = b -{- x' sma -\- y' sin a'. 2. If a' — a = 90°, we have, sin a' = cos a ; cos a' = — sin a ; * substituting these values in the last equation, we have, X = a •{• x' cos a — 2/' sin a, y = b -{• x' sma -{- y' cos a, which are the formulas for passing from one system of rectangular co-ordinate axes to another. To pass from an oblique to a rectangular system. 30. The first set of formulas of the last Article, for a common origin, are, X = x^ cos a + 2/' cos aj y •= x' sin a -f y' sin a'. * Trig. Art 63. ' BOOK I.J TRANSFORMATION OF CO-ORDINATES. 55 If "we regard the oblique as the primitive axes, it be- comes necessary to find the co-ordinates of points referred to these axes, in terms of the rectangular co-ordinates, and the angles a and a' ; that is, we must find the values of x' and y'. If we multiply both members of the first equation by the sin cl\ and both members of the second, by cos a', and then subtract them, and remember that, sin (a' — a) = sin a! cos a — sin a cos a', y' will be ehminated ; and if x be eliminated, in a similar manner, we shall obtain, , _ a; sin a' — y cos a' t _ V ^^^ a — a; sin a sin (a' — a) > ^ gjj^ ^a' — a) 1. K the origin be changed, at the same time, to a point whose co-ordinates, with reference to the oblique system, are a and J, we shall have, . ic sin a' — y cos aJ , -, V cos a — a; sin a sm (a' — a) ' ^ sin (a' — a) REMARKS. 31. The primitive co-ordinates of any point, determined with reference to a new system, depend for their values, Ist. On the position of the new origin: 2d. On the angles which the new axes make with the primitive axes : iind, 3d. On the co-ordinates of the same point, referred to the new system. 32. The transformation of co-ordinates embraces two distinct classes of propositions: 1st. To transfer the reference of points from one system of co-ordinate axes to another system, which is known. In this case, the co-ordinates of the new origin, and the I 56 ANALYTICAL GEOMETRY. [boOK I. angles which the new axes make with the primitive axes, are known. 2d. So to dispose of the new origin, and to give such directions to the new axes, as to cause the resulting equar tions to fulfil certain conditions, or to assume certain forms. In this case, the conditions imposed, determine the position of the new origin, and the directions of the new axes. 33. Since the primitive co-ordinates of points are always deteiTnined in linear functions of the new co-ordinates, that is, by equations of the first degree, the substitution of their A^alues in the equation of any line, will not alter the degree of that equation ; hence, A given equatioii of a line^ and its equation ichen re- ferred to a new system of co-ordinate axes^ will always he of the sa)ne degree. 34. We shall terminate this subject by a single example. Having given, the equation of a straight line, y z=z a'x + h\ referred to rectangular co-ordinates, it is required to find its equation when the line is referred to oblique co-ordi- nates having a difierent origin. We have (Art. 29 — 1), X rr: a 4- a' cos a -f 2/' COS a', y = h + x' mi a -\- y' sin a'. Substituting these values for x and y, in the equation of the line, we have, 5 4- x' sin a 4- y' sin a' = a' {a + x' cos a 4- y' cos a'J 4- h' ; or, by reducing, , a' cos a — sin a , aa' -'- h' — h ^ ~ sin a' — a' cos a' sin a' — a' cos a' ' BOOK I.] POLAR CO-OEDIK^TES. 57 wliich is the equation of the straight line, referred to the oblique axes. The coefficient of a;', is the sine of the angle which the line makes with the axis of JT', divided by the sine of the angle which it makes with the axis of Y' (Art. 13). The second term, in the second member, is the distance cut off from the axis of Y' (Art. 13 — 3). POLAR CO-ORDINATES. 85. We have seen, that the relative position of points and lines may be determined, analytically, by referring them to two co-ordinate axes. There are also other methods, by which they may likewise be detei-mined. Assume, for example, any point, as A, and through it draw any straight line, as AX^. If we suppose a straight y^ line, as AJS, to be turned around the ^^\ point A, so as to make with AJC all ^ ■ possible angles, from to 3G0°, and suppose, at the same time, the line AB to increase or dimmish at pleasure, the extremity Jj, may be made to occupy, in succession, every point of the plane. Under this hypothesis, there are two variable quantities considered: 1st, the variable angle XAJ3; and 2d, the va- riable distance A£'y and every point, in the plane, may be determined by attributing suitable values to these vari- ables. The fixed line AJC, is called the initial line; the fixed point A^ the pole; XAB the variable angle^ and AB the radius-vector. This method of determining the position of points, is called the systerii of polar co-ordinates. I I 58 ANALYTICAL GEOMETKY, Designate the variable angle JCAJ^, by V, the radius-vector AJS, by r, and the co-ordinates of the point ^, referred to rectangular axes, by x and y ; then, if the origin ' of the rectangular axes be at A^ we shall have, X — r cos V, and, y = r sin v.* From the first equation, we have, [book I. r = cos V Now, since x, and the cos v, are both positive in the first and fourth angles, and both negative in the second and third, they will always be afiected with the same sign ; and hence, the sign of r will be constantly positive ; con- sequently, A negative value of the radius-vector can never enter into the analysis. If, therefore, such a value should be obtained, we infer, that incompatible conditions have been introduced into the equations ; and hence, all negative values of the radium vector must he rejected. To pass from a rectangular to a polar system. 36. Let A be the origin of the co-ordinate axes. A' the pole, A'B^ parallel to AX^ the initial line. Legendre, Trig. 37. BOOK I.J POLAR CO-ORDINATES. 59 Then, the line AF will be the radius-vector of the point P. Let the co-ordinates of the pole A', be denoted by a and b. Then, and, But, and, hence, and, D X A'R = r cos V, Pi2 = r sin v. AD = AB + BB, FB = BB + BB; X = a -\- r cos Vj y z= 6 + r sin V ; which are the required formulas. 1. If the pole A\ be placed at the origin A^ the equa- tions will become, X =z r cos V, y =: r sin v. 2. If, instead of estimating the variable angle v from the initial line A'R^ parallel to AJC^ it be estimated from A'R\ making with AX a given angle, ±a, the equa- tions will become, X = a -\- r cos {v ± a), y = b + r sm {v dz a). BOOK II. OF THE CIKCLE. 1. The equation of a line expresses the relation which exists between the co-ordinates of every 'point of the line (Art. 12). 2. Lines are divided into different orders, according to the degree of their equations. For example, the right line is a line of the first order, since its equation is of the first degree. The circumference of the circle is a line of the second order, its equation being of the second degree; and if the equation of a line were of the third degree, the line would be of the thu-d order. 3. Tlie Interpretation of an equation, consists in. class- ing the line which the equation represents; in determining its position, its form, its limits, and the points iu which it intersects the co-ordinate axes. Equation of the circvunference of a circle. 4. Let A be the origm of co- ordinates, and ^l-3r, A Y, the co- ordinate axes. It is required to find the equa- tion of a curve such, that all its points shall be at a given dist- ance from the origin A. Let i? BOOK n.J THE CIRCLE. 61 denote the distance, and x and y, the co-ordinates of any point of the curve, as P. The square of the distance from the origin to any point, whose co- ordinates are x and y, is, a;2 ^ 2/2 . hence, x^ -\- y^ — i?^, which is the equation required. Interpretation. 1, To interpret the equation, we begin by determining the points in which the circumference cuts the co-ordinate axes. The co-ordinates of these points must satisfy, at the game time, both the equation of the circle, and the equa- tions of the axes. The equations of the axis of X (Bk. L, Art. 9), being X indeterminate, and y = ; if we make y = 0, in the equation of the circle, the corresponding values of x will be the abscissas of those points which are common to the circumference and the axis of JT; that is, X ^ ±R\ which shows that the curve cuts the axis of abscissas in two points, one on each side of the origin, and each at a distance from it equal to the radius of the circle. 2. To find the points in which the circumference cuts the axis 3^ make ic = 0, and there results, 2/ = ±i?; 62 ANALYTICAL GEOilETEY. [bOOK II. the axis of Y, therefore, intersects the circumference in two points, equally distant from the origin, one above the axis of -Zj and the other below it. 3. To trace the curve between these points, find the valu6 of y from the equation of the circle, which gives, y = ± V-^ - x\ Now, since every value for jc, gives for y two equal values, with contrary signs, it follows that the curve is symmetrical with respect to the axis of SS'; and in the same manner, it may be shown to be symmetrical with respect to the axis of Y. Beginning at the point where x = 0, we have, y = ± i?. The values of y then decrease, numerically, as x increases numerically ; and when x becomes equal to ± H, we have, 2/ = 0; hence, the curve intersects the co-ordinate axes in four points, at a distance from the origin, equal to H. 4. If X becomes greater than ± i2, the values of y become imaginary, which shows that the curve is limited both in the direction of x positive, and of x negative. By placing the equation under the form, X = znV^^-yS we may show, that the circumference is also limited in the direction of y pcJsitive, and in that of y negative. BOOK II.] THE CIRCLE 63 5. By attributing a particular value to either of the variables, in the equation, 2/ = ±V^^ the corrcq)onding values of the other variable may be found. If we suppose i2 = 1, and then make. a; = 0, we have, 2/ = ± 1. 1 gives. y^±^J\ =i^. 3 gives. y-^A-V^' &c.. &c., &0. 6. If, in the equation. x^-^y'-. = -R', X and y denote the co-ordinates of a point within the cir- cumference, the equality will be destroyed, and x^ -f y^, will be less than 72^, and we shall have, a;2 + 2/2 — i22 < ; that is, negative. For a point on the curve, a.2 + 2/2 _ 722 _. 0; and for a point without the curve, a;2 -f y2 _ 722 > ; that is, positive. 7. The equation, 2/2 = i22 - x\ may be put under the form, y2 = {li + x) (H-^ 64: ANALYTICAL GEOMETRY. [book II. in which the factors, H + x, and It — x, are the two segments into which the ordinate y divides the diameter; this ordinate is, therefore, a mean proportional between the two segments.* 8. The equation of the circle may also be placed nnder another form, by transferring the origin of co-ordinates, fi'om the centre to a point of the circumference. For this transformation, we have the Formulas (Bk. I., Art. 2§), X = a + x\ and y = h ■\- y\ Let the origin be transferred to B. The co-ordinates of this point are, a = — B^ and 5 = 0; hence, a; = — jR -f a', and y = y\ Substitutmg these values in the equation, y-i- - JEC- - x\ we obtain, y'2 = 2Ex' - iB'2 ; or, omitting the accents, 2/2 = 2Rx — a;2 ; which is the equation of the circle when the origin of co- ordinates is at the left extremity of the initial diameter. ' 9. When the absolute term in the equation of a line is wanting, the line will pass through the origin of co- ordinates. Legendre, Bk. IV. Prop. 23. Cor. BOOK n.J THE CIRCLE. 65 For, the co-ordinates of the origin are, X = 0, and y = ; these values being substituted in any equation wanting the absolute term, wiU reduce both members to 0; hence, the co-ordinates of the origin will satisfy the equation of the line; and, therefore, the line will pass through the origin. 5. There is yet a more general form under which the equation of a circle may be expressed. The characteristic property of the circumference of a circle is, that all its points are at uu equal distance from the centre. To express this pro- perty, ajialytically^ and in a gen- eral manner, designate the co-or- dinates of the centre by x' and y' ; the co-ordmates of any point of the circumference, by x and y, and the radius by M. The distance from any point, whose co-ordinates are x\ y\ to a point whose co-ordinatca\ are x and y (Bk. I., Art. 19), is, {x - xy + (y - y'Y = R\ This, therefore, is the most general equation of the circle re- ferred to rectangular co-ordinates. By attributing proper values and signs to x' and y\ the centre may be placed at any point in the plane of the co-ordinate axes. 1. To find the points in which 66 ANALYTICAL GEOMETRY. [book II. the circumference intersects the axis of Jl, make y = 0, and we Lave, x' ± yJi^ y from which we see, that the values of X will become imaginary when y' exceeds i?, and it is plain that in that case there will be no in- tersection. 2. To find the points in which the circumference inter- sects the axis of I^ make a; = 0, and we have, y = y' ± V^^2 _ r^'2^ in which the values of y wiU be imaginary, if x' exceeds M, 3. If the co-ordinates of the F centre of a cu'cle are, x' = — 2, and y' = — 4, and the radius equal to 6, its equation will be, {x -f 2)2 + (y -f 4)' 36, N ' / A V c~ y 1 from which the cii'cumference may be readily described. Find the points in which it cuts the co-ordinate axes. Supplementary chords. 6. ScTPLEMENTARY Chokds, are pairs of chords drawn through the extremities of a diameter, and intersecting each other on the curve. / BOOK n.] THE CIRCLE. 67 Supplementary chords of the circle are at right angles. 7. Let A be the origin of co-ordinates, and JB and JB\ the extremities of a diameter. The equation of a straight line passing through a given point, is of the form (Bk. I., Art. 20), y — ^' = a{x - x'). If the line passes through the point iS, whose co-ordinates are £c' = + i?, and y'= 0, its equa- tion will be, y = a(x — 72) ^.Y (1-) For a like reason, the equation of a straight line passing through i?', whose co-ordinates are x' = — JR, and y' = 0, is, y = aXx + H) (2.) If these two lines intersect each other, the co-ordinates of their point of intersection will satisfy both equations. Hence, if we suppose x, in one equation, to be equal to x in the other, and y equal to y, and then combine the equations by multiplying them together, member by mem- ber, the resulting equation, 2/2 = aa'ix' ■- E-'), .... (3.) will express the condition, that the two straight lines shall intersect on the plane of the co-ordinate axes. But, if the point of intersection is in the circumference of the circle, a and y must satisfy the equation. a.2 4. 2/2 M\ (4.) 68 ANALYTICAL GEOMETRY. [bOOK n. or, y2 ^ i22 _ 2,2 ^ _ i(a;2 _ i22). Hence, aa' = — 1, or, aa' + 1 = 0. The two supplementary chords, therefore, are at right angles (Bk. I., Art. 23—2). 1. In the equation of condition, aa' -\- I = 0, the two tangents a and a', are undetermined; there are, therefore, an infinite number of values which may be at- tributed to either of them, that will satisfy the equation ; hence, there is an indefinite number of supplementary chords that may be drawn through the extremities of the same diameter, each pair of which will be at right angles. 2. If one of the supplementary chords makes a given angle with the axis of -Z", its tangent, a or a', is known ; and tlien, the value of the other tangent may be found from the equation of condition. 3. If either a or a' is equal to 0, the other will be in- finite ; which shows, that if one of the chords coincides with the axis of -X^, the other will be perpendicular to it. Tangent line to the circle. 8. Two points of a curve are consecutive^ when there is no point of the curve between them. A straight line is tangent to a curve when it has two consecutive points in common with the curve. To find the equation of a tangent line to a circle. Let A be the origin, and the equation of the circle, x^ +i/^=Ji^, (1.) BOOK II.] THE CIRCLE. 69 Take any point of the circumference, as P, and de- signate its co-ordinates by x'\ y". Through this point draw a secant line ; its equation will be of the form (Bk. I., Art. 20), y - y" = a(x - x") (2.) It is required to find the value of a, when the secant 'line PB becomes tangent to the circumference. Since the point P is in the circumference, its co-ordinates will satisfy the equation of the circle, and we shall have, JB"2 4. y"2 ^722 (3) Subtracting Equation ( 3 ) from ( 1 ), member from mem- Ijer, we obtain. or, {x 4- a;") {x - cc") -f (y -f y") (y - y") (4.) in which equation, x and y are the co-ordinates of any point of the circumference. If Equation (4) be combined with (2), a; and y, in the resulting equation, will be the co-ordinates of the points in which the secant intersects the circumference. The equa- tions are most readily combined, by substituting for y — y", in Equation (4), the value of y — y" in Equation (2). Making the substitution, we obtain, (X + a;") {x - JK'O -f (y + y") a (aj - x") = 0, and, by fiictoring, we have, 70 ANALYTICAL GEOMETRY. [bOOK H. {x - x") X [ic + x" 4- a[y + y")] = 0, which is satisfied by making, x — x" = 0, or, a; + aj" + a{y + y") = 0. In the first equation, x denotes the abscissa of the point P; in the second, of P'. If we suppose the secant PP to turn around the point P, the point P will approach P ; and when P' shall become con- secutive with P, the secant line will become tangent to the circumference. When this takes place, we shall have, X z= x'\ and y = y'\ and the second equation will give, a = r,, y a;", y'\ being the co-ordinates of the point of contact. ion (2), we Substituting this value in Equation (2), we have, y-y =-y. or, by reducing, yy" _ 2/"2 _ _ a-'^a; ^ aj'/a^ or, or, yy" + xx" = 2/"2 + aj BOOK n.] THE CIRCLE. 71 in which X and y are the general co-ordinates of the tan- gent line. 1. For the point in which the tangent intersects the axis of J^ we have a; = 0, and, if yn 2. For the point in which the tangent intersects the axia of X^ we have, y = 0^ and, a; = ^, = AT. x" Normal line. 9. A Normal Line to a curve, is a line perpendicular to the tangent at the point of contact. Every normal line, in a circle, passes through the centre. 10. The tangent of the angle which the tangent line to a circle makes with the axis of JT (Art. 8), is. The equation of any straight line passing through the point of tangency will be of the form, y - y" = «'(« - ^")' The equation of condition requiring this line to be perpen- dicular to the tangent (Bk. I., Art. 23 — 2), is, aa' -f 1 = 0, or, a' = . . (2.) 72 ANALYTICAL GEOMETRY. [bOOK II. Substituting for a its value in Equation ( 1 ), we have, a'=K,. The equation of the normal, therefore, becomes, y - y" = J;'(« - ^"), or, by rtducmg, yx" -y"x = 0; or, y = ^,x', and since this equation has no absolute term (Art. 4 — 9), the line passes through the origin of co-ordinates. We have thus proved a property well known in Elementary Geometry, viz. : that a line perpendicular to the tangent, at the point of contact, passes through the centre of the cii'cle. Polar equation. 11, The polar equation of a curve, is the equation which is obtained by referring the curve to a fix^d point and a given straight hne. The fixed point is called the pole ; the variable distance, from the pole to any point of the curve, is called the radius-vector ; and the angle which the radius- vector makes with the given straight line, is called the vari- able angle. Polar equation of the circle. 12. Let it be required to find the polar equation of the circle, when the pole is in the circumference. The equation of the circle, referred to rectangular co-or- dinates, when the origin is in the circumference, as at £\ (Art. 4—8), is, y'^ z= 2Rx ^ x^ , ^ , , . (1.) BOOK n.] THE CIRCLE. 73 If B' is the pole of the polar co-ordinates, we have (Bk. I., Art. 36—1), a;=rcosv, and y^zrsinv. _ B Substituting these values of X and y, in Equation ( 1 ), we have, r^ sin^v = 2i?r cos v — r^ co^v. Transposing, and remembering that, sin^y -f cos^y = 1, we have. 2Er cos?; 0; which is the polar equation of the circle when the pole is at B\ and the angle v^ estimated from the axis of -Xl Interpretation of the equation. 13. Since the polar equation, r^ — 27?r cos v = 0, has no absolute term, one of the roots is equal to ; * which ought to be the case, since the pole is on the curve (Art. 4—9). ' Dividing by this value of r, we obtain for the other value, r = 272 cos V. This value of r will be positive, when the cosv is posi. tive ; and negative, when the cos v is negative. But the * Bourdon, Art. 251. University, Art. 193. 74 ANALYTICAL GEOMETRY. [bOOK U. negative values of the radius-vector must be rejected, since they cannot enter into the analysis (Bk. I., Art. 35). The figure also indicates the same result. For, the cosw is positive in the first and fourth quadrants ; hence, the radius-vector is positive when it falls in the first or fourth angle. The cosr, is negative in the second and third quadrants ; hence, the radius-vector is negative when it fells in the second or third angle. For V = 0, the cos u = 1, and we have, r = 2R = JB'B. When V increases from to 90°, the radius-vector con- tinues positive, and determines all the points in the semi- circumference BCJB'. Tliis may also be verified. For, in the right-angled triangle B' CB, B'C z= B'B cos BB'O;* that is, r = 2i2 cos v. When V becomes equal to 90°, cosv = 0, and r is 0. The radius-vector then becomes tangent to the circumfer- ence, since the two points in which it before cut it, have united. From V = 90°, to t? = 270°, the cosy is negative; and there is no point of the curve either in the second or third angle. From V = 270°, to v = 360°, the cos v is positive, and the radius-Vector will determine all the points of the semi-cu'cumference, below the axis of abscissas. * Trig. Art. 37. BOOK n.] THE CIRCLE, 75 2. If the pole be placed at the point 7?, whose co-ordi- nates are, a = + By 6 = 0, the equation will become, r = — 2R cosw. In this equation the ra- dius-vector will be negative, when cos v is positive, and positive, when the cost; is negative. Hence, the radius-vector will not give points of the curve from v = 0, to v = 90°. It will give points of the curve from v = 90°, to v = 270° ; and it Avill again fail to determine a curve from v = 270°, to v = 360^. The ficj- ure verifies these results. 3. If we place the pole at the centre, the equations for transformation, will become, r cos Vy y = r sm 0. BOOK III. D p PD' OF THE ELLIPSE. 1. An Ellipse is a plane curve, such, that the sum of the two distances from any point of it, to two fixed points, is equal to a given distance. Thus, if F and i^be two fixed points, and AB a given dis- tance ; then, if i^P + Pi^, is constantly equal to AB^ for every position of the point P, the curve APBP will be an ellipse. 1. The fixed points, J?" and F, are called foci of the ellipse. 2. The line AB^ passing through the foci, and limited by the curve, is called the transverse axis; and the extremities A and P, the vertices of the transverse axis. 3. The point C, on the transverse axis, and equally dis- tant from the foci F' and F^ is called the ceiitre of the ellipse. 4. The line DD' drawn through the centre, perpendi- cular to the transverse axis, and limited by the curve, is called the conjugate axis^ and I> and B' are its vertices. Construction of the Ellipse. 2. — 1. We can easily cc»nstruct an ellipse when its trans- verse axis and foci are given. BOOK III.] THE ELLIPSE, 77 Let F' and F be the foci, and AB the transverse axis. Take a thread, equal in length to AB^ and fasten its two ex- tremities, the one at F\ and the other at F. Press a pencil against the thread, and move it around the points F^ F\ keeping the thread constantly stretched : the point of the pencil will describe an ellipse ; for, in every position of the pencil, we shall have, F'P ■{- PF = AB, which is the characteristic property of the curve. 2. When the pencil is at j5, we have, AB = BF' -f BF\ but, BF' = FF' + FB\ hence, AB = FF + 2BF .... (1.) When the pencil is at A, we have, AB = AF+ AF; but, AF = FF' + AF' ; hence, AB = FF'+2AF .... (2.) Equating the second members of Equations (1) and (2), cancelling the common term FF\ and dividing by 2, we have, BF=AF; tKat is, the distance from either focus to the nearest vertex of the transverse axis is equal to the distance from the other focus to the other vertex. Since the centre G is the middle point of FF{Kvi. 1-3), it follows, that the centre C is also the middle 2^oi?it of the transverse axis. 78 ANALYTICAL GEOMETRY. [bOOK HI. 3. We may also construct the ellipse by points^ when the transverse axis and foci are given. Let AB be the transverse axis of an ellipse, andi^' and F the foci. Take, in the di- viders, any portion of the trans- verse axis, as AD^ and with the focus F\ as a centre, de- scribe the arcs p and q. With JBD, the remaining part of the transverse axis, as a radius, and the other focus F, 2i^ 2i. centre, describe two other arcs intersecting the former; the points of intersection will be points of the curve. For, the sum of , the distances from p or q^ to F' and F^ is equal to AB. If with the radius AJD^ two arcs be described from the focus jP, and with the radius BD two arcs be described from the focus F\ these arcs will also determine, by their intersections, two points of the curve; so that, for each time we take a part AD of the transverse axis, we shall determine four points of the curve. 4. Construct an ellipse when its axes are given. If from either vertex of the conjugate axis, as i>, the lines DF\ DF^ be drawn to the foci, they will be equal to each other. For, in the two right-angled triangles, F' CD, FCD, OF' is equal to CF, and CD is common; hence, the hypo- thenuse DF' is equal to DFJ^ » Legendre, Bk. 1. Prop. V. . D ^■\ c yrf BOOK in.] THE ELLIPSE. 79 But F'D -I- DF is equal to AB\ hence, DF\ or JDF, is equal to OB. If, therefore, with either vertex of the conjugate axis as a centre, and with a radius equal to half the transverse axis, the circumference of a circle be described, it mil intersect the transverse axis at the foci. Having found the foci, the ellipse may be constructed, as in the last case. Equation of the ellipse. 3. Let F' and F be the foci, and denote the distance between them by 2c ; then, GF or CF' = c. Let F be any point of the curve. Designate the distance FF, by r\ and FF, by r. Let 2^ = AB, denote the the given line, to which the sum, F'F + FF, is to be constantly equal. Through C, the middle point of F'F, draw OB perpen- dicular to F'F, and let C be the origin of a system of rect- angular co-ordinates, of which AB, I)D\ are the axes. De- note the distance CB by B^ and let x and y denote the co-ordinates of any point, as P. The square of the distance between any two points, of which the co-ordinates are jc, y, and x\ y' (Bk. I., Art. 19), is, (y - y'Y + (aJ - cc')2. If the distance be estimated from the point F\ of which the co-ordinates are y' = 0, and x' = — c, we shall have, 80 ANALYTICAL GEOMETRY. [bOOK ITI. F'F^ = r'2 =z y'^+ {x + cy . . . (1.) K it be estimated from the point I\ of wHch the co-ordi- nates are y' = 0, and x' = -i- Cy we shall have, r^= y^+ (x-cy .... .(2.) Since the lines intersect each other, the co-ordinates of P will satisfy Equations ( 1 ) and ( 2 ) ; hence, the equations are simultaneous. If we add and subtract them, we obtain, r'2 -f- 7-2 = 2(2/2 + a;2 + ^) . , , (3,) and r'2- r2 = 4ciB (4.) Equation (4) may be placed under the form, (r'-f r) {r' - r) = 4cx ... (5.) But we have, from the property of the ellipse, r' + r = 2A (6.) Combining (5) and (6), we have, ^ -^ = -3- O') Combining (6) and (7), by addition and subtraction, we obtain, r'= A + ^. . (8.) and r = A-^ . . (9.) Squaring both members of Equations (8) and (9), com- bining the resulting equations, and substituting the values of r'2 and r-, in Equation (3), we obtain. ^' + ^' = y' + «^' + «'• BOOK ni.] THE ELLIPSE. 81 Substituting for c^ its value, A"^ — JB^, (Art. 2 — 4), we have, A'' + ^^—j^ 052 = 2/2 + a;2 4. ^2 _ ^ . or, A^ 4- A^X^ - i?2a,2 _ ^2y2 ^ ^23.2 4. ^4 _ J[2^2^ Cancelling and transposing, we have, ^2y2 + J52a.2 _ ^2^2^ which is the equation of the ellipse, referred to its centre and axes. Interpretation of the equation. 4. 1. If, in the equation of the ellipse, we make y = 0, the corresponding values of x will be the abscissas of the points in which the curve intersects the axis of X (Bk. n., Art. 4—1), viz.: X = -\- A, for J3, and X = — A, for A, 2. If we make cc =r 0, the corresponding values of y will! be the ordinates of the points in which the curve intersects- the axis of IT: viz. : y + B, for J9, and 3. If we place the equation of «■ the ellipse under the form, y = ± ^V^^ - x% we see, that for every value of X, as Clly whether plus or 6 82 ANALYTICAL GEOMETRY, [book hi. minus, there will be two values for y, numerically equal, with contrary signs ; hence, the curve is symmetrical with respect to the transverse axis. If' X be made greater than A^ whether it be taken plus or minus, the values of y will be imaginary ; hence, the curve will be limited both in the direction of x positive and X negative. 4. If we place the equation under the form, a; = ± -^^B^ y\ we see, that for every value of y, whether positive or negative, as CX^ there will be two equal values of ic, with con- trary signs ; hence, the curve will be symmetrical with re- spect to the conjugate axis. If y be made greater than j5, either positive or negative, the values of x will be imagin- ary ; hence, the curve will be limited in the direction of y positive and y negative. 5. The equation of the ellipse, may be put under the form, and this equation will be satisfied, 80 long as X and y denote the co-ordinates of points of the curve. If we take any point, as P', without the curve, its ordi nate JP'Z^, will be greater than the ordinate of the curve BOOK in.] THE ELLIPSE. 83 If we denote this ordinate by y, the first member of the equation, instead of reducing to 0, will be equal to a posir live quantity. If, on the contrary, we take a point P", within the curve, its ordinate F"JD, will be less than the ordinate of the curve ; and if we designate this line by y, the first member of the last equation will be negative. Therefore, the following analytical conditions will deter- mine the position of a point, with respect to the curve of the ellipse, viz.: Without the curve, In the curve, Within the curve, ^2y2 ^ ^23.2 _ ^2^2 ^ 0. J2y2 ^ J^2rf.2 _ ^2^2 <- 0. .^, Equation when the origin is at the vertex of the transverse axis. 5. If we transfer the origin of co-ordinates from the centre (7, to A^ one extremity of the transverse axis, the equations of transformation (Bk. I., Art. 28), will reduce to, X =z - A -{- x\ y = y'- Substituting these values in the equation of the ellipse, it -reduces to, ^2y'2 ^ ^2a.'2 _ 2B''Ax' = 0, which may be put under the form, B ^ y'' = jjS."^^^' - «'")> or> y' = 3-2(2-40; - x^) 84 ANALYTICAL GEOMETRY, [book III. by omitting the accents. This is the equation of the ellipse refeiTed to the vertex of the transverse axis, as an origin of co-ordinates. In this equation, the absolute term is want- ing, as it should be, since the origin of co-ordinates is in the curve (Bk. 11., Art. 4—9). Eccentricity. Polar equation. 6. The ECCEXTEicTTY of an ellipse, is the distance from the centre to either focus, di\'ided by the semi-transverse axis. If c denotes the distance from the centre to either focus (Art. 3), and e the eccentricity, then, ^/A^^B' and c = Ae (1.) 1. Resuming Equations (8) and (9) (Art. 3), we have, r' =z A + ex . . (2.) and r = A — ex . . (3.) For the value of r', the pole is at F' ; for r, it is at J^ and the origin of co-ordinates is at the A\ centre of the ellipse. Let us transfer the origin of co- ordinates to the focus F'. For this point we have (Bk. I., Art. 28), X = — Ae ■\- x\ and Substituting this value of a; in Equation ( 2 ), we have, r' = A - Ae^ + ex'. But x' = r' cosv (Bk. I., Art. 35). Substituting this value of x\ we have, y = 2/ BOOK III.] THE ELLIPSE. 85 r' = A — Ae"- + er' cos v ; whence, r' = -—^ -i (4.) 1 — e cos V which is the polar equation, when the pole is at F\ 2. If ?; = 0, cosw = 1, and we have, ^, ^ A{\ - e^) ^ ^n _^ ^) ^ ^ ^ ^ ^ ^rj^ 1 — e If V increases from to 360°, the corresponding values of r\ will give all th6 points of the curve. 3. K we had transferred the origin of co-ordinates to the focus JFJ we should have had (Bk. I., Art. 28), X = Ae + x\ and y — y'. Substituting this value of cc in Equation (3), we have, r = A — Ae^ — ex'. But x' = r cos V. Substituting this value of x\ we have, r = -4 — Ae^ — er cos v ; , A(l - e^) whence, r = — ^^ ~ (5.) 1 + e cos V ^ ' 4. We see, from Equations (2) and (3), that when the pole is at either focus, the radius-vectors will be expressed in ratio7ial functions of the abscissas of the points in which they intersect the curve. It may be easily shown, that the foci are the 07ily points in the plane of the curve^ which enjoy this remarkable property. 5. If u = 0, cosy = 1, and we have, r = ^[} - ^') = A{l-e) = A-c z= FJB. I + e ^ 86 ANALYTICAL GEOMETRY. [book m. If V varies from to 360°, the corresponding values of r will give all the points of the curve. The difference be- tween Equations (4) and (5) is this: for y = 0, the value of r' begins at the remote vertex j while in Equation (5), under the same supposition, the corresponding value of r, begins at the nearest vertex. Diameters. 7, A DIAMETER of an ellipse, is a line drawn through the centre, and limited by the curve. The points in which it' intersects the curve, are called vertices of the diameter. Every diameter is bisected at the centre. 8, The equation of the ellipse, referred to its centre and axes, is, ^2^2 _,_ ^2^.2 ^ ^2^2 .... (1.) Since every diameter, as II' CH^ passes through the ori- gin of co-ordinates, its equa- tion will be, y = ax . , (2.) If Equations ( 1 ) and ( 2 ) be combined, the values of x and y, in the resulting equation, will be the co-ordinates of the vertices II and H'. Combining and eliminating, we have, " = ± ^^V3?+^' y = ± ^^« V2S + jp BOOK III.] THE ELLIPSE. 87 If we denote the co-ordinates of the point 11^ by x\ y\ and the co-ordinates of ZT, by x'\ y'\ we shall have, Since the co-ordinates of these points are the same, with contrary signs, it follows that, that is : Every diameter is bisected at the centre, Ordinates to diameters. 9. An ordinate of a point, on a curve, to a diameter, is its distance from the diameter, measured on a line paral- lel to a tangent at the vertex. The parts into which the ordinate divides the diameter, are called seyme?its. Relation of ordinates to each other. 10. The equation of the el- lipse referred to the vertex A, as the origin of co-ordinates (Art. 5), is. 2/' ' - J>^ X)X. If we designate a particular ordinate by y', and its ab- scissa by x' ; and a second ordinate by y'\ and its abscissa by x'\ we shall have. 88 and, ANALYTICAL GEOMETRY. y'^ = ^(2^ - x')x', . y'-- = J(2^ - x")x" . [book m. . (1.) . (2.) Dividing Equation ( 1 ) by ( 2 ), we obtain, y" (2A - x')x' ^ {2 A - x")x"' or. y"2 : ; (24 - x')x' : (2A - x'')x''. But 2A denotes the trans- verse axis A^, and since x' = AD, 2A - x' = DB; ^ therefore, {2A — x')x\ is the rectangle of the segments AD, DB. In like manner, (2JL — x")x", is the product of the segments AE, EB, Since the same may be shown for the conjugate axis, we conclude that. The squares of the orclinates, to either axis of an el- lipse, are to each other as the rectangles of the correspond- ing segments into which they divide the axis. Parameter. 11, The Paea:meter of the transverse axis, is the double ordinate passing through the focus. To find its value, let us take the Polar Equation (5) (Art. 6), A / ^(1 - c2) VIZ r = 1 -\- e cos V BOOK III.] OF THE ELLIPSE. 80 If we make v = 90°, the radius-vector will be per- pendicular to the transverse axis, and r will be equal to the ordinate. Under this supposition, cos v = 0, and we Bhall have, r = A(l — e^). In Art. 6 we have, 92 _ A^- JB^ Substituting this value for e\ we have, = '■(■ - ^^) = Hence, parameter = —j- z= -^—r = 2r A' YA A If we write this in a proi^ortion, we have, 1A : IB '.'. 2B : parameter ; that is. The parameter of the transverse axis is a third jyro- portional to the transverse axis and its conjugate. 1. In the polar equation, when the pole is at the focue, the numerator^ in the value of r, is equal to half ttic parameter, ' Ellipse and circumscribing circle. 12, If on the transverse axis AB^ the cii'curaference of a circle be described, it is re- quired to find the relation be- Al tween any ordinate GD^ of the \ circle, and the ordinate IID^ of the ellipse, corresponding to the same abscissa CD. 90 ANALYTICAL GEOMETRY [book ni. Let Y' denote any ordinate of the circle, and y' the corresponding ordinate of the ellipse, and x' the common ab- scissa. We shall then have (Bk. n. Art. 4—5), J-/2 A-'- x' (1.) We also hare from the equa- tion of the ellipse referred to its centre and axes (Art. 3), ^2 (2.) Dividing Equation (2) by (1), member by member, we have Y'2 - ^2' or Y, = B A' B\ that is, Tlierefore, Y' \ y' w A Any ordinate of the circle, is to the corresponding ordi- nate of the ellipse, as the semi-transverse axis, to the semi- conjugate, , 1. , It follows, from the above proportion, that every or- dinate of the circle is greater than the corresponding ordi- nate of the ellipse : hence, every point of the ellipse, ex- cept the vertices of the transverse axis, is within the cir- ■cumference of the circle : therefore, the tra}is verse axis is greater t/ia?i any other diameter, 2. If -S is made equal to A, the ellipse becomes the circumference of the circle described on the transverse axis. BOOK in.] THE ELLIPSE. 91 Ellipse and inscribed circle. 13. If, on the conjugate axis, DD\ the circumference of a circle be described, it is re- quired to find the relation be- tween any ordinate, PN^ of the circle, and the ordinate J/iV, of the ellipse, corresponding to the same abscissa CN. Denote the ordinates, NP and NM^ by X' and x\ and designate CN^ by y'. We shall then have, X'2 ^ _2J2 y'\ and, ^•^ = ^{B^ - yn) hence, If therefore. X'2 - B^' x' A '' X' = b'^ . X' : x' : : B : A\ that is. Any ordinate of the circle is to the con^esponding ordi- nate of the ellipse^ as the seini-co72Jugate axis is to the semi-transverse. 1. Since ^ > i?, every ordinate of the ellipse is greater than the corresponding ordinate of the circle : therefore, 'every point of the ellipse, except the vertices of the con- jugate axis, is without the circumference of the circle. Hence, the conjugate axis is less than cmy other diameter. 2. If ^ is made equal to J?, the elhpse becomes the circumference of the circle described on the conjugate axis. 02 ANALYTICAL GEOMETEY. [book in. (1) J^ Equation of the Tangent. 14. It is required to find tlie equation of a tangent line to the ellipse. Take any point of the curve, as P, and designate its co- ordinates by jc", 2/". Through this point, di-aw a secant line ; its equation Tsdll be of the form, y -y" ~ a{x - x") It is now required to find the value of a, when the secant line PP' becomes tangent to the curve. The equation of the ellipse is, ^2y2 + ^2^2 ^ J^2JS2 .... (2.) Since the point P is in the curve, we shall have, A^y"^ + B'~x''^ = A^J^ . ... (3.) Subtracting (3) from (2), we have, A'~{y' -]/"') + P-(^' - aj"^) = 0; or, A^{y^y"){y-y")-^J^'ix-\-x''){x-x") = 0. (4.) In this equation, x and y are the co-ordinates of any point of the ellipse. If Equation (4), be combined with Equation (1), the co-ordinates x and y, in the resulting equation, will be the co-ordinates of the points in which the secant intersects the ellipse. These equations are most readily combined, by substituting for y — y'\ m Equation ( 4 ), the value in Equation ( 1 ). Substituting, we obtain. BOOK ni.] THE ELLIPSE. 93 or, by factoring, we have, {x - x") X [A^a{y + y") + B\x + x")-] = 0, an equation, which may be satisfied by making, x-x'' = 0; or, A''a{y + y") + J3^{x + x") = 0. In the first equation, x is the abscissa of P; in the second, x and y are the co-ordinates of jP\ Suppose P' to move towards P ; when they become consec- utive, we shall have, x = cc", and y = y'' ; which will give, from the last equation, a = — JPx" AY' Substituting this value in Equation (1), we have, y -y or, by reducing. A'^yy" - AY"" = - B'^xx" + B'^x''^ ; Dr, A'yy" + B^xx" = Ay^ + PV^ ; or, A^yy'' + B'^xx'' = A^B\ ^hich is the equation of the tangent line, and in which, y and x are the general co-ordinates of its points. Sub-tangent. 15. A SUB-TANGENT is the projection of the tangent on the axis of abscissas, or on the axis of ordinates; that is, 9^ ANALYTICAL GEOMETEY. [bOOK 111. it is the part of either axis, from the point of intersection, to the foot of the ordinate through the point of tangency. 1. To find the sub-tangent, take the equation of the tan- gent, A^yy" + B~xx" = A^B^ If, in this equation, we make y = 0, we find. X = A^ which is the line OT. If from CT, we subtract CHy which is designated hj x", we shall have the sub-tangent, x' x" 2. This expression for the sub-tangent TR^ is independent of the conjugate axis, and will, therefore, be the same for all ellipses having the same transverse axis AB^ and the points of tangency in the same perpendicular BP. Hence, if the circumference of a circle be described on the trans- verse axis, and the ordinate RP be produced till it meets the curve at §, the tangent, at this point, will pass through the common point T. The angle CPT^ formed by the tangent line and the diameter passing through the point of contact, will be obtuse. 3. If we determine, in like manner, the sub-tangent on the conjugate axis, it will be independent of the transverse. Equation of the normaL 16. Since the normal passes through the point of tan- gency (Bk. II., Art. 9), its equation will be of the form, THE ELLIPSE, 95 BOOK m.] y-y"= a\x-x"), . . . (1.) and since it is perpendicular to the tangent, we sliall have, aa' -h I = 0, But we have found (Art. 14), Ay ' hence, a' ^2 x" Substituting this value in Equation (1), we have, A^ v" y-y" = % |-,(a: - «"), which is the equation of the normal Hne. Sub-normal. 17. A sub-nok:mal is the projection of the normal on the axis ; that is, it is the part of the axis which lies di- rectly under the normal. 1. To find the sub-normal AT?, take the equation of the normal, A^ v" y-y" = ^ h^'^ - '^")- and make y = ; this will give, X = CJSr = ^i!-^— V' = e'^x'^ (Art. 6). If we subtract this value from (7i?, which is denoted by a?", we shall have the sub-normal. JV7? = J?2 A^ 96 ANALYTICAL GEOMETRY. [book in. Normal bisects the angle of the two lines drawn to the focL 18. If from P, any point of the curve, we draw two lines to the foci F' and F^ and recollect that CF\ or CF^ is equal to c = Ae (Art. -6), we have, by nsing the value of CN - e^x" (Art. 17), F'K = F'G ^- CN = Ae + e'x" = e[A + ex") ; and, FN = CF - CN = Ae- e^x" = e{A - ex"). Hence, FJ^ : FN : : A + ex" : A- ex". By referring to the values of r' and r (Art. 3, Equations (8) and (9), and recollectmg that -j = e, "we have, r' : r : : A -{- ex : A — €x\ hence, r' : r : : F'2T : FN; therefore, FN bisects the angle FFF,* Tangent line and lines to the focL 19. Let C be the cen- tre of the ellipse, FT a tangent, and FF, FF, two lines drawn to the IbcL Draw the normal, FN. Then, since NFJI and NPT are right angles, they are equal. From each, ♦Legendre, Bk. IV. Prop. 17. BOOK III.] THE ELLIPSE. 97 take the equal angles, NPF' and JSfPF, and there T\all remain F'PIT, equal to FPT. Hence, If a line he drawn tangent to an ellipse at any pointy and two lines he drawn from the same point to the two foci-i the lines drawn to the foci will make equal angles with the tangent. Supplementary chords. 20. Let AP be the trans- verse axis of an ellipse. If a straight line be drawn through the point A^ whose co-ordinates are, x' = - A, y' = 0, its equation wiU.be, y = a'{x -f- A). If a line be drawn through P, whose co-ordinates are^ x' = A, and y' = 0, its equation will be, y = a{x - A). If these lines intersect each other, we have, 2/2 _ aa\x^- J.2); . . . . (1.) and if they intersect on the curve of the ellipse, x and y must satisfy the equation. y' p^ (.12 _x'') = - -'{x''^ - ^P). . . (2.) 98 ANALYTICAL GEOMETRY. [bOOK IIL By combining Equations (1) and (2), we have, aa = — — ; that is, If^ through the vertices of the transverse axis^ two sup- jilementary chords he draicn^ the iy)^oduct of the tangents oj the angles lohich they form with it, icill he negative^ and equal to the square of the ratio of the semi-axes, 1. Since the product of the tangents is negative, the angles to which they correspond will fall in diiferent quad- rants.* 2. In the equation, aa = -^,, there are two undetermined quantities, a and cc' ; hence, an infinite number of pairs of supplementary chords may be drawn through the extremities of the diameter A^. If, however, a value be assigned to «, or a', that is, if one of the supplementary chords be given in position, the equation of condition will determme the other, and thei'e- fore, the corresponding supplementary chord may also be drawn. 3. If the ellipse becomes a circle, we shall have, aa' = — 1, or, aa' -|- 1 = ; which shows, that the supplementary chords are perpen- dicular to each other, a property bofore proved (Bk. II., Art. 7). * Legendre, Trig. Art. 59. BOOK III.] THE ELLIPSE. 99 Supplementary chords. Tangent and diameter. 21. Let PT be a tan- gent line to the ellipse, and denote the co-ordinates of the point of contact by a;", y". Then (Art 14), If a diameter be drawn through the point P, its equation will be satisfied for the co-ordinates a;" y'\ giving, and 2/" = a'a", x' (2.) Multiplying Equations ( 1 ) and ( 2 ), member by member, -52 aa' = — A^ When a and a' denoted the tangents of the angles which the supplementary chords make with the transverse axis, we had (Art. 20), hence. If, in this equation, we make, a = a, we shall have, a a'; that is,, If owe chord is parallel to the tangent, the other will be parallel to the diameter passing through the point of conr tact. 100 ANALYTICAL GEOMETKY. [bOOK HI. Or, if we make, we shall have, a a ; that is, If one of the chords he made parallel to the diameter^ the other icill he parallel to the tangent, 1. Since the co-ordinates of the points P and P', are the same with contrary signs (Art. 8), the value of «, in Equa- tion ( 1 ), will be the same, whether we consider the tangent at P or P' ; hence. The tangents drawn through the extremities of the same diameter are parallel. y Construction of tangent lines to the ellipse. 22. Construct a tangent Hne to an ellipse, at a given point of the curve, when the axes are given. First Method. Let P be the given point. On the transverse axis ^1j5, describe a semi-circumfer- ence, and through P, draw PjR perpendicular to AB^ and produce it till it meets the circumference at P. Through P', draw a tan- gent line to the circumference of the circle, and from 7J where it meets AJB produced, draw 7P, and it will be tangent to the ellipse at P (Ait. 15). 1. The angle GPTh^mg aright angle, the angle OPT, which lies withm it, is obtuse. Hence, the angle formed BOOK III.] THE ELLIPSE. 101 by a tangent line, and the diameter passing through the point of contact, is, in general, obtuse. If the point of tangency be at either vertex of the trans- verse axis, the tangent line to the ellipse will coincide with the tangent line to the circle, and will then be perpendic- ular to the transverse axis. Or, if the point of contact is at either vertex of the conjugate axis, the tangent line to the ellipse will become parallel to the tangent line to the circle, and, consequently, perpendicular to the conjugate axis. Second method. 23. Let C be the centre of an ellipse, AB the trans- verse axis, and P the point of the curve at which the tangent is to be drawn. Through P, draw the semi-diameter P(7, and through A, draw the supplementary chord AII^ parallel to it. Then draw the other supplement- ary chord JBII^ and through P, draw FT parallel to BH\ then will FI" be the tangent required (Art. 21). Third method. 24. Let P be the given point. Find the foci F' and F. (Art. 2—4.) From P, draw the lines FF' and FF to the foci. Produce F'F^ un- til FM shall be equal to FF, and draw F3L Then draw FT perpendicular to FM^ and it ^\\\\ be the tangent required, fiinc^ it makes equal angles with the lines FF' and FF [Xvi. la^. 102 ANALYTICAL GEOMETRY. [book m. To draw a tangent parallel to a given line. 25. Let AB he the transverse axis, and M the given line. Through the vertex JB draw the supplementary chord £G, parallel to Jtf. Then draw AG, and through the centre C draw CF parallel to AG, and produce it till it meets the ellipse again at P'. Through P, or P, draw a parallel to GB, and it will be the tangent required. 1. We see, from this construction, that if two tangents be drawn to the ellipse through the two extremities of the same diameter, they will be parallel to each other. To draw a tangent through a point without the curve. J7 ,-' 26. Let M be the given point. With either focus, as jF", as a centre, and a radius equal to the trans- verse axis, describe the arc AW£r', Then, with M as a centre, and a radius equal to MJ\ the distance to the other focus, describe the arc FKHK\ intersect- ing the former in K and K\ Through K, draw KF' \ and through P, where it BOOK ni. THE ELLIPSE. 103 intersects the ellipse, draw the straight line MPT^ and it will be tangent to the ellipse, at P. For, since P is a point of the ellipse, F' P + PF is equal to the transverse axis. But F' P -f PK is equal to the transverse axis, by construction. Hence, PF — PK. Further, since the arc FK is described from the centre M, MF = MK; hence, the line MP has two of its points each equally distant from the pomts F and PT; it is, therefore, perj^endicular to FF* ; and since the triangle FPF is isosceles, M T will bisect the vertical angle P. The opposite angle F'PM, being equal to TPF, is equal to FPT; hence, MT is tangent to the ellipse (Art. 19). 1. The two arcs KHK ., KNK'., will intersect each other in two points, K and K' . There will, therefore, be two lines, KF\ K' F\ drawn to the focus F ; hence, there will be two points of contact, P, P\ and, consequently, two tangent lines, il/P, MP\ CONJUGATE DIAMETERS. St. Two diameters of an ellipse are said to be conju- gate to each other, when either of them is parallel to the two tangents drawn through the vertices of the other. Since two supplementary chords may be drawn, respect- ively parallel to any diameter and the tangent through its vertex (Art. 2i), it follows, that two supplementary chords lufiy always be drawn respectively parallel to any two con- jugate diameters. If, therefore, we designate by a and a', the tangents of the angles which two conjugate diameters make, re- Bpectively, with the transverse axis, these tangents will * Legendre Bk L Prop. 16. Cor. 104 A>-ALTTICAL GEOifETEY. f BOOK UI. fulfill the condition of sup- plementary chords, and sat- isfy the equation, act' = A} Let us designate the cor- responding angles by a and a.' "We shall then have, sm a cos a and a' = sm a' cos a' Substituting these values in the last equation, and reducing, we obtain, A"^ sin a sin a' 4- -Z>- cos o cos a.' = 0, and dividing both members by cos a cos a', we have, A^ tan a tan a' + B- = 0, an equation, which expresses the relation between the angles which two conjugate diameters form with the transverse axis. It is called, the equatioji of condition of conjugate diameters. In the equation of condition, a and a' are undetei-rained. Hence, any value may be assigned to either of them ; and when assicjned, the value of the other can be determined from the equation of condition. If a = 0, we shall have, sin a = 0, and cos a = 1. Hence, JP cos a' = 0, and, consequently, cos a' = ; or, a' = 90°. Therefore, when one of the conjugate diame- ters coincides with the transverse axis, the other will coin- cide with the conjugate axis. Tlie axes, therefore, fulfill the condition of conjugate diameters, as they should do, since each is parallel to the tangents drawn through the vertices of the other. i THE ELLIPSE. 105 BOOK IIl] Ellipse referred to its centre and conjugate diameters. 28. The equation of the ellipse, referred to its centre Ahf + i?2ic2 = A^B^ B and axes, is. Let B'B and DD\ be two conjugate diameters. It is required to refer the el- lipse to these as a system of oblique axes, and to find its equa- tion. The formulas for passing from a system of rectangular to a system of oblique co-ordinates, the origin remaining the same (Bk. I., Art. 29), are, X z= x' cos a -f y' cos a', y = ic' sin a + y' sin a'. Squaring these values of x and y, and substituting in the equation of the ellipse, we obtain the equation of the curve, referred to conjugate diameters ; viz., ( +2(^2 gin a sin a'-f B' cos a cos a')x'y' ) A^B--. But the equation of condition, that the new co-ordinate axes shall be conjugate diameters (Art. 27), is, A^ sin a sin a' 4- J>2 cos a cos a' = ; tcnce, the equation reduces to, (.12 sin2 a'-f J52 cos2 a')y'2_(_ (^^2 ^^^2 a -{■ B"^ cos2 a)x^= A^B^. To find the semi-diameter (7-Z>, make y' = ; then, A^B' A"^ sin2 a -f J52 cos2 a 5* = CB^ = CB'^ = A' 106 ANALYTICAL GEOMETllY. [cOOKIH. If we make x' — 0, we shall have, ^ A- sin^ a' -f M^ cos2 a' The denominators, in the two last equations, are the coefficients of a'^, y'^^ {^ i}^q equation of the curve, referred to oblique axes. Finding their values, and substituting them in that equation, Ave have, y'2 ^.'2 ;|7i + -^ = 1 ; hence, or, omitting the accents of x and y, since they are general variables, ^'2?/2 + J5'2a.2 _ A'^j^'^, which is the equation of the ellipse, referred to its centre and conjugate diameters. This equation, being of the same form as the equation of the ellipse, referred to its centre and axes, it follows, that every value of x will give two equal values of y, with contraiy signs; and every value of y, two equal values of x, with contraiy signs; hence, the ellipse is symmetrical with respect to either of its conjugate dia- meters ; that is, Either diameter bisects all chords drawn parallel to the other and terminated hy the curve. Relation of ordinates to each other. 29. The equation of the ellipse, referred to its conjugate diameters, is, THE ELLIPSE. 107 BOOK III.] A'Y + ^'^^^ = A'^B'\ If we designate any two or- dinates to the diameter AB^ by y\ y'\ and the corresponding abscissas by x\ x'\ we shall have, y^ _ {A' + x'){A' ~x') y"2 - {A' -{- X"){A' - X")' 7/2 : 2/''2 .. {A' + x'){A' - X') : {A' -\- x"){A' - x"). If the ordinates be drawn to the conjugate diameter, it may be readily shown, that, x"' : x"^ :: {B' ^ y'){B' - y') : (B' + y"){B' -^ y") Hence, tlie squares of the ordinates to either of two conjugate diameters^ are to each other as the rectangles of the segments into which they divide the diameter. Parameter. 30. The Parameter of any diameter is a third propor- tional to the diameter and its conjugate. Thus, if P desig- nate the parameter of the diameter 2A\ we shall have, 2A' : 2B' '.: 2B' : P, 2B'^ A' ' or. P = Relations between the axes and conjugate diameters. 31. The equation of the ellipse, referred to its con- jugate diameters, which are oblique axes, is 108 ANALYTICAL GEOMETKY. [jJOOK lU. It is required to refer the ellipse to its transverse and conjugate axes, which is a rectangular system. The formulas for passing from oblique to rectangular axes, the origin remaining the same (Bk. I., Art. 30), are, a sin a' — 2/ cos a' y cos a — x sin a X =■ 2/'= sin (a' — a) sin (a' — a) Substituting these values of y\ x\ we have, (^'2 cos^a +^'2 cos2a') 2/2 4. (^'2 ^\^2a, + ^'2 sin2a')a;2 — 2{A'^ sin a cos a + B'^ sin a' cos a') xy = A'^ JB'^ sin2 (a'— a), which is the equation of the ellipse, referred to its centre and axes, and, hence, must be of the form, Jl^^ -\-B^ x^ =AB^; .... (1.) consequently, A'^ sin a cos a + J?2 gJQ q^' qq^ ^ _ q^ and we have, (.tt'2 cos2a + B"^ cos2a')2/2 + ^A'"^ sin2a + B'^ sin2a')a;2 = ^'2i?'2 fein2 (a' - a) (2.) If we multiply together, the values of A"^ and -S'2 (Art. 2§), we have, A'^B"^ equal to A'^B^. ^< sin'a' ein'a + J? B\^\x:?aQ,Q'S?a' + cos'asin'a') + ^*cos'aCOsV, which may be put under the form, A^BK {A? sin a' sin a + B"^ cos a! cos a) 2 -f ^2^2 31^2 (ct'— a) after adding and subtracting, in the denominator, 2^^-52~(sifr»^^€0S»rt^^ii- <;os^^sin^a'). BOOK III.] THE ELLIPSE. 109 But the first terra ol the denominator is (Art. 27) ; hence, ^/2 £'2 ^ £_J? =—^^ . ^2J52 sin2 (oc'- a) sin2 (a'- a) Whence, A^^^ = A'^B'^ sin2 (a' - a). The second members of equations (1) and (2) are, therefore, equal, and by making x and y each equal to 0, in succession, in equation (2), we prove that the co-efiicients of y^ and a^, in equations (1) and (2), are also equal, each to each; hence, ^''cos^a + ^'^cosV^^^ . . . (1.) A'^ sin'^a + B" sinV'z= B' . . . (2.) A" B" sin*^ (a'- a) = A'B\ or, A'B' sin (a' - a) = AB . . (3.) The equation of condition, of conjugate diameters (Art. 27), is, A' tan a tan a' + B^ = 0. If we add Equations (1) and (2), member to member, and recollect, that cos'^a + sin'a = 1, we have, A"-\-B'' = A'+B\ Tabulating the last three equations, and changing their order, for convenience of interpretation, we have, A' tan a tan a'+ jB^ = (1.) A'B' sin (a'- a) = AB (2.) A'' + B" = A' + B' ... (3.) Tliese three equations express the relations that exist be- tween a and a', the semi-axes, A and B, and any two semi-conjugate diameters. A' and B'. 110 ANALYTICAL GEOMETRY. [bOOK 111. Interpretation of A'^ tan a tan a'+W = 0. 1. If we know the angle which the conjugate diameters make with each other, it will be equivalent to knowing a or a'. For, denote the known angle by /3; then a'— a = i3; or a' = (3 -\- a- hence, , _ . tan /3 + tan a tan a' = tan (/3 + a) = ^— ^ ^ 1 — tan p tan a Substituting this value of tan a', in Equation (1), we have, A' tan'^a + (A' - B') tan a tan i3 + ^* = ; from which we can find tan a, and, consequently, a, in terms of the axes and the known quantity, tan ^S. Interpretation of A'B' sin (a'— a) = AB, 2. Let us suppose the ellipse, whose centre is (7, to be circumscribed by a rectangle, formed by draw- ing tangents at the verti- ces of the axes, and also by a parallelogram, formed by drawing tangents at the vertices of the conjugate diameters. Denote the semi-conjugates, CP and CiV, by A' and B'. From Jf, Draw MK perpendicular to CN'. The angle JSTCP is designated by a' — a, and since MN^G is the BOOK III.] THE ELLIPSE. Ill supplement of NCP^ its sine will be equal to sm (a'— «). Further, NM =z CF=A'. Therefore,* 3fK= A' sin (a'- a). Hence, AB' sin (a'- a) = GP3IN.\ The second member of Equation (2), ^ x ^, is equiva- lent to the rectangle GJBHD. But the parallelogram CPMN is one-fourth the parallelogram 3/Z, and the rect- angle CBIID is one-fourth the rectangle IIF\ hence, Equation (2) expresses the following property: The rectangle which is formed hy drawing tangents through the vertices of the axes^ is equal to the paral- lelogram which is formed hy drawing tangents through the vertices of two conjugate diameters. Interpretation of J.'' 4- B'' — A ■\- B\ 3. If we multiply both members by 4, we have, 4^" + ^B" = W + ^B\ which expresses the following property: The sum of the squares described on the axes of an el- lipse^ is equal to the sum of the squares described on any two conjugate diameters. " The area of the ellipse is found in the Calculus, p. 75. * Legendre, Trig. Art. 37»v'/j^% f Mens. Art. 95. BOOK lY. d — c jB OF THE PARABOLA. 1. The Pakabola is a plane curve, such that any point of it is equally distant from a fixed point and a given straight line. The fixed point is called the fo- cus of the parabola, and the given straight line, the directrix. Thus, if i^ be a fixed point, and £^D a given line, and the point P be so moved, that Pi^ shall be constantly equal to P(7, the point P T\-ill describe a parabola, of which 7^ is the focus, and DU the di- rectrix. 1. This property of the parabola afibrds an easy metliod of describing it mechanically. Let PX be a given line, and JLCD a triangular ruler, right-angled at C. Take a thread, the length of which is equal to the side (7P, and attach one extremity at P, and the other at any point, as P. Place a pencil against the thread and the ruler, making tense the parts of the thread PP. PP. Then, if the side BOOK TV,] THE PARABOLA, 113 (7X of the ruler, be moved along the line J5X, the pencil will describe a parabola, of which JF is the focus, and J5i the directrix ; for, the distance JPJ^ will be equal to P C, for every position of the ruler. Equation of the Parabola. 2. Let JF^ be tbe focus, and DC . the directrix. Denote the distance i^, from the focus to the direc- trix, by p, and let the point .1, equally distant from 2^ and J^ be assumed as the origin of a system of rectangular co-ordinates, of which AX, AY, are the axes. The dis- ♦) tance AJ^, will be denoted by -• z Let P be any point of the curve, and denote its co-ordi- nates by X and y. Then, the distance between any two points (Bk. 1. Art.. 1»), is, v^" - ^r + (y" - ^y Substituting for x'\ y", the co-ordinates of the point P, which are x and y, and for x', y', the co-ordinates of F^ which are, a/ = "^j and y' — 0, we have. FP = yV+(7-|) But, by the definition of the curve, FP = PC = PA -{- AB 1 + ^ '(I.) I 114 ANALYTICAL GEOMETRY. [bOOKIV. Hence, sj y-^ _f. ^a; - 1^ = | + aj , or, 2/2 + a.2 _ j^ ^ ^ _ ^ ^ ^ _j_ a.2. hence, y^ = Ipx, which is the equation of the parabola, referred to its vertex and the rectangular axes, AX^ and -4 Yi Interpretation of the equation. 3. The axis of abscissas, AX^ is called the axis of the parabola, and the origin -4, is called the vertex of the axis, or principal vertex. 1. The equation of the parabola gives, y = ±: ^2px. from which we see, that for every positive value of aj, there will be two equal values of y, with contrary signs. Hence, the parabola is symmetrical with respect to its aa^is. 2. "We see, further, that y will increase with a;, and will have real values so long as x is positive. Hence, the curve extends indefinitely^ in the direction of x positive. If we make a; = 0, we have, 2/ = ±0, which shows, that the axis of I^ is tangent to the curve, at the origin. If we make x negative, we shall have. y = ± -/- 2/)aj; BOOK IV.] THE PAEABOLA. 115 or, y imaginary ; which shows, that the curve does not pass the axis of Y, and extend on the side of x negative, 3. By a course of reasoning similar to that in Bk. IIL, (Art 4 — 5,) we have the conditions for determining the posi- tion of a point, with respect to the cm*ve. They are, For a point without the curve, y^ _ 2px > 0. For 'a point in the curve, y^ — 2px = 0. For a point within the curve, y^ — 2px < 0. "^^JL Parameter. 4, The Parameter of the axis, is the double ordinate through the focus. 1. If, in the equation of the parabola, y"^ = 2px, we make, x = ^, the corresponding value of y, will be the ordinate through the focus. Under this supposition, we have, y2 = 2p X I = jp2 ; or, y = p. Hence, 2p = the parameter. 2. In the ellipse, the parameter of the transverse axis is ar third proportional to the axes (Bk. m.. Art. 11); in the parabola, it is a third proportional to any abscissa and the corresponding ordinate. For, from the equation, 2/2 = 2px, we have, X : y :: y : 2p. 3. If the parameter and axis of the parabola are known, 116 ANALYTICAL GEOMETKY. [bOOK IV. we have a simple construction for detennining points of the curve. Let ^JT, AYj be the co-or- dinate axes. The equation of the curve is, From the origin A, lay off a distance AB, on the nega- tive side of abscissas, equal to 2p. Then, from A, lay off any distance, as AP, and draw PM perpendicular to AJi. On -SP, as a diameter, de- scribe a semi-circumference, and through ; that is. The squares of the ordinates are to each other as their corresponding abscissas. BOOK IV.] THE PAKABOLA. 117 Polar Equation. 6. Let US resume the consideration of Equation (1) (Art. 2), which is, FP = r =^_^x (1.) T =l + X, and in which the origin of co-ordi- nates is at the vertex of the axis. The formulas for transferring the origin to the focus, whose co-ordi- nates are, a P , and h (Bk. L, Art. 2§), are, X =^ + x\ and y y'^ Substituting this value of ic, in Equation (1), we Lave, p ■{- x' . . . (2.) -1+1 + -' If we denote the variable angle which the ladius-vector makes with the axis, by v, we have, x' = r cos V ; hence, r = p ■\- r coo c ; whence. P 1— cosv (3.) which is the polar equation of the parabola, when the pole is at the focus. 1. We see, from Equation (2), that the radius-vector is expressed, rationally, in terms of the abscissa of the point in which it intersects the curve. This property is peculiar to the focus. 118 ANALYTICAL GEOMETEY. [bOOK IV. Interpretation of the polar equation. 7, In the polar equation, P 1 — cosv' as well as in the corresponding equation of the ellipse, which is expressed under a similar form (Bk. IH., Art. O-l), the values of the radius-vector begin at the remote vertex, that is, in the case of the parabola, at an infinite distance from the focus. If we make v = 0, we have, r - - = CO. If we make v = 90°, we have, r = p. that is, half the parameter. If we make v = 180°, we have, 1. If it is desirable that the values of r should begin at the nearest vertex, make v = 180° — v\ and we shall have, cosv = — cosu'. Substituting, — cos v' for cos v, the equation becomes, P 1 + cos v' ' in which equation, the values of r begin at the nearest vertex, and v' increase from to 360°. BOOK IV.] THE PARABOLA. 119 Tangent line to the parabola. 8. Let us designate the co-ordi- nates of any point of the curve, as P, by jc", y" \ the equation of a straight line passing through this point will be, y-y" = a{x-x") . . (1.) It is required to determine a, when the right line is tangent to the para- bola. The equation of the parabola is, y^ = - tain, (y + y"){y-y") = 2p{x-x") . . (2.) Combining Equations ( 1 ) and ( 2 ), we have, {y + y") a{x- X") = 2p{x - W') ; or, transposing and factoring, {x-x'')[a{y + y^')^2p] = 0; an equation which may be satisfied by making, X — x" = 0, or, a(y + y") — 2p = 0. In the first equation, x is the abscissa of P; in the second, y is the ordinate of 7*';\vlKn P' becomes consecutive 120 ANALYTICAL GEOMETRY. [bOOK IV. with P, we have, n- P a = —rr y Substituting this vahie, in the equation of the line passing through jP, we ha^e, y-y"= ~r,{^ -''")■' and, by reducing, and observmg that y""^ = 2pa5", yy" = p{^ + «")> which is the equation of the tangent. Sub-tangent. 9. If, in the equation of the tan- yy" = p{^ 4- X"), we make y = 0, we shall have, = 2^^ + »") ; but since the factor p, is a constant gent. quantity, T A X -\- x' ; or, jc' rr that is, AD^ the abscissa of the point of tangency, is equal to — AT \ or, the sub-tangent 7!Z>, is bisected at the vertex A. The analytical condition, expressed by x + x" = 0, or, AT + AD = 0, indicates, that the quantities are numerically equal with contrary signs ; hence, they are estimated on different sides of the oriirin. BOOK IT.] THE PA R ABOL A. Normal and Sub-normaL 121 10. Let x'\ y"^ be the co-ordmates of the point of tangency. Then, the equation of the normal will be of the form, y-y" = a\x - x"\ and since it is perpendicular to the tangent, aa' + 1 — 0. But we have already found (Art. 8), V y' therefore, we have, hence. r p y y-y"= - ^('^ - «"), which is the equation of the normal. 1. If, in the equation of the normal, y-y"= - ^'(^ - «"), * we make y = 0, and then find the value of a; — a", we shall have, X — x" = p. But, X is equal to the distance AN"^ and x" to the dis- tance AR ; hence, x — x" = RN' = p ; that is, the sub- normal is constant, and equal to half the parameter. 6 1-J2 ANALYTICAL GEOMETRY. [book rv. Ferpendiciilar from the focus to the tangenL 11. The equation of a line passing through the focus, whose co-ordinates are, a' = ^ , and y' = 0, is, y = 4-f) The condition, that this line shall be perpendicular to the tangent, gives, aa' -^ 1 = 0; hence, P y the equation of the perpendicular HF^ therefore, becomes. » = -*-!) P Combining this with the equation of the tangent TP, which is, and substituting for y""^^ its value 2/>aj", and reducing, we find, x{1x" + ^) = ; an equation which can only be satisfied when a; = ; hence, the point H^ at which the perpendicular meets the tangent^ is 071 the axis of Y. 12, Through P, the point of contact, draw PD perpen- dicular to the axis ; then (Art. 9), TA = AD ; and hence, TIT = HP, THE PARABOLA. 123 HOOK IV.] Therefore, the two right-angled triangles TFII and IIFP^ have the two sides about the right angle equal ; consequent- ly, the triangles are equal, and the angle FTP is equal to TPF\ that is. The tangent to the parabola at any point of the curve, makes equal angles with the axis and with the line drawn from the point of tangency to the focus. 13. In the right-angled triangle TFH, in which AS is perpendicular to TF^ we have,* FH = FTx FA', or. FH FP X FA, But, FA is equal to —p\ hence, it is constant for every position of the point of contact ; thbrefore, FlP varies as the distance FP \ that is, The square of the perpendlcidar drawn from the focus to the tangent, varies, as the distance from the focus to the point of contact. CONSTRUCTION OF TANGENT LINES, Tangent line at a given point of the curve. 14.— 1. — First Method. Let P be the given point. Draw PR perpen- dicular to the axis. Then, from the vertex A, lay oS AT equal to AE, and join T and P ; TP will be tan- gent to the curve (Art. 9). » Legendre, Bk. IV. Prop. 23. 124 ANALYTICAL GEOMETRY [book IV. 2. — Second Method. Draw the or- dinate Pi? to the axis, and from the foot P, lay off a distance Hy = p, and join P and JST. Then, draw TF perpendicular to PiV at P, and it will be the tangent required (Art. 10). 3. — Third Method. Join P and the focus P' (next figure). Then lay off from JF] on the axis, a distance FT, equal to PP, and join P and T; FT will be the required tangent. (Art. 12.) Tangent parallel to a given line. 15. Let P(7 be a given line, to which a tangent is to be dra^-n parallel. At the focus F, lay off an angle XFF, equal to twice the angle which the given line makes with the axis of X. Through P, the point at which FF intersects the curve, draw FT parallel to P(7, and it will be the tangent required. For, the outward angle FFX is equal to the sum of the angles T and PPP* But FTF being equal to the angle which BC makes with the axis of X, is equal to one-half of FFX \ hence, the angle FTF is equal to half of FFX\ therefore, the triangle FTF is isosceles, and, consequently, FT is tangent to the curve at P (Art. 12). *Legendre, Bk. I. Prop. 25. Cor. 6. BOOK ^V^] THE PARABOLA. ""'^ent through a given point without the curvew 125 Let 6^ be a given point, through which a tangent is to be drawn. With (r, as a centre, and a radius equal to GF^ the distance to the focus, describe the arc of a circle intersecting the directrix at C and C. Through C and C\ draw two lines parallel to the axis BX^ inter- secting the parabola in F and F . Through G^ draw GF and GF^ and they will be tangents to the parabola, at F and F' . For, join F and the focus F. Then, since P is a point of the parabola, FF— FC\ and, by construction, GF= GC\ hence, the line GF has two points, G and P, each equally distant from C and P; it is, therefore, perpendicular to (7P* Since the triangle OFF is isosceles, FG bisects the angle (7PP; therefore, TFF = FTF ', hence, TF is tangent to the curve (Art. 12). It may be proved that GF' is tangent at F', PARABOLA REFERRED TO OBLIQUE AXES. ,17. We have thus far deduced the properties of the parabola, from its equation, obtained by referring the cui-ve to a system of rectangular co-ordinates, having their origin at the vertex. We now propose to develop some of the properties of the curve, by referring it to a system of oblique co-ordinates. * Legendre, Bk. I. Prop. 16. Cor. 126 ANALYTICAL GEOMETEY. [bOOK IV. Sqoation when referred to oblique axes. 18. The formulas for passing from rectangular to ob- lique co-ordinates, when the origin is changed (Bk. I., Art. 29 — 1), are, X = a + ic' cos a + y' cos a', y = J + a' sin a + y' sin a' Substituting these values of x and y, in the equation, y2 = 2px, it becomes, (1.) y'2 sin2a'+ 2x'y' sin a sina'+ x'^ sin^a + b^— 2ap ) __ + 2(6sina'— ^cosa')y'-f 2(5 sin a — j9C0sa)aj' ) In this equation, there are four arbitrary constants, a, b, oL and a', to which we can assign values at pleasure. By giving a fixed value to either, we introduce one con- dition into the coefficients of Equation ( 1 ), and by assigning values to all, we introduce four conditions. Let the values assigned to these arbitrary constants be such, that Equation (1) shall contain only the second power of y, and the fii'st power of x; that is, reduce to the form, 2/2 — 2/XB. This requires that, b^ - 2ap = (1.) sin^a = (2.) an a sin a' = (3.) ft sin a' —jo cos a' = . . . . . (4.) BOOK IV.] THE PARABOLA, 127 Having introduced these conditions, the equation becomes, ,'2 - y ^^ sin^a' Let us interpret these four conditions, separately. Interpret the equation, 52 — 2ap = 0. 1. This equation of condition is of the same form as the equa- tion of the parabola, referred to the primitive axes. Therefore, the co-ordinates of the new origin will satisfy the primitive equation, and hence, the new origin is on the curve at some point as A\ y Interpret the equation, sin^a = 0. 2. In this equation of condition, we have, sin^a = ; hence, a = 0, which shows, that the new axis of abscissas, -X"', is parallel to the primitive axis AX^, Interpret the equation, sin a sin a' 0. 3. This equation of condition, is satisfied by virtue of the sin a = ; hence, it is nothing more than the second. 128 ANALYTICAL GEOMETRY. [bOOK IT. Interpret the equation, h sina' — p cosa' = 0. 4. This equation of condition, gives, P tan a' = Y ; o and since this value of tana' is the same as that found in (Art. 8), for the tangent of the angle which the tan- gent makes with the axis of JT, we conclude that the new axis y, is tangent to the parabola at the new origin, A'. Interpret the equation, y^ = . ^ / ^'' 19. To simplify the form, put, we shall then have, by omittmg the accents of the variables, 2/2 = 2p% for the equation of the parabola, referred to the new axes. 2/? The coefficient, 2p', or its equal, . ^ ^ , is called the 2)arameter, of the new diameter A'^'. In this equation, every value of x will give two equal values of y, with contrary signs ; hence, the curve is symmetrical with respect to the axis A'Jl' ; or, this axis bisects all chords of the parahola ichich are parallel to the tangent A' Y'. 1. Diameter^ as a general term, designates any straight line which bisects a system of chords drawn parallel to BOOK rv.] THE PARABOLA. 129 the tangent at the vertex, and terminating in the curve; and the curve is said to be symmetrical with respect to the diameter, whether the chords are oblique or perpen- dicular to it. In this sense, therefore, every line drawn parallel to the axis AX, is a diameter of the parabola; hence, all diameters of the parabola are parallel to each other, a 'property lohh-h shows that the centre of the curve is at an infinite distance from the vertex. For the area of the parabola, see Calculus, page 72. EOOK Y. OF THE HYPERBOLA 1, An Hyperbola is a plane curve, such that the dif- ference, of the distances from any point of it to two fixed pomts, is equal to a given distance. The fixed points are called the foci. The characteristic property of the hyberbola, gives rise to the following constructions of the curve. First — By a continuous movemenL 2. Let F' and F, be the foci, W and FF^ the distance between them. Take a ruler, longer than the distance F'F^ and fasten one of its extremities at the focus F' . At the other extremity, J?i at- tach a thread of such a length, that the length of the ruler shall exceed the length of the thread by a given distance AB. Attach the other extremity of the thread at the focus F, Press a pencil, P, against the ruler, and keep the thread constantly tense, while the ruler is turned around F'^ as a centre. The point of the pencil will describe one branch of the curve. For, FF + PH is equal to the length of the thread, to BOOK v.] THE HYPERBOLA. 131 which if we add AB, we shall have the length of the ruler. Hence, or, F'JP + JPJI = IT+ PjEr+ AB; F'P- FP = AB; therefore, P is a point of the hyperbola. If the extremity of the ruler, attached at the focus F^ be removed, and attached at F, the second branch of the hyperbola, WAP^ may be described in a similar manner. 3. The TRANSVERSE AXIS is that part of the line drawn through the foci, lying between the two branches of the curve, as AB. The points A and J5, in which the trans- verse axis intersects the curves, are called vertices, 4. The CENTRE of the hyperbola, is the middle point (7, of the transverse axis. 5. The line (7Z>, drawn through the centre, perpendicu- lar to the transverse axis, and equal to the square root of OF — CB , is called the semi-conjugate axis ; and DD\ is the conjugate axis. Second — Construct the curve by points. 6. Let AB be a given line, ani F' and F^ two given points. It is required to describe an hyperbola, of which AB shall be the transverse axis, and F' and F^ the foci. 132 ANALYTICAL GEOMETBY. [book V. From the focus F, lay off a distance F'N^ equal to the trans- verse axis, and take any other distance, as F'H^ greater than F'B, With F' as a centre, and F'H as a radius, describe the arc of a circle. Then, with F as a centre, and NH as a radius, describe an arc intersecting the arc before described, at p and q ; these will be points of the hyperbola ; for, Fq — Fq^ is equal to the transverse axis AB. If, -rt-ith jP as a centre, and F'H as a radius, an arc be described, and a second arc be described, with F' as a centre, and NH as a radius, two points in the other branch of the curve will be determined. Hence, by changing the centres, each pair of radii wiU determine two points in each brunch. I Third— When the axes are given. 7. Since the square ot the semi-conjugate axis is equal to the square of the distance from the centre to the focus minus the square of the semi-transverse axis, the square of the distance from the centre to the focus (Art. 5), is equal to the sum of the squares of the semi-axes. Let AB and DB\ be the axes of an hyperbola. At the vertex B^ draw BH perpendicular to AB^ and make it equal to the semi-conjugate axis (7X>, or CD', Join H and the centre C, Then, with C BOOK v.] THE HYPERBOLA 133 as a centre, and CH as a radius, desciibe a semi-circum- ference, intersecting AB produced in F and F' ; these points will be the foci. The curves may then be described as before. y Equation of the Hjrperbola. 8. Let F and F\ be the foci, and denote the distance between them by 2c. Denote the semi-transverse axis CB^ by -4, and the semi-conjugate, (7Z>, by B. Let P be any point of the curve, and designate the distance F' P, by r\ and FP^ by r\ then, 2^1 will denote the given line AB^ to which the difference, F'P — PF^ is to be equal. Through the centre C, draw CD perpendicular to F'F^ and let C be the origin of a system of rectangular co-or- dinates. Let X and y denote the co-ordinates of any point, as P. The square of the distance between any two points of which the co-ordinates are a;, y, and x\ y' (Bk. I., Art. 19), (y - y'Y + (^ - a')'. Jf the distance be estimated from the point F\ of w^hich the co-ordinates are, cc' = — c, y' = 0, we shall have. F'P = r'2 = 2/2 _p (aj ^ c)' (1.) and if it be estimated from the point F^ of which the co-ordinates are cc' = -f c, and y' = 0, we shall have, 134 ANALYTICAL GEOMETRY, [book V. FP^ - r^= y^+ {x-cy. . (2.) If we add and subtract Equa- tions (1) and (2), we obtain, r'2 + 7-2 = 2(y2 + iB2 + c2). . (3.) and, r'2 — r2 = 4cx . , (4.) Equation (4) may be put under the form, {r' + r) {r' — r) = 4cx . . . . (5.) But we have, from the property of the hyperbola, r'—r = 2A ..... (6.) Combining (5) and (6), we have, ..... (7.) Combining { 6 ) and ( 7 ), by addition and subtraction, we obtain, (8.) r'=A + ^ ... and, r = — ^ + ex (9.) Squaring both members of Equations ( 8 ) and ( 9 ), com- bining the resulting equations, and substituting the values of r'2 and r^, in Equation (3), we have. ^2-H c^x^ 2/2 + a;2 4- c^ BOOK v.] THE HYPERBOLA. 135 Substituting for c^, its value, A^ + JP (Art. 7), we have, A^ 4- AW + ^23.2 ^ ^2y2 _|_ ^2a;2 + ^* + A^B^ ; whence, ^V^ - -^^^^ = — ^^^^ which is the equation of the hyperbola, refen-ed to its centre and axes. ^^ Interpretation of the equation. 9. — 1. If in the equation of the hyperbola, ^2y2 _ ^2^2 _ _ J^2J^2^ we make y = 0, the corresponding values of x will be the abscissas of the points in which the curve intersects the axis of X (Bk. 11., Art. 4—1); viz.: X = -\- A, for B; and, x = — A, for A. 2. If we make x = 0, the corresponding values of y will be the ordinates of the points in which the curve intersects the axis of Y^ viz. : y =z + ^/^, for D; and, y = - J? /^", for D' ; and since these values are both imaginary, the curve does not intersect the conjugate axis (Introduction, p. 24). 3. If we place the equation of the hyperbola under the form, J3 , y = ± ~jv^^ ~ ^^^ ^^® ^^®» 1st. That, for every value of ar < A, whether positive 136 ANALYTICAL GEOMETRY. [bOOK V. or negative, tlie corresponding values of y will be im- aginary. 2d. That, for every value of ic > ^1, whether positive or negative, there will be two values of y, numerically equal, with contrary gigns. Hence, we see, 1st. That both branches of the curve are symmetrical with respect to the axis of X. 2d. That they do not approach nearer to the centre, than the ver- tices B and A, 3d. That from the vertices, they extend indefinitely in the direction of x positive and x negative. 4. By a course of reasoning similar to that pursued in (Bk. III., Art. 4 — 5), we find the following analytical con- ditions for determining the position of a point with respect to the hyperbola: Without the curve, ^2^2 _ jpy.% _}_ j\2jp. y q^ In the curve, A^y"^ — B'^x^ + A'^B'^ — 0. Within the curve, -4^ __ J522.2 ^ ^2^52 ^ q. 5. By comparing the equation of the hyperbola with the equation of the ellipse, referred to its centre and axes, it is seen, that the two are identical., except in the sign of ^, which is minus in the hyperbola, and plus in the ellipse. We may, therefore, pass from one equation to the other, by substituting for B^ B^/— 1. Hence, it follows, that Every result obtained from the equation of the ellipse, will become a corresponding result from the equation of the hyperbola, by changing B into B^— 1. BOOK v.] THE HTPEKBOLA. 137 Equation when the origin is at either vertex of the transverse axis. 10. If ^Ye transfer the origin of co-ordinates from the centre C to A, one extremity of the trans- verse axis, the equations of trans- formation (Bk. I., Art. 2§), will reduce to, X = - A -\- x\ y = 2/'. Substituting these values in the equation of the hyper- bola, it reduces to, Ay^ - i?2ic'2 + 2B^Ax' which may be put under the form. 0; A' y" = -ji[^" 2Ax% which is the equation of the hyperbola, referred to the ver- tex A^ as an origin of co-ordinates. K we refer it to the vertex ^, as an origin, the equation ■vsill become. B^ y'2 = jrpAx^ -f a;'2). >< Conjugate and equilateral hyperbolas. 11. If on the conjugate axis DD\ as a transverse, and a focal distance equal to ^/A^ -f jB^, Tve construct the two branches of an hyperbola, their equation will be, j52^2 _ ^2^2 _ ^AW; 138 ANALYTICAL GEOMETRY [book V. or. ^2y2 _ J52^2 ^ J[2^^ in whicli JB denotes the semi-trans- verse axis, and A the semi-conju- gate. T^o hyperbolas, thus con- nected, are called conjugate hyper- bolas. The conjugate axis of the one, is the transverse axis of the other, and the focal distances are equal. 1. If the transverse and conjugate axes are equal, the hyperbolas are called Equilateral. The equation then be- comes, 2/2 _ a.2 _ _ j2^ when A is the transverse axis, and y JB\ w-hen £ is the transverse axis. These correspond to the case in which the ellipse becomes the circle. Eccentricity. 12. The EccENTKicrrY of the hyperbola, is the distance from the centre to either focus, divided by the semi-trans- verse axis, and is denoted by e; hence, c = Ae. Polar Equation- 13. Resuming Equations (8) and (9), (Art. 8), we have, »•' = ^ + ^, and r = - J. 4- 2 J or, r' = A -\- ex, , , (1.) and r = — J. + «b . . (2.) BOOK v.] THEHYPERBOLA. 139 In the first of these equations, the pole lies without the curve, and in the second, within it. 1. If the origin of co-ordinates be transferred to the focus F\ whose co-ordinates are, — c = — Ae^ and 0, we have the formulas (Bk. I., Art. 2§), X — — Ae + x', and y = 2/'* Substituting this value of cc, in Equation (1), and r'cosv, for x\ we have, r' = A — Ae^ + er' cos v ; Ail — e^) whence, r' = -— ^ ^ , (3.) 1 — e cos V which is the polar equation of a branch of the hyperbola, in terms of the eccentricity and variable angle, when the pole is without the curve. 2. If the origin of co-ordinates be transferred to the focus ^ whose co-ordinates are, -f c = Ae, and 0, (Bk. I., Art. 28), we have the formulas, X = Ae + x\ and y = 2/'- Substituting this value of a;, in Equation (2), and rcosv, for x'j we have, r = — A -{- Ae^ -f er cos v ; •• whence, r = — -— ^ , (4.^ 1 — e cos V ^ ' which is the polar equation of a branch of the hyperbola, in terms of the eccentricity and variable angle, when the pole is within the curve. 14:0 XALYTICAL GEOMETRY, [book V. 3. "We sec, from Equations (1) and (2), that the values of '»•' and r, are expressed, rationally, in terms of the ab- scissas of the points in which the radius-vector intersects tJie curve. This property is peculiar to the focus, as a pole. Diameters. 14. A Diameter of any hyperbola is a line drawn through the centre and limited by the curve. The points in which it intersects the curve, are called vertices of the diameter. Every diameter is bisected at the centre. 15. If, in the equations ex- pressing the values of x', y\ x'\ y" (Bk. m., Art. 8), we substi- tute for B, B^ — 1, we have, •='= ^^\i^^^^ y'= '-^^"vS^i ■=-^^vkS' y"=-^'"'\S^- Hence, the co-ordinates of H' and H^ are equal, with con- trary signs, therefore, CS = CH'. Relation of ordinates to each other. 16. The equation of the hyperbola, referred to the ver tex B of the transverse axis (Art. 10), is, BOOK v.] THE HYPERBOLA. 141 y. i?2 If we designate a ^^articular ordinate by y', and its abscissa by x\ and a second ordinate by y", and its abscissa by a;", "we shall have, ^^ y'2 = -^I'lAx' + cc'2), and y'"^ = 2^(2.iic" + a;"^). Dividing the first equation by the second, we obtain, {1A + x' )x' (2^ + x")x" or, ,//2 2/'^ : y^ : : (2^ + x')x' : {2A + a;")a;", in which the segments, are, •» 2A 4- ic^ a;', and 2A + a;", a;". Hence, the squares of the ordinates are to each other as the rectangles of the segments. Parameter. ly. The Paeameter of the transverse axis, is the double ordinate passing through the focus. To find its value, let us take the Polar Equation (4) (Art. 13), I — e cosy If we make v = 90°, we have (Bk. III., Art. li), 14:2 ANALYTICAL GEOMETRY. [book V. Substituting for e^^ its value A^ ' and reducing, i?2 _, ^ 2^ 4^ r = —r\ hence, Parameter = —-r- = ——r • A^ A 2A If we write this in a proportion, we have, 2A : 2B : : 2B : Parameter; that is, The parameter of the transverse ojxis^ is a third propor- tional to the transverse axis and its conjugate, 1. The numerator, in the value of r' or r (Art. 13), is equal to half the parameter. -i Equation of the tangent. Sub-tangent. 18. The equation of a tangent to an ellipse, at a point whose co-ordinates are a", y" (Bk. m., Art. 14), is A'yy" + B'xx" = A^B^. This will become the equa- tion of a tangent to the hyperbola, if we substitute for jB, B'^— 1 (Art. 9 — 5) ; this substitution gives, A^yy" - B^xx'' = - A^B^; which is the equation of a tangent to the hyperbola, 1. I^ in this equation, we make y = 0, we have, CT=x = ~r BOOK v.] THE HYPERBOLA. Subtracting this from CB =z x" we obtain, TB = ~. 143 •which is the value of the sub-tangent. Equation of the normal. Sub-normaL 19. The equation of a normal line to an ellipse, at a point whose co-ordinates are x'\ y" (Bk. HI, Art. 16), is -^ y-y"=B^'i---")- This becomes, by changing the sign of jB^, y-y"= - W'(=^ - '"")' which is the equation of the normal, PK. 1. To find the point in which the normal intersects fhe axis of -3r, make y = 0, and we have. cjsr = X A^A- B^ x'\ and by eubtractuig ic", we find the sub-normal, BN^ 4- —W-' Ui ANALYTICAL GEOMETRY. [bOOK V. Tangent bisects the angle of the two lines drawn to the focL 20. If from P, any point of the curve, we draw two lines to the foci F' and I\ and recollect that CF' or CF is equal to c — Ae (Art. 13), we have, by using the value o^ CT = — (Art. 18), X ^' Aex"-A^ A{€x"-A) ^ and TF=Ae---r, = -7, = -^^>7 ; hence, F'T : TF : : ex" + A : ex" - A, By referring to the values of r' and r (Art. 8), and re- memhering that -j = ^' ^^ have, r' ; r : : ex" ■\- A : ex" - A; hence,* F'T : TF : : r' : r. Therefore, PT bisects the angle F'FF\ Hence, i^ a ?jwe he drawn tangent to an hyperbola at any point, and two lines he drawn from the same point to the foci, the lines drawn to the foci xcill malce equal angles with the tangent. _____^ * Leg., Bk. II. Prop. 4. Cor. f Leg., Bk. lY. Prop. 17. BOOK V.J THE HYPERBOLA 145 Supplementary Chords. ai. The equation of condition of supplementary chords, in the ellipse (Bk. in., Art. 20), is, aa' z = - J52 Substituting for ^, B^ 1, we have, aa' = A' ir^ 1. If the chords are drawn to any point P, in the branch IIBP^ the tangents a and a' will be both positive ; if drawn to a point in the other branch, they will both be negative. 2. If tne hyperbola is equilateral, A = B, and there will result, aa' = 1, wliich shows, that the sum of the two acute angles formed^ by the supplementary chords xoith the transverse axis, o?i the same side, is equal to 90°.* Supplementary Chords. Tangent and diameter. 22. In the ellipse, the relation between the tangents of the angles which a tangent and the diameter passing through the pomt of con- tact make with the transverse axis, is expressed by the equation * Lcgendre, Trig. Art. 61. 10 146 A>'ALYTICAL GEOMETRY. [BOOK V. (Bk. m, Art. 21), aa B^ ^2*' hence, in the hyperbola, it is. aa = A' But the equation of condition of supplementary chords, is aa = hence, aa' = aa' ; that is. If one chord is parallel to a diameter, the other wiU be parallel to the tangents drawn through its vertices. Construction of tangent lines to the hyi)erbola. 23. The property proved in Art. 22, enables us to draw a tangent to an hyperbola, at a given point of the curve. Let (7 be the centre of the hy- perbola, AB its transverse axis, and P the given point of the curve at which the tangent is to be drawn. Through P, draw the serai- diameter PC, and through A, draw the supplementary chord AH, parallel to it. Then, draw the other supplementary chord BIT, and through P, draw PT parallel to BB:-, then will l^T bo tho tangent xequired. •-V H \^ ■^ A c-r\B ■ / \ BOOK v.] THE IIYBEKBOLA. 147 24. The property proved in Art. 20, enables us to draw a tangent to an hyperbola, through a given point without the curve. Let H be the given point. "With this point as a centre, and IIF^ as a radius, describe the arc of a circle. With F\ as a centre, and a radius equal to the transverse axis, describe the arc of a circle intersecting the former at G and 6^'. Draw i^' 6r, cutting the curve in P. Through P, draw UPT^ and it wdll be tangent to the hyperbola at P. For, if we draw 7£F, IIG, we shaU have HF = JIG, by construction ; and since jP is a point of the hyperbola, and F'G equal to the transverse axis, we shall have PF = PG; hence, PT is perpendicular to FG;* and since the triangle FGP is isosceles, PT will bisect the angle F'PF, and will, therefore, be tangent to the hyperbola. 1. The tw^o arcs described with the centres F' and JI, intersect each other in two points, G and G'; a, line may, therefore, be drawn through F' and either of these points, thus giving two points of tangency, and two tangents. Conjugate diameters. 25, Two diameters of an hyperbola are said to be con- jugate to each other, when either of them I'S parallel to the two tangents drawn through the vertices of the other. * Legendrc, Bk. I. Prop. 16. Cor. 148 ANALYTICAL GEOMETKY. [bOOKV. 26. The equation of condition of conjugate diameters in the elKpse (Bk. III., Art. 27), is, A^ sin a sin a' 4- j5^ cos a cos a' = 0. Hence, for the hyperbola, it is, A^ sin a sin a' — J52 cos a cos a,' = 0, or, A^ tan a tan aj — B^ = 0. Hyperbola referred to its centre and conjugate diameters. 27. The equation of the hyperbola, referred to its centre and axes, is ^2^2 _ ^.^2 _ _ ^2J52^ The fonnulas for passing from rectangular to oblique co-ordinates, the origin remaining the same, are, X = a' cos a + 2/' cos a', y = x' sina. -\- y' sin a'. Squaring these values of x and y, and substituting in the equation of the hyperbola, we have, (.42sin2a'— ^2cos2a')2/'2+(.4=sin2a-J52cos2a)a;'2 I _ _j^2j^ -\- 2 {A'^sia asm a' — J3- cos a cos ci')x'y' )~ But the condition, that the new axes shall be conjugate diameters, gives, A^ sin a sin a' — i?2 ^os a cos a' = ; hence, the equation reduces to (J.2 8in2a'-^2cos2a')y^2 ^ (^2sui2a _ J52cos2a)a;'2 _ _ A^B^, If we suppose, in succession, y' =0, a' = 0, and denote BOOK v.] THE HYPERBOLA. 149 by A' and D\ the corresponding abscissas and ordinates, we find, A'^ = —— , B"- A? sin^a — IP- cos^a ' A? sin^a' — IP cos^a' If ^'2 is positive, A' will be real, and the diameter will intersect the curve. Under this sujDposition, we shall have, A? sin^a < JS^ cos^a, or, tan « < -7 • But, tan a tan a! = -j^ (Art. 26) ; hence, tan a' > — ; or, A? sin^a' > B"^ cos^a' ; hence, B"^ will be negative. The supposition, therefore, w^hich renders A'"^ jDOsitive, or A' real, gives B'"^ negative, or B' imaginary; which shows that only one of the diameters intersects the curve. At- tributing to -B'2^ itg proper sign, we have, A'^ = - .„ . / ^ — ^ , -B'^ = A"^ sin^a — J?2 cos^a ' A"^ sm^cc'—B'^ cos^a' Findmg the values of the denominators in these equations, substituting them in the general equation, and reducing, we obtain, A'Y^ - J?'2^'2 _ _ ^'2^'2 . or' omitting the accents of x and y, since they are general variables, we obtain, which is the equation of the hyperbola, referred to its centre and conjugate diameters. 150 ANALYTICAL GEOMETRY, [book V. Interpretation. 1. We have already seen (Art. 9—2), that when the transverse axis A£ is real, the conjugate axis DD' will be imaginary, and recipro- cally; that is, the two axes will not intersect the same branch of the hyi^erbola. The last proposition proves the same property for any two con- jugate diameters. If then, 2A' designates the diameter IT'IT, 1B' will designate the conjugate diameter G' G^ terminating in the conjugate hyperbola ; and each will be parallel to the two tangent lines drawn through the vertices of the other. If B' were made real, A' would be imaginary, and the equation would represent the cuiwes FJDG^ F'D'G\ 2. The equation of the hyperbola, referred to its centre and conjugate diameters, being of the same form as when referred to its centre and axes, it follows, that every value of a;, will give two equal values of y, with contrary signs ; or, if B' were real, every value of y, would give two equal values of a*, with contrary signs ; hence, each hyperbola is symmetrical with respect to the diameter which it inter- sects; that is. Either diameter bisects all chords drawn parallel to t/ic otha', and terminated by the curve, 3. It may be readily shown, that the squares of the or- dinates to either diameter, are proportional to the rect- BOOK v.] THEHYPERBOLA. 151 angles of the corresponding segments, from the foot of the ordinates respectively, to the vertices of the diameter. 4. The equations of the hyperbola and ellipse, referred to their centres and conjugate diameters, are identical^ except in the sign of jB'^, which is minus in the hyper- bola, and plus in the ellipse. We may, therefore, pass from one equation to the other, by substituting for B\ B'\^ — 1. Hence, it follows, that, every result obtained from the equation of the ellipse^ loill become a correspond- ing result from the equation of the hyperbola^ by changing B' into ^V - 1- Relation between the azes and the conjugate diameters. 28. In the ellipse (Bk. III., Art. 3l), we have, A^ tan a tan a' + i?^ = (1.) ^'^' sin (a'- a) = ^^; and, . . . (2.) A' -^ B" = A' -{- B' ... (3.) J>y substituting for B^ B -y/"^^ and for B . ^^/TTT, we have, for the hyberbola, -l'^ tan a tan a'— J5' = .... (1.) A'B' sin (a'- a) = AB , . . (2.) and, A" - B" = A' - B' , (3.) *■ Interpretation. Equation (2). — Construct a rectangle on the axes, and a parallelogram on two conjugate diameters. Di-aw from G a perpendicular to CII; this perpendicular will be equal to B' sin (a'— a). Hence, the area of the paral 153 ANALYTICAL GEOMETRY, [book V. lelograra CGJPJI, is equal to ^'i?'sm(a'_ a) = AB', hence, tlie whole parallelo- gram is equal to the whole rectangle. Therefore, the parallelogram formed by drawing tangents at the four vertices of conjugate diameters^ is equivalent to the rectangle formed by drawing tangents through the vertices of the axes. 2. Equation (3). — This equation, or, 4^ '2 _ 4J5'2 ^ 4^2 _ 4^2^ expresses this property: TJie difference of the squares of two conjugate diameters is equal to the difference of the squares of the axes. Hence, there can be no equal conjugate diameters, unless A = B\ in which case, the hyperbola is equilateral, and then, every diameter will be equal to its conjugate. (A/^ The hyperbola referred to its asymptotes. 29. The Asymptotes of an hyperbola, are the diagonals of the rectangle described on the axes, indefinitely produced in both directions. Thus, II' 11^ G'G^ are as}'mptotes of the hj'perbola whose transverse axis is AB, and also of the conjugate BOOK v.] THE HYPERBOLA 153 hyperbola wLose transverse axis is DD', If we designate the angle estimated from CB around to Cn^ by a, and the angle BGG^ by cfJ \ or, what is equivalent, designate the angle BGG\ by -a', tan a = or, tan^a = B_ A j52 (1.) tan a A^ sin^a— ^2 cos^a =: 0. ( 3.) B^ A^' -A-^'y or, tan^a' = ^2sinV— ^2cosV=0. (4.) Equations (3) and (4) express the relations between the angles which the asymptotes form with the transverse axis. They are called, equations of conditio^i, 1. If the hj-pcrbola is equilateral, A = B, and Equor tions (1) and (2), give, tan a =r 1, and tan a' = — 1 ; which shows, that the asymptotes make equal angles xcith the transverse axis — one lying in the first and tJiird angles, BOOK v.] or, by putting Jf, for THE HYPEKBOLA ^2 + JJ2 155 , and omitting the accents, xy = M. Interpretation of the equation. 31. If lines be dra^vn through the vertices of the axes, they will form the rhombus AD'BD. The di- agonals (7P, (7§, of the rectangles described on the semi-axes, are equal to each other, and each is equal to V!Z2~+TB2. But these are also equal to the diagonals BD^ BD\ and each pair mutually bisect each other at H and N. Hence, CH = lv^+~B'^, and (7iV = l^^A^+B^; therefore, CH X CN = A^B^ ^ 4 If we designate the angle included between the as}Tnp- totes by /3, we shall have, ^ CB^ X CJSTsm [3 = xi/ sin (3; the first member of the equation is equal to the rhombus CHBlsf'^ the second, to any parallelogram, as CQMK^ whose sides are denoted by x and y; that is, TTie rhombus described on the abscissa and ordinate of the vertex of the hyperbola, is equivalent to the paralklO' 15f> ANALYTICAL GEOMETRY. [bOOKV. gram described on the abscissa and ordinate of any point of the curve. 1. The rhombus CSBK^ described on the abscissa and ordinate of the vertex of the hyperbola, is called the power of the hyperbola. It is one-eighth of the rectangle described on the axes. Asjrmptotes approach the curve. 32. Let us resume the equation of Art. 30, xy =z M, from which, y = — • Since M is constant, if x increases continually, y will diminish, and if x becomes infinite, y will become ; hence, the hyperbola continually approaches the asymptote, and as y cannot become negative so long as x is positive, it foUows that the curve T\ill touch the asymptote when y is 0. The same may be shown with respect to the other asymptote. Hence, The asymp^totes continually approach tne hyperbola^ and become tangent to it at an infinite distance from the centre. Asymptote, the limit of tangents. 33. The equation of the tangent, when the curve is is referred to its centre and axes (Art. 18), is, A^yy" _ B^-xx" = - A^JBK If we make y = 0, we find, A^ 25 = ZT5 BOOK v.] ALGEBRAIC CUR YES. 157 which is the distance from the centre to the point in which the tangent intersects the transverse axis. If x" increases, x diminishes, and if x" be made infinite, X will be equal to 0; that is, the tangent line Avill pass through the centre, and since both the tangent and asymp- tote touch the curve at a point infinitely distant from the centre, they will coincide. 1. The asymptotes have been defined as the diagonals, prolonged, of the rectangle described on the axes. It is easily proved, that they are also the common diagonals of all 'parallelograms formed hy drawing tangent lines through the vertices of conjugate diameters. ALaEBRAIC CUEVES. Classification. 34. An ALGEBRAIC CURAT: is one in which the relation between the co-ordinates of all its points are expressed only in algebraic terms. 35. We have seen that every equation of the first degree, between two variables, is the equation of a straight line (Bk. L, Art. 18). We have also seen, that the equations of the circle, the ellipse, the parabola, and the hyperbola, are all of the second degree; and analogy would lead us to infer, that every equation of the second degree heticeen two variables, represents one or the other of these curves. This is now to be proved rigorously. 36. The general equation of the second degree between two variables, is, A- 158 ANALYTICAL GEOMETEY. [book V, A%f + Bxy -^ Cx^-^- Dy + Ex ■\- F = 'y . (i.) which contains the first and second powers of each variable, their product, and an absolute term, F, The coefficients, A^ J5, (7, 7>, jEJ and F^ are entirely independent of the variables y and x. They are called con- stants ; or arbitrary constants, since values may be attri- buted to them at pleasure. 37. Let us suppose that the co-ordinate axes are rect- angular; this supposition will not render the discussion, or the results, less general. For, if the co-ordinate axes were oblique, we might readily pass to a system of rect- angular co-ordmates, without affecting the degree of the equation^ since the equations for transformation are always linear. Change of direction of the axes. 3§. To pass from a system of rectangular to a system of oblique co-ordinates, the origin remaining the same, we have (Bk. I., Art. 29), sc = tc' cos a -}- 2/' cos a', y — cc' sin a -f y' sin a'. Substituting these values of x and y, in Equation (1), we have. A sin^cc' B sina'cosa' C cosV (2.) y'2 -\- 2 A sin a sin a' _S sin a cos a' B sin a' cos a 2 C cos a cos ol' x'y' -f A sin^a ^ sin a cos a C cos2a «'» + Z) sina' JS'cosa' y' -h i>sina \x'-\-F=^ EcosoL I BOOK v.] ALGEBRAIC CURVES. 159 Since a and a' are entirely arbitrary, we may assign U> either of tbera, such a value as will reduce the co-efficient of x'y' to 0. This supposition gives, 2 A sin a sin a' + J? (sin a cos a' + sin a' cos a) -f 2 (7cos a cos a' zr 0. If we suppose, a' — a = 90°, the new axes will be rect- angular, and we shall have, sin a' = cos a, and cos a' = — sin a. Making the substitutions, we have, 2^ sin a cos a + ^(cos^a — sin^a) — 2 (7 sin a cos a = 0; or, (A — (7)2sina cos a + _B(cos% — sin^a) = 0. But, 2 sin a cos a = sin 2a, and cos^a — sin^a = cos 2a;* hence, . (^ — (7) sin 2a + j5 cos 2a =z 0. Dividing both members of the equation, by cos 2a, we have. tan 2a (3.) c Therefore, when the new axis SZ\ makes with the prim- itive axis -Z", an angle equal to half the angle whose tangent IS -j -^, the coefficient of ic'y' will reduce to 0. Equation ( 2 ) will then take the form, omitting the accents, A'y^ 4- (7V + Z>V + ^'x + J^ = . . (4.) * Legendre, Trig. A.rt. 66. 160 ANALYTICAL G E O M E T K Y. [bOOK V. Change of the origin of co-ordinates. 39. The formulas for passing from a system of co-ordi- nates to a parallel system (Bk. I., Art. 28), are, X = a -\- x\ and y = 5 -f y'. Substituting these values of x and y, iu Equation (4), we have, Ay"' + C'x'-' + ^A'h y' + 20 'a + A'h'' + C'a^ + D'h + E'a ■\- F = , . (5.) In Equation ( 5 ), a and h are entirely arbitrary. If we attribute to them such values as make, 2A'b + ^' = 0, whence, J =_-_,..( c.) and, iG'a -{- E' =z 0, whence, a = — —j^,^ . . (7.) and put, - {A'b'' + Co' -\- D'b + Ea + E) = F\ Equation (5) will become, omitting the accents, ^y -f- C'x^ =z F' (8.) Interpretation. 40. — 1. The ti*ansformation, from Equation (1) to (4), is always possible. For, such a value may be given to a or a', as shall render the coefficient of x'y\ in Equation (2), equal to 0. 2. The transformation, from Equation (4) to (8), is BOOK V.J ALGEBRAIC CURVES. 161 always possible, except in the cases when b and o^ in Equa- tions ( 6 ) and ( 7 ), are both infinite, or vrhen either of them is infinite. Under the first supposition. A' = 0^ and C = 0, which causes the second powers of the variables to disap- pear, in Equation (4), and the equation then becomes, D'y + E'x + F = 0, which is the equation of a straight line (Bk. I., Art. IS). K only one of them is infinite, as a, for example, then, C = 0, and Equation ( 5 ), after making this supposition, takes the form, ^y -f Ex = F". If we now transfer the origin of co-ordinates to a point on the axis of JT, such that, _ F" X - -^-x, we shall have, A'y^ -\-E'{^, - x'\ = F" ; E' or, 2/^ = ■j,x\ or, y^ = ^px, which is the equation of a parabola. Every curve, denoted by an equation of the form, y = Ipx, in which n is any positive number, except 1, is called ai parabola. If n = 2, we have the common parabola. If n = 3, tho cubic parabola, &c. 3. Let us interpret Equation (8), A' if -j- Cx'' = F\ 11 162 ANALYTICAL GEOMETRY. [bOOK V. When A'^ C\ and I^\ are all positive, this is the equation of an ellipse, referred to its centre and axes (Bk. HE., Art. 8) ; then. A' = A\ C = ^, and JF" = A''B\ If A' = C\ it becomes the equation of the circumference of a circle. 4. 1^ A' is negative, and C and F' positive, then, by changing ,the signs of both members, A'lf- CV = -F\ which is the equation of an hyperbola referred to its centre and axes ; then, A' = A'', C = J^\ and F = A^B^ (Bk. lY., Art. 8). 5. If JL' is negative, €' positive, and F' negative, then, by changing the signs, we have, which is the equation of a conjugate hj^erbola ; C = J3\ A' = A^, F = A^JB^, and 2B the transverse axis (Bk. v.. Art. 11). 6. If, in Equation ( 2 ), we attribute such values to a and a', as shall reduce the coefficients of the second powers of the variables to ; and then transfer the origin of co-ordi- nates, so as to get rid of the first powers of the variables, the equation wWi take the form, xy = M, which is the equation of an hyperbola, referred to its centre and asymptotes (Bk. Y., Art. 30). Hence, Every equation of the second degree between two varh- hles^ will, under every hypothesis, represent either a cir- cle, an ellipse, a parabola, or an hyperbola. BOOK v.] ALGEBRAIC CURVES. 163 41. — 1. Lines are classed into orders, according to the degree of their equations. 2. Straight lines are represented by equations of the first degree, between two variables, and are called, lines of the first order, 3. The circumference of the circle, the ellipse, the para- bola, and the hyperbola, are represented by equations of the second degree, between two variables ; hence, they are called, lines of the second order, 4. And lines denoted by an equation of the third de gree, are lines of the third order ; and similarly, for the higher degrees. Equations when the origin is in the curve. 42. — 1. The equation of the circle, when the origiB is in the curve, is, y^ = 2JRx — x\ 2. The equation of the ellipse, when the origin ia at the vertex of the transverse axis, is, y^= J(2^aj-c«2). 3. The equation of the parabola, under the same hy- pothesis, is, y^ = 2px. 4. The equation of the hyperbola is, y^= ~{2Ax-\-x% 164 ANALYTICAL GEOMETRY. [bOOK V. These equations may all be put under the form, in which m is the parameter of the curve, and n the square of the ratio of the semi-axes. In the circle and ellipse, n is negative ; in the hyperbola it is positive, and in the parabola it is 0. 2. The curves, whose properties have been discussed in the last four books, are precisely those which are obtained by intersecting the surface of a cone by planes, as is shown in Bk. VI., Art. 45—50. For this reason they are called, Conic jSections. BOOK YI. SPACE— POINT AND LINE — PLANE — SURFACES. a. Space is indefinite extension, and is entirely similar in all its parts. The geometrical magnitudes are portions of space. Their absolute places cannot be determined, either by construction or by the algebraic analysis, since there is nothing fixed to which they can be referred. Their relative positions may, however, be easily found, and these enable us to discuss and develop their pro- perties. 2, Thus far, the analysis has been limited to points and lines lying in the same plane. These have been re- ferred to two axes, making a given angle with each other. The analysis is now to be extended to points and lines in space, which will be referred to three planes, at right angles to each other. 3. Through any point, as -4, conceive a horizontal plane to be drawn. Through the same point, conceive a ver- ticai plane, ZAJ^, to be dra^\Ti : this is the plane of the paper and intersects the horizontal plane in the line X'AX. Through the same point conceive a second ver- z / X / A y Z' 166 ANALYTICAL GEOMETBY, [book VL tical plane to be drawn, per- pendicular to the plane ZAX. This plane will intersect the horizontal plane in the line YA Y\ and the first vertical, in the line ZAZ'. These three planes are called, co- ordinate planes 4. Since the co-ordinate planes are respectively at right angles to each other, the line of intersection of either two will be perpendicular to the third: and this line of inter- section is called the axis of that plane to which it is per- pendicular. For example : Z is the axis of the horizontal plane YX"; y; the axis of the first vertical plane ZAX\ and -X" the axis of the second vertical plane ZAY. The three are called, the co-ordinate axes, and their point of intersection Ay the origin of co-ordinates. 5. The co-ordinate planes are supposed to be indefinite, and hence, they will divide all space into eight equal parts, or triedral angles, having the origin A, for a common vertex. Four of these angles are above the hori- zontal plan^ YAX^y and four below it. They are thua designated iZAX" is called the 1st angle. YAX' " 2d " X'AY' T'AX 3d «« 4th «* BOOK VI.] CO-OEDINATE PLANES. 167 The fifth angle is directly beneath the first, the sixth beneath the second, the seventh beneath the third, and the eighth beneath the fourth. This manner of naming the angles difiers from that adopted in the plane, where the first angle is beyond the axis of abscissas, and where we pass round from the right to the left ; both methods are now too well established to be changed, merely for the purpose of producing uni formity. 6. The distance of any point, in space, from either of the co-ordinate planes, is estimated on the axis of that plane, or on a line parallel to the axis. 7. If from any point, in space, a line be drawn per- pendicular to either of the co-ordinate planes, the foot of the perpendicular is the projection of the point on that plane. 8. The line in which any plane intersects either of the co-ordinate planes, is called its trace on that plane. 9. If, through a straight line, in space, a plane be passed perpendicular to either of the co-ordmate planes, its trace is called, the projection of the line on that plane. 10. Let us suppose that we know the distance of a poiJit from the three co-ordinate planes, viz. : from TZ = a, from ZX = b, from YJl = c. 168 ANALYTICAL GEOilETEY [liOOKVl. From the origin A, lay off on the axis of JT, a dist- ance Ap = a, and through p pass a plane parallel to the co-ordinate plane YZ. Its traces jt?P', pP, will be re- spectively parallel to the axes Z and I" Lay off, in like manner, on the axis of 1% a distance Ap'= J, and through p' pass a plane parallel to the co-ordinate plane ZJC. Its traces jo'P, /)'P" will be parallel, respectively, to the axes -X' and Z. Since the point must be in both planes, at the same time, it will be in their common intersection, which is perpendicular to the horizontal plane at P. Lay off, from the origin of co-ordinates, on the axis of Z, a distance Ap" = c, and through />'', pass a plane parallel to Y^: its traces p"P\ p'^P"-, ^iH be parallel, respect- ively, to the axes JST and Y^ and the point in w^hich this 2>lane is pierced by the perpendicular to the horizontal plane at P, will be the position of the required point. The point will, therefore, be projected on the first vertical plane ZX^ at P', and on the second vertical plane ZY, Sit F". Its co-ordinates, are P/, pP^ and pP'. The distances of a point, from the co-ordinate planes, are expressed, algebraically, by a = a, y = h z=: c, and since these conditions determine the position of the point, they are called, the equations of the p^oint. Hence, the equations of a p>oint are the equations which express its distances from the three co-ordinate planes. BOOK VI.] CO-ORDINATE PLANES. 169 11. Let us consider, separately, the conditions wliich de- termine the distances of a point from the co-ordinate planes. The conditions, a = ± a, limit the point to one of two planes drawn parallel to the co-ordinate plane YZ^ on different sides of the origin, and at a distance from it equal to a. The conditions, y = zh b, limit the point to one of two planes drawn parallel to the co-ordinate plane ZJT, on different sides of the origin, and at a distance from it equal to b. If these conditions exist together, the point will be lim- ited to four straight lines, parallel to the axis of Z. The conditions, 2 = ± c, limit the position of the point to one of two planes drawn parallel to the co-ordinate plane YJT, on different sides of the origin, and at a distance from it equal to c. If all the conditions exist together, the point will be either one of the eight points in which the two last planes are pierced by the four parallels before drawn ; and each of these points will be found in one of the eight angles, formed by the co-ordinate planes. By attributing to the co-ordinates of these points the signs plus and minus, the position of any one of them may be exactly determined. Thus, 1st angle, x =z -{- a. y = -\- b^ z = -\- Cy 2d angle, x = — a, y z= + b, z = + c, 3d angle, x z= — a, y = - b, z = + c, 4th angle, x = + a, y = — b, z = -{- e, 8 170 ANALYTICAL GEOMETRY. [bOOK VI. 5th angle, x = -{■ a, 2/=+ 5, 2=— c, 6th angle, x = — a, y = ■\- h^ z = — c, 7th angle, x = — a, y = — b, z = — c, 8th angle, jc=+a, y =z — h^ z = — c, 12. Smce either co-ordinate denotes the distance of a point fi'om a co-ordinate plane, it follows, that when this distance is 0, the point will be found in the plane. Hence, we have the following for the equations of the co-ordinate planes : For the co-ordinate plane YAX^ whose axis is Z^ 2 = 0, X and y indeterminate ; that is, X and y must be indeterminate, in order that they may represent the co-ordinates of every point of the plane. For the co-ordinate plane JCAZ^ whose axis is YJ y = 0, X and z indeterminate. For the co-ordinate plane YAZ^ whose axis is X^ X = 0^ y and z indctenninate. 13. Since either axis Hes in two of the co-ordinate planes, we shall have, for the equations of the axis of JC^ 2/ = 0, s = 0, and x indeterminate. For the equations of the axis of !FJ as = 0, s = 0, and y indetermmate. BOOK VI.] LINES IN SPACE. For the equations of the axis of Zy a; = 0, 2/ == 0, and z indeterminate. And for the origin, which lies in the three axes, ic = 0, y = 0, and 2 = 0. 14. We also have, for a point in the axis of X^ y = 0, s = 0, and jb = ± a. For a point in the axis of YJ a; = 0, 2 = 0, and y = db 5. For a point in the axis of Z^ a; = 0, 2/ = 0, and 2 = ± c. 171 Distance between two points. 15. Let (ft Q\ Q",) be one of the points, and (P, r, P",) the other. Denote the co-ordinates of the first point, by x\ y\ z\ those of the second by x'\ y'\ z'\ and the length of the required distance, by D, The line §P, is the projec- tion of the given line on the co-ordinate plane of YX^ Q' P Its projection on ZX, and F" Q" its projection on YZ , The distance i>, will be the hypothenuse of a triangle, of which the base is §P, and altitude p'F\ 172 ANALYTICAL GEOMETRY. [bOOKVI. But, Qp ^x"-x\ Fp = y"-y\ and p' T = z" " z' . In the right-angled triangle QPp>i ^^^ have, Qf= {x"-xy+ {y"-y'y\ hence, i>2 ^ ^^» _ r^y + (y// _ y')2 4. ^^n _ ^ry^ and, D= y^x" - x'Y + (y" - y')' + (2" - ^Y- 1. The projection of a line, on either of the co-ordinate axes^ is that part of the axis intercepted between the two perpendiculars drawn through its extremities. Hence, if the line whose length is 2>, be projected on the three co- ordinate axes, x" — x\ y" — y\ z" — z\ will represent, respectively, the length of the projection on each axis; hence, it follows, that the square of any line in space^ is equal to the sum of tJie squares of its three projections on the co-ordbiate axes. 2. If one of the points, the one, for example, of which the co-ordinates arc x\ y\ z\ be placed at the origin,- we shall have. which expresses the distance from the origin of co-ordinates to any point in space. Line and co-ordinate axes. 16. The three lines Fp^ Fp\ Pp" drawn perpendicular to the co-ordinate planes, may be regarded a? the three BOOK VI.] LINES IN SPACE. 173 edges of a parallelopipedon, of which the Ime drawn to the origin is the diagonal. We have, therefore, verified a pro- position of geometry, viz. : the sum of the squares of the three edges of a rectangular parallelopipedon is equal to the square of its diagonal. 1. This last result offers an easy method of determining a relation that exists between the cosines of the angles which a straight line makes with the co-ordinate axes. Let US designate the length of the line, passing through the origin of co-ordinates by r, and the angles which it forms with the axes, respectively, by ^, Y, and Z. We shall then have for the lines Ap, Ap\ Ap'\ which are respectively designated by x'\ y'\ s", the following values, viz.: • x" =z r cos JT, y" = r cos rj 2" = r cos Z, By squaring these equations, and adding, we obtain, rf.fn 4_ y"2 4. ^"2 _ r2(cos'-X + cos2 T -f cos^Z). But we have already found, Hence, cos^X-f cos^F-f cos^Z = 1 ; that LS, the sum of the squares of the cosines of the three angles which a straight line forms with the three co-or- dinate axes J is equal to radius square, or 1. Equation of a straight line in space. IT. Let C'P' be the projection of a straight line on 174: ANALYTICAL GEOMETKT. [book VI. the co-ordinate plane ZX, and C'P", its projection on the co-ordinate plane YZ. Since C'P' is the pro- jection of the line on the co-ordinate plane ZX, the line itself, in space, is in the plane passing through C 'P,' and perpendicular to the co-ordinate plane, ZX (Art. 9). Since C"P" is the projection of the line on the co-ordinate plane ZY^ the line itself, in space, is in the plane passing through C"F" and perpendicular to the co-ordinate plane YZ ; hence, it must be the common intersection of these two projecting planes. The conditions, therefore, which fix the projections of a line, will determine the line in space. Let a = a2 -|- a, be the equation of the projection C'P', and y = bz i- (3, the equation of the projection C"P", In these equations, a denotes the tangent of the angle ADP', a the distance AC', b the tangent of the angle JP"FZ, and /3 the distance AC". The angles in the co- ordinate plane ZX, are estimated from the axis Z to the right, and in the co-ordinate plane YZ, 'they are estimated from the axis Z, towards the left. If we suppose a, a, b, and /3, to be given, the two pro- jections C'JP'i C"F", will be determined; and hence, the BOOK VI.] LINES IN SPACE. 175 line, of which they are the projections, will he determined in space. Hence, az -f ct, y z= bz -h (3, are the equations of a straight line. 1. Since the projections of a straight line on the two co-ordinate planes Z^, ZY, determine the position of the line in space, they ought, also, to determine its projection on the third co-ordinate plane, YX^. This indeed may be easily proved. For, through P' draw a parallel to the axis of Z, and from the point in which it intersects the axis of JT, draw a parallel to axis of Y. Through P" draw a parallel to the axis of Z, and through the point in which it intersects the axis of Y, draw a parallel to the axis of -Z"; then will P be the projection of the point (P, -P"), on the co-or- dinate plane YJT. Find, in a similar manner, the projection of a second point, as (C, C"), and draw the projection CP. V V Interpretation of the equations of a line. 18. — 1. Let us now consider the equations, aj = az -f a, y = ^2 -f /3, separately. 17G ANALYTICAL GEOMETRY. [bOOK VX The equation, aj = as + a, being independent of y, will be satisfied for every point of the plane passing through C'F', and perpendicular to the co-ordinate plane ZX; hence, it may be regarded as the equation of that plane. In like manner, the equation, y = 52 + /3, being independent of ic, will be satisfied for every point in the plane passing through CF"^ and perpendicular to the co-ordinate plane YZ\ hence, it may be regarded aa the equation of that plane. 2. Let us now consider the conditions which would be unposed upon the straight line, by supposing a, 5, a, and jS, to become known, in succession. When a, J, a, and ^, are aU undetermined, the equations, JB = as + a, 2/ r= J2 4- /3, may be made to represent every straight line which can be drawn in space, by attributing suitable values to a, J, «, and i3. And when a, J, a, and ^, have given values, the equations wiU designate but a single straight line. If we suppose a to be given, the line may have any position in space, such, that its projection on the co-or- dinate plane ZX^ shall make an angle with Z, of which the tangent is a. If we suppose a also to be given, the projection of the line on the co-ordinate plane ZX^ will intersect the axis of -X" at a given point, and the two conditions, wiU limit UOOK VI.J LINES IN SPACE. 177 the line to a given plane. Its position in the plane will btill be entirely undetermined. If we now suppose h to be given, the direction of the line will then be determined, but it may still have an indefinite number of parallel positions in the given plane. If finally, we attribute a value to ^, the projection on the plane of YZ^ will mtersect the axis of Y at a given point ; and hence, the position of the line will become known. The letters a and S represent the co-ordinates of the points in which the line mtersects the co-orduiate plane rx. The resolution of problems involving the straight line in space, consists in finding such values for the arbitrary constants a, 5, a, and /3, as shall satisfy the required con- ditions. Equations of a line passing through two points. 19. Let x\ y\ z\ and a;", y'\ 2", be the co-ordinates of two given points. The required equations will be of the form, a; = as + a . . (1.) y = ^»s -f /3 . . (2.) in which it is required to find such values for a, a, J, and' /2, as shall cause the right line to fulfill the required con- ditions. Since the straight line is to pass through a point, of which the co-ordinates are x\ y\ z\ we shall have, a;' = «2' -i- a . . (3.) y' = hz' ^ f^ . . (4.) 12 178 ANALYTICAL GEOMETEY. [bOOK \1, and since it is also to pass through a point, of which the co-ordinates are ic", y", s", we shall likewise have, x" = az" 4- a . . (5.) y" = hz" + ^ . . (6.) The last four equations enable us to determine the four constants, a, a, J, /3. By subtracting Equation ( 5 ) from Equation ( 3 ), and ( 6 ) from (4), we obtain, X' - x" = a(z' - z'% and y' - y" = h{z' - z'% fi-om which we find, ^ --'--" ana j> = y^--y:^^ z' -z'" ^ ~ z' -z"' hence, a and h are detennined. If these known values be substituted, respectively, in Equations (3) and (4), or in (5) and (6), the values of a and /3 will become known, and either set would represent the required line. But it is more convenient to have the equations under another form. Subtract Equation ( 3 ) from Equation ( 1 ), and ( 4 ) from ( 2 ) ; we then have, X — x' = a{z — s'), y — y' = b(z — z'), which are the equations of a straight line passing through a given point. Substituting for a and b, their known values, we have, ■x-x' = J^-^; {z - z'), y-y'= 5--^' {z - z\ which are the equations of a straight line passing througli the two given points. BOOK VI.] LINES IN SPACE. 179 Lines intersecting and paralleL 20. It is required to find the conditions which will cause two lines to intersect each other. Let X = az -{- n, y =z hz -\- ^,' and X = a'z -{■ a', y = h'z + P\ be the equations of the lines, in which the arbitrary con- stants a, a, b, (3, a\ a', h\ /3', are undeteraiined, If these lines intersect each other in space, they must have one point in common, and the co-ordinates of this • point will satisfy the equations of both lines. If wc de- signate the co-ordinates of the common point by x\ y\ z\ we shall have, x' = az' + a . . (1.) y' = hz' -\- ^ . . (2.) x' = a'z' -fa'.. (3.) y' = h'z' -f /3' . . (4.) Eliminating x' and y' from these equations, we find, {a — a')z' -f a — a' = . . . . (5.) (5_ j^y +/3 _/3'= . . . . (6.) and if s' be eliminated from the two last equations, we have, (a - a') (/3 - ^') - (a - a') {h - h') = 0, which is called the equation of condition, since it must always be satisfied in order that the two straight lines may intersect each other. ^ There are eight arbitrary constants entering into this 180 ANALYTICAL GEOMETRY. [bOOKVI. equation. It may, therefore, be satisfied in an infinite num- ber of ways. Indeed, if values be attributed, at pleasure, to seven of the constants, such a value may, in general, be found for the remaining one, as will satisfy the equation, and, consequently, cause the lines to intersect each other. When parallel. 21. Let us now find the co-ordinates of the point of intersection. "We find, from Equations ( 5 ) and ( 6 ), , a' -a /3'_^ 12! — ; , or Z' — -J- jj . a — a — Substituting this value of 2', in Equations (1) and (2), we have. These values of the co-ordinates of the point of inter- section, become infinite, when a = a\ and h = V \ that is, when the projections of the lines on the co-ordi- nate planes ZX and ZY, are parallel. 1. If we have, at the same time, a' = a, and /S' = /?, the co-ordinates of the point of intersection will become - , or indeterminate ; as, indeed, they should do, since the two lines would then coincide throughout their whole extent. BOOK Vl.J LINES IN SPACE. 181 Angle between two lines. 22. Let a; = as + a, y = bz -\- (3^ be the equations of the first line, and aj = a's + a', f/ = b'z-h (3\ be the equations of the second. It has been shown,* that two straight lines which cross each other in space, may be regarded as forming an angle, although they do not lie in the same plane. They are sup- posed to make the same angle with each other as would be formed by one of the lines, and a line drawn through any point of it, and parallel to the other ; or, as would be formed by two lines drawn through the same point, and respect- ively parallel to the given lines. If, then, two lines be drawn through the origin of co-ordi- nates, respectively parallel to the given lines, the angle which they form with each other will be equal to the required angle. The equations of these lines will be, X = az, y = hz^ X = a'z, for the first, b'z, for the second. Let us take, on the first line, any point, as -P, and desig- nate its co-ordinates by x\ y\ z\ and its distance from the origin, by r'. Take, in like manner, on the second line, any * Legendre, Bk. VL Prop. 6. Sch. 182 ANALYTICAL GEOMETBY. point, as P", and designate its co-ordinates by Jc", y", s", and its distance from the origin, by r', and let D denote the dis- tance between the points. If we designate the angle included between the lines, by F", we shall have, in the triangle AP'P"^ cosF= ^n , • • • (1) and we have now only to find r\ r", and 2>. Let ns designate the three angles which the first line forms with the co-ordinate axes, respectively, by -Z", J^ and Z, and the angles which the second line forms with the 8ame axes, by X\ Y\ and Z' ; we shall then have (Art.. 16), sc' = r' cosJT, y' = r' cosYi ^ =. r' cosZ, x" = r"cosX', y" = r"cosF', s" = r"cos^. But the square of the distance between two points is, i)2 = {x' - x"Y + (y' - y"Y -f {z' - z"Y (Art. 15), or, 7>2 = x"'-\-y'^+ z'24- x"^-\- 2/"2+ z"^- 2{x'x"-^ y'y" ^ z'z") ; or, by substituting for the co-ordinates of the points, their diiitances from the origin into the cosines of the angles which the lines make with the co-ordinate axes, we have, r'=(cos'X+ cos'Y+cos^Z) + r"='(cos'X' -f- cos»r + cos'Z')] 2r'r"(cosXcos^' + cosFcoaF' + cos Z cos 2') * Leg., Mens. Art. 97. BOOK VI.] LINES IN SPACE. 183 But it has been shown (Art. 16 — 1), that, C0S2X+ C0S2 Y+ COS^Z = 1 , C0S2 JP + 0082!^ + cossZ' = 1 ; and hence, j)2 _ ^2 1.//2_ 2rV'XcosXcosX''+cosFcosF'+cosZcosZ'). If this value of J)^ be substituted in Equation (1), cosF= ^r, , we shall find, after dividing by 2rV", cos Y = cos X cos X' 4- cos Y cos Y' -\- cos Z cos Z' ; that is, the cosine of the angle included hetioeen two I'mes^ is equal to the sum of the rectangles of the cosines of the angles which the lines in space form with the co-ordinate a'jr4^s. Angle under another form. 23. Having found the cosine of the angle included be- tween two lines, in terms of the angles which they form with the co-ordinate axes in space, we shall, in the next place, find the same value in terms of the angles which the projections of the lines on the co-ordinate planes ZX and YZ^ form with the axis of Z. The equations of the parallel lines drawn throuc^h the orii^in, are, X = az^ y •= bz, X = a'z, y — b'z, y^ Let us desi ornate the co-ordi- I Si ANALYTICAL GEOMETRY. [bOOK TT. natos of tlie point P, on the first line, by x\ y\ z' ; we shall then have, ic' = az\ y' = hz'-, and for the A'alue of r', From these three equations we find, y' = But we have already found (Art. 16), x' = r' cos X, y' — *'' cos J^ z' = r' cos Z. Substituting these values, and dividing by r', we obtain, co&X= . :, cos JF— . -, cosZ=- 'v/r-fa2_|_J2 yi+«2^j2 0_|_a2^J2 If we reason, in the same manner, on the equations of the second line, we shall find, a' h' 1 cos A" = =:, cosF' = ■ ; , cosZ' If these values be now substituted in equation for cos F", we shall have, 1 4- aa' + hh' cosT^ = ± ^/l~+~d' + 62 yi + a"" + h"" The cos "F will be plus or minus, according as we take the signs of the radical factors in the denominator, like or unlike. The plus value of cos Y will correspond to the acute angle, and the minus value, to the obtuse angle. BOOK VI.] LINES IN SPACE. 185 1. If we make, v = 90°, cos"F = 0, and 1 + aa' + bb' = 0, which is the equation of condition, when the two lines are perpendicular to each other in space. EXAMPLES. 1. What is the distance between two points of which the equations are, aj' = 5, y' = 5, z' = S; x" = - 1, y" =0, s" = - 5 ? Ans, 11.18 -f 2. Find the equations of a line which shall pass through a point whose co-ordinates are, a;' = 3, y' = — 2, and z' = 0, and be parallel to a line whose equations are, X = z -\- 1, and y ■— \z — 2. A71S. CC = 2+3, 2/ = -i2 — 2. 3. Required the equations of a line passing through the two points whose co-ordinates are, x' =2, y' = 1, ^' = 0, and aj" = - 3, y" = 0, z" = — 1. A71S. X = 5z + 2, y = z -{- 1. 4. Required the angle included between two lines, whose equations are, X = dz -\- 5 ) p ^, , ^ {. of the 1st, y = 5z -i- 3) and X = z + 1 of the 2d. y = 2z ) Ati^. 14° 58'. 186 ANALYTICAL GEOMETRY [cook VI. 5. Required the angles which a straight line makes with the oo-ordinate axes, its equations being, X = —22-1-1, y = s + 3. Ans. "144° 44' with X, 65° 54' with Y, 65° 54' with Z, 6. Having given the equations of two straight lines, 2s+ 1 ) X y =z 22 + 2 f and X = 2 + 5 y = 42 + /3 ■1 of the 1st, of the 2d, required the value of /3' so that the lines shall intersect each other, and the co-ordinates of the point of intersection. /3' = - 6, Ans. OF THE PLANE. X' = 1 y' = 10, U' = 4. 24, The EQUATION of a plane is an equation, express- ing the relation between the co-ordinates of every point of the plane. To find the equation of a plane. 25. A line is said to be perpendicular to a plane when it is perpendicular to every line passing through its foot BOOK VI.] THE PLANE. 187 and lying in the plane: and, conversely, the plane is said to be perpendicular to the line.* A plane may, therefore, be generated by drawing a line perpendicular to a given line, and then revolving this per- pendicular about the point of intersection. K the perpen- dicular be at right angles to the given line, in all its posi- tions, it will generate a plane surface. Let X = az + a, y = 52 -f /5, be the equations of a given line. If we designate the co-ordinates of a particular point, by x\ y\ z\ the equations of the line passing through this point, will be, X -x' = a{z - z') . (1.) y - y' = h{z - z') . (2.) The equations of a second line passing through the same point, of which the co-ordinates are x\ y\ z\ are of the form, X — x' = a' {z — z') . . . . (3.) y -y' = b'{z-z') . . . . (4.) But the two lines will be at right angles to each other, if their equations fulfill the condition (Art. 23 — 1). I + aa' + bb' = (5.) ^ K we now attribute to a' and b\ all possible values that will satisfy this equation, we shall have all the perpen- diculars which can be drawn to the given line, through the point whose co-ordinates are x\ y\ z' , These perpen- diculars determine the plane. * Legendre, Bk. YI. Def. 1. 188 ANALYTICAL GEOAIETRY. [bOOKVI. It id necessary, however, to find the equation of the plane in terms of the co-ordinates of its different points MVe find from Equations ( 3 ) and ( 4 ), z — z'^ z — z' ' Substituting these values in Equation (5), 1 + aa' + hb' = 0, and reducing, we find, z — z' + a{x — x') + b(y -^ y') = 0; but, since a, b, z\ x\ y\ are known quantities, we may denote the constant part of the equation by a single letter, by making, — z' — ax* — by' — — c. hence, the equation of the plane becomes, z -\- ax -\- by — c = 0. 1. Since the equation of a plane contains three vari- ables, we may assign values, at pleasure, to two of them, and the equation will then make known the value of the third. For example, if we assign known values, denoted by x' and y\ to x and y, the equation of the plane will give, z = c — ax' — by', and hence, the co-ordinate z becomes known. Traces of planes. 26, The lines in which a plane intersects the co-ordi- liOOK VI.] THE PLANE. 189 nate planes, are called the traces of the plane. These traces are found by combining the equation of the plan© with the equations of the co-ordinate planes. Thus, if in the equation, z •\- ax + hy — c = 0, we make y = 0, which is the characteristic of the co- ordinate plane ZJT^ the re- sulting equation, s + aic — c = 0, ij/ will designate the trace CD, common to the two planes. The equation may be placed under the form, 25 = — ax -{- c, and hence, the trace may be dra^vn. Or, if we make, in succession, X = Oy and s = 0, we shall find. = c AD. and = AO, and the trace may then be drawn through the points C and D, 1. We likewise find, for the trace j57>, z = ~- by + c; a c and for the trace BC, y=— ^-aj-l-r-* We also find AD = c, by making y = 0, in the 190 ANALYTICAL GEOMETRY. [bOOK VI. equation of the trace BD^ and AB = -, by making jc = 0, in the equation of the trace BC^ or by making 2 = 0, in the equation of the trace BB. 2. By comparing the equa- „^ tions of the traces with the equation of a straight line, in Bk. 1., Art. 13, we see that, — o, is the tangent of the angle Tviiich the trace CD makes with the axis of X; — b, the tangent of the angle which the trace BB makes with the axis of Y; and — 7 , the tangent of the angle which the trace B C makes with the axis of X. 3. The equation of a plane may be written under the form, Ax + Bj/ -{' Cz -\- B = 0, in which A, J5, (7, and B, are constant for the same plane, but have different values when the equation represents dif- ferent planes. The coefficients A, B, and (7, are arbitrary functions of the angles which the traces of the plane form with the co-ordinate axes, and Z> is an arbitrary function of the distances from the origin to the points in which the plane cuts the co-ordinate axes. If the plane passes through the origin of co-ordinates, its equation takes the form. Ax -{- Bi/ -h Oz = 0. BOOK VI.] THE PLAIJE 191 Ziine perpendicular to a plane. 27. The equations of the straight line, to which the plane has been drawn perpendicular, are, JC - a' = a(2 — Z'), y - y' - l){z- Z') J and the equations of the traces (7Z>, j5Z>, may be placed un- der the form, 1 c \ c X =z 2 + -, 2/ = — t2 + 7 • a a By comparing the coeffi- cient of 3, in the equation of the projection of the line on ^^ the co-ordinate plane ZX^ with the coefficient of s in the equation of the trace (7Z>, we find, that their product plus unity is equal to ; hence, the lines are at right angles to each other. The same may be shown for the trace ^2>, and the projection of the line on the plane YZ ', and also for the trace jBC, and the projection on the plane YX. Hence, this property, viz. : If a line be perpendicular to a plane in space, the projections of the line will he re- spectively perpendicular to its traces. EXAMPLES. 1. Find the traces of a plane whose equation is, 2 — 92/ + 11a; — 12 = ,0. 2. Find the traces of a plane perpendicular to a line, whose equations are, a; = 32 + 5, and y = — 2z — 4, 192 ANALYTICAL GEOMETKY. [bOOK VI. 3. Find the traces of a plane "whose equation is, 2x — 3y — z = 0. SURFACES OF THE SECOND OEDER. 28. The EQUATION of a sukface, is an equation express* ing the relation between the co-ordinates of every point of the surface. It has been shown (Bk. I., Art. 18), that every equa- tion of the first degree, between two variables, represents a straight line ; and in (Bk. V., Art. 40 — 6), that every equation of the second degree, between two variables, represents a circle, an ellipse, a parabola, or an hyperbola. It has also been shown (Art. 26 — 3), that an equation of the first degree between three variables represents a plane, and analogy would lead us to infer what may be rigorously proved, viz.: that everi/ equation of the second degree, between three variables, represe7its a curved surface. 29. Surfaces, like lines, are classed according to the degree of their equations. The plane, whose equation is of the first degree, is a surface of the first order ; and every surface whose equation is of the second degree, is a surface of the second order, 30. The equation of a surface, is an equation which ex- presses the relation between the co-ordinates of every point of the surface. Although the equation determines the sur- face, yet it does not readily present to the mind, its form, its dimensions, and its limits. To enable us to conceive of these, we intersect the surface by a system of planes, parallel to the co-ordinate planes. If then, we combin BOOK VI.] THE PLANE, 193 the equations of these planes with the equation ot the sur- face, the resulting equations will represent the curves in which the planes intersect the surface. These curves will indicate the form, the dimensions, and the limits of the surface. 31. To give a single example, let us take the equation, iC2 + 2/2 ^ 22 _ 722, Let us intersect the surface re- presented by this equation, by a plane parallel to ZX, and at a distance from it equal to c. The equation of the plane will be (Art. 11), z = ± c. Combining this with the equa- tion of the surface, we shall have, jc2 + 2/2 ^ 7^2 _ ^2^ which is the equation of the curve of intersection. This^ equation represents the circumference of a circle, whoso centre is in the axis of Z, and radius, ^H^ — c'^. The ra- dius will be real, for all values of c less than J2, whether- c be plus or mmus. It is zero, when c is equal to i?, and' imaginary, when c is greater than JR. Tlius, in the first case, the intersection will be the circumference of a circle,, in the second case, it will be a point, and in the third, it will be an imaguiary curve ; or, in other words, the plane will not intersect the surface. Since the given equation is symmetrical with respect 13 194: ANALYTICAL GEOMETRY. [bOOK VI. to the tliree variables x, y, and 25, we may obtain similar results by intersecting the surface by planes parallel to the co-ordinate planes, YZ and ZJT. The co-ordinate planes intersect the surface in circlep, whose equations are, (C2 + y2 = Ji2^ X^ + Z^ = I^, ^^ _|_ ^2 _ J^^ These results indicate that the surface whose equation is, JC2 _|_ 2/2 ^_ 22 _ 7^2^ is the surface of a sphere ; but, to prove it rigorously, it would be necessary to show, that every secant plane would intersect it in the circumference of a circle. Surfaces of revolution. 32. Every surface which can be generated by the revo- lution of a line about a fixed axis, is called a surface of revo- lution. The revolving line is called the generatrix j the line about which it revolves, is called the axis of the surface^ or the axis of revolution. In all the cases considered, we shall suppose the generatrix, in its first position, to be in the co-ordinate plane ZX, and to be revolved about the axis of Z. 33. When the generatrix is a straight line, and not per- pendicular to the axis of Z, the surface described is called a surface of single curvature. When the generatrix is a curv^c, the surface is called a surface of double c^'rrature. The section made by a plane passing through the axis, ia called a meridian section; or a meridian curve, when the surface is of double curvature. BOOK VI.] SURFACES OP REVOLUTION. 195 34. It is plain, from the definition of a surface of revo- lution, that every point of the generatrix will describe the circumference of a circle, the centre of which is in the axis. Equations of the surfacen. 35. Let DC he any curve, in the co-ordinate plane Z^, and let it be revolved around the axis of ^; it is required to determine the equation of the surface which it will describe. If we designate the abscissa of any point of the generatrix, as J), by r, and the ordinate by z, the equation of the generatrix will be expressed in terms of r and z ; and may be written under the form, r = n^); (1.) which is read, r, a function of 2, and means, that r may be expressed in terms of z, and when so expressed, the equa- tion of the generatrix is known. We have now to express, analytically, the conditions which will cause this point of the generatrix to describe the circum- ference of a circle around the axis of Z. To do this, we have only to consider, that the circumference described by the point D, will be projected, on the co-ordinate plane YJT, into an equal circumference. If the co-ordinates of the points of this circumference be designated by x and y, we shall have, r = ya^jTp (2.) If we now suppose r to take all possible values in equa- tion (1), that will satisfy the equation of the generatrix. 196 ANALYTICAL GEOMETRY. [bOOK VI. and then combine this equation with Equation ( 2 ), we shall have, r = F{2) = v^M^^ . . . (3.) which is the equation of the surface of revolution. An examination of the construction of Equation ( 3 ), will indicate the method of applying it, in finding the equation of any surface. Equation ( 1 ) is the equation of the generatrix from which r is found, in terms of z and constants. This confines the point D, whose co-ordinates are r and 2, to the generatrix. Equation (2) requires, that the curve traced by Z>, be the circumference of a circle; hence, the combination of those two equations, gives the equation of the surface. 36. As a first application, let it be required to find the surface generated by the semi-circumference of a circle, whose centre is at the origin of co-ordinates. The equation of the generatrix will be, r^+ z" = JR^l hence, r ^ -/i^^ _ g2^ • Substituting this value of r, in Equation (3), we have, V-^' - 22 = V'a;2 + 2/2 . or, x^ + 2/2 + s2 _ j^2^ which is the equation of the surface of the sphere, when the centre is at the origin of co-ordinates. 37. The volume described by the revolution of an ellipse about either axis, is called, an ellipsoid of revolution. It BOOK VI.] SURFACES OF REVOLUTION. 197 is also, sometimes, called a spheroid. It is called a prolate spheroid^ when the ellipse is revolved about its transverse axis, and an oblate spheroid^ when it is revolved about the conjugate axis. | 38. Let it be required to find the equation of the surface of a prolate spheroid. Since the transverse axis of the ellipse coincides with the axis of Z, the equation of the generatrix will be, jBV ^ ^2^2 ^ ^2^2 . , /32J52 _ ^2^2 hence, r = ^ -^^ Substituting this value, in the general equation of the surface of revolution, Equation ( 3 ), we obtain, which is the equation of the surface of a prolate spheroid. 39. We find, by a similar process, the equation of the surface of the oblate spheroid, to be, ^V ^ J?2(a;2 + 2/2) _ ^2J52^ If in either of these equations, we make -4 = ^, we obtain, a;2+ y2_{. g2 _ J22, the equation of the surface of a sphere. 40. If an hyperbola be revolved about its transverse axis, each branch will describe a volume. The surface of each volume is called a nappe^ and the twp volumes taken 198 ANALYTICAL GEOMETET, [book VL together, are called, an hyperboloid of revolution of two nappes. The volume described by the revolution of an hyj^erbola about its conjugate axis, is called, an hypei'holoid of re- volution of one nappe. 41. If the transverse axis of an hyperbola coincides with the axis of Z, the centre being at the origin, ita equation will be, If the conjugate axis coincides with Z, we have, ^2^2 _ ^2^2 _ _ ^2^2^ In the first case, the equation of the surface is, ^2^2 _ ^2(^:2 + 2/2) — ^2^2 . and in the second, ^2^2 _ ^2(^2 + 2/2) = - A'^B^. 42. If a parabola be revolved around its axis, the volume described, is called a paraboloid of revolution. The equation of the generatrix being, the equation of the surface will be, aj2 ^ 2/2 _ 2^. Surfaces of single curvature. 43. Let the generatrix be a straight line parallel to the axis of Z. Equation (1) will then become, r = l\z) z= cf , an arbitrary constant ; y Vhat is, for every value of z, r will be constant. /^ BOOK VI.] SURFACE OP THE CONK. 199 Equation (2) will become, r = -y/x^ + y\ or, a;2 _|_ y2 __ ^2 . lieuce, the equations of the surface, are, r = an arbitrary constant, and tP- -{- y'^ — r^. The first condition indicates, that every point of tlu; generatrix is at the same distance from the axis of Z; hence, each point will describe an equal circumference. The second condition indicates, that all these circumferences will be projected into the same circumference, on the co- ordinate plane YX\ hence, the surface is that of a right cylinder with a circular base. Surface of the cone. 44. Let the generatrix be any straight line oblique to the axis of Z, as BC. Denote the distance A (7, by c. Tlien, since BC passes through the point (j\ whose co-ordinates are, z' — c, and x' — 0, its equa- ticm (Bk. I., Art. 20), is, * z — ax + c; if r denote the abscissa of any point of the generatrix, and 2 the ordinate, its equation (Art. 35) will be, ar -f- whence. r = 200 ANALYTICAL GEOMETRY whence, vre have (Ait. 35), [book Vl z— c V^M-y'; or. (z — cy = a^ix^ + 2/^; (1.) In this equation, a is the tangent of the angle CBX. Denote its supplement, CBA^ by v ; then,* or. a^ = tan^y a = — tan v ; hence, Equation (l') becomes, (a;2 + y2) tan^y = (2 — c)2 ... (2.) This is the equation of the surface of the cone generated by the line B (7, revolving about the axis Z. It is a right cone^ with a circular base. C is the vertex of the cone, AB^ the radius of the circle in which it is intersected by the co-ordi- nate plane yiY", and v, the angle which the generatrix makes with the base. If the generatnx, BC, be prolonged beyond the point C, the prolongation will generate an equal conical surface, lying above the vertex C. The conical surface below the point (7, is called the loicer nappe of the cone, and the surface above (7, the upper nappe of the cone. * Legendre, Trig. Art. 63. BOOK YI.] SUKFACE OP THE CONE. 201 45. Let the surface of this cone be now intersected by a piano, passing through the axis of Y, and consequently perpen- dicular to the co-ordinate plane ZJC. Designate by it, the angle DAJC, which the secant plane makes with the co-ordinate plane yX. The equation of this plane will be the same as that of its trace AD (Art. 18—1); that is, z = X tan u. If we combine this equation with the equation of the sui-face, we shall obtain the equation of their curv^e of intersection. This equation is of the simplest form, when the curve is referred to two axes in its own plane. Let us, then, refer it to the two axes, A Y^ AD^ in the plane of the curve, and at right angles to each other. If we designate the co-ordinates of any point, referred to these axes, the one, for example, which is projected at -5, by cc', y\ we shall have, AC =^ X = a' cos I/, BC — z = x'sinu; and since the axis of Y is not changed, y = y'- Substituting these values in equation of the surface, (Equa- tion 2), we shall obtain, after reduction, the equation of intersection, y'2 tan^y -f x"^ co8-i« (tan-?; — tan^w) -f lex' sin u = c^; 9* 202 ANALYTICAL GEOMETRY. [bOOK VI. or, omitting the accents, y2 tan^y -f- x"^ cos'^u (tan^y — tan^w) + 2caj sin u — c^. This equation is of the same form as Equation (4), Bk. v., Art. 38 ; hence, what was proved of that, may be proved of this. Therefore, 1st. When the coefficients of y^ and a^, have the same sign, the curve will be an ellipse : 2d. When the coefficient of jc^, is zero, the curve will be a parabola : 3d. When the coefficients of y"^ and jc^, have unlike sfigns, the curve will be an hyperbola. Since the tan^y is always positive, the change of signs in the coefficients of y'^ and x\ must arise from the change of sign in the coefficient of x^\ and since coshi is positive, the sign of this coefficient wdll depend on the relative values of v and u. When v > w, it will be posi- tive ; when v = u^ it will be zero ; when w > ^^, it will be negative. 46. In order to obtain the forms and classes of the curves which result from the intersection of the cone and plane, it might, at first, seem necessary to cause the angle u to vary from to 3G0°. But since the sui*iace of the cone is s}Tnmetrical with respect to its axis, it is plain that all the varieties will be obtained by varying u from to 90°. 47, Let us then resume the equation of intersection, 2/2 tan^y + x^ cos^w (tan^y — tan^w) + 2caj sin w = c-, BOOK VI.] SURFACE OF THE CONK. and begin the discussion of it, by supposing, u = 0, which will cause the secant plane to coincide with the co-ordinate plane ICT. The equation of the curve will then become, 208 tan^y ' hence, the curve is the circum- ference of a circle, of which A IS the centre, and AD equal to c tany , the radius. 48. If we now suppose u to increase, the curve of intersection will be an ellipse, so long as u rve of intersection loill be an ellipse. 49. When u becomes equal to V, the cutting plane becomes parallel to a generatrix of the cone: hence. If a right cone with a circular base be intersected by a plane 204 ANALYTICAL GEOMETRY. [bOOK VI. parallel to the generatrix^ the curve of intersection wiU be a parabola. ♦50. When u becomes greater flian V, the cutting plane will in- tersect both nappes of the cone ; hence, If a right cone with a circular base be intersected by a plane making with the base of the cone an angle greater than the angle formed by the generatrix and base, both nappes of the cone will be intersected, and the curves of intersection are hyperbolas. DIFFERENTIAL AND INTEGRAL CALCULUS, DESIGNED FOR ELEMENTARY INSTRUCTION. By CHAELES DAVIES, LL.D., PB0FEB80K OF HIGHEB MATHEMATICS, COLrMBIA OOLLEQB. Emtbbei), according to Act of Congress, in the year Eighteen Hundred and Sixty. Bt CHAELES DAVIE? In the Clerk's Office of the District Court for the Southern District of New Tork, PREFACE. The Diffeeentiai, aitd Integrai. Calculus is too impor- tant a part of Mathematical Science to be entirely omitted in a course of Collegiate instruction. The abstract quantities, Number and Space, are pre- sented to the mind, in the elementary branches of Mathe- matics, as of definite extent, and as made up of parts; and the value or measure (how much?) in any given case, is expressed by the number of times which the quantity contains one of its parts, regarded as a unit of measure. But we do not attain to a clear apprehension of their quantitative nature^ until we regard them as of indefinite extent, as possessing continuity, and as capable of chang- ing from one state of value to another, according to any conceivable law. The Difierential and Integral Calculus embraces all the processes necessary to such an analysis. It regards quan- tity as the result of change. It examines established laws of change and determines their consequences. It supposes IV • PREFACE. laws of change and traces the results of the hypothesis. In short, it embraces -within its gras}D — in the Material, everything from the minuest atom to the largest body — ^in Space, all that can be measured, from the geometrical point to absolute infinity — in Time, the entire range of duration — and in Motion, every change from absolute rest to infinite velocity. The German, the French, the English, and even the American press, has been prolific in the number of Trea- tises recently published on this subject. The effort to furnish better Text-Books, proves at once the value of the knowledge, and the great difficulty of presenting it in the best possible form. In regard to the Treatise now presented to the public, I have simply to say, that it is an Elementary Text-Book for the use of College Classes, and other classes of about the same grade. The Treatise of Professor Courtexay, late of the Virginia University, and that of Professor Church, of the Military Academy, may be advantageously read by those who wish to advance further; and it is due to Columbia College to state that this Treatise is used in the Course prescribed to all the pupils, and is not an exponent of the higher course pursued by those who make Mathematical Science a special branch. Columbia Collegb, June, 1860. CONTENTS. SECTION I. DEFINITIONS AXi) FIS ST PRTKCIPLE8. UtTIOLS Dcfinitiona 1 Uniform and Varying Changes 2-4 Function and Variable 4-9 Algebraic and Transcendental Functions 9 Geometrical representation of Functions 10 Language of Numbers inadequate 11 Consecutive Values and DifiFerentials. ... 12 Differential Coefficient 13 Form of difference between two states of a Function 14-15 Differential Coefficient and Differential 15 Equal Functions have equal Differentials 16 Converse not True 17-1 Signs of the Differential Coefficient 19 Nature of a Differential Coefficient, and of a Differential 20 Rate of Change 21-24^ Nature of Diflferential Calculus 24-26 . SECTION II. D IFr EBE NTI AL3 OF ALGEBRAIC KUNOTIONa. Differential of Sum or Difference of Functions 26 Differential of a Product 27-29 Differentials of Fractions 20 Differentials of Powers and Formulas 30 Differential of a Particular Binomial 30 Riite of Clian-e of a Function 31 • Partial DiiTcrentiaLs 32-.-34 14 VI CONTENTS SECTION IIL INTEGRATION AND APPLICATIONB. ▲BTIOLa Integration and A pplications 34 Integration of Monomials 35-41 Integration of Particular Binomials 41 Integration by Series 42 Equations of Tangents and Normals 43-50 Asymptotes 60-52 Differential t f an Arc 52 Rectirteation of Plane Curves 53 Quadratures 54 Quadrature of Plane Figures 55 Nature of the Integral 56 Area of a Rectangle 57 Area of a Triangle 58 Area of a Parabola 69 Area of a Circle 60 Area of an Ellipse 61 Quadrature of J^urfaces of Revolution 62 Surface of a Cylinder 63 Surfiice of a Cone 64 Surface of a Sphere 65 Surface of a Paraboloid 66 Surface of an Ellipsoid 67 Cubature of Volumes of Revolution 68 Examples in Cubature 68-71 SECTION rv. BT70CE89IVK D T F F E K E X T I A L S — SIGNS OF DIFFERENTIAL CO- EFFICIENTS — FORMULAS OF DEVELOPMENT. Succepsive Differentials 71 Pigns of the First Differential Coefficient 72 Si'jns of the Second Differential Coefficient 73 Applications 74 Maclaurin's Theorem 75-76 Taylor's Theorem 77-81 CONTENTS. Vii SECTION V. MAXIMA AND MINIMA. ABTrCLP Maxima and Minima 81-86 Points of Inflection 84 SECTION VI. DIFFEKENTIALS OF TRANSCENDENTAL FUNCTIONS. Differentials of Logarithmic Functions 86 Relation between a and k 87-90 Differential Forms which have Known Logarithmic Integrals 91 Circular Functions 92-99 Differential Forms which have Known Circular Integrals 99 Applications 100 SECTION vn. TBANSCENDENT AL CURVES — CURVATURE — BADIU8 OF CURVA- TURE — INVOLUTES AND EV0LUTE3. Claa8i6cation of Curves 101 Logarithmic Curve — General Properties 102-106 Asymptote 106 Sub-tangent 107 The Cycloid 108 Transcendental Equation of the Cycloid 109 Differential Equation 110 Sub-tangent — Tangent — Sub-normal — Normal Ill Position of Tangent 112 Curve Concave 113 Area of the Cycloid 114 Area of Surface generated by Cycloidal Arc 116 Volume generated by Cycloid 116 Spirals, or Polar Lines 117 General Properties 118 Spiral of Archimedes 119 Parabolic Spiral 120 Hyperbolic Spiral 121 Viii CONTENTS. ▲XTICU Logarithmic Spiral 122 DirectioD of the Measuring Arc 123 Sub-tangent in Polar Curves 124-127 Angle of Tangent and Radius-vector 127 Value of the Tangent 128 Diflferential of the Arc 129 Differential of the Area 130 Areas of Spirals v 131-135 CURVATURE. Curvature of a Circle inversely as the Radius 135-136 Orders of Contact 137 Oscillatory Curves 138 Osculatory Circle 139 Limit of the Orders of Contact 140 Radius of Curvature 141 Measure of Curvature 142-144 Radius of Curvature for Lines of the Second Order 144-149 Evolute Curves .< 149 A Normal to the Involute is Tangent to the Evolute 150 Evolute and Radius change by Same Quantity 151 Evolute of the Cycloid '. 152 Equation of the Evolute Curve 153 Evolute of the Common Parabola 154 INTEGRAL CALCULUS. Nature of Integration 1 55 Forms of Differentials having known Algebraic Functions 156-159 Forms of Differentials having known Logarithmic Functions 159 Forms of Diflferentials having known Circular Functions 1 60 Integration of Rational Fractions • 161 Integration by Parts 162 Integration of Binomial Differentials 163 When a Binomial can be Integrated 164 Formula ^ 165 Formula ^ 166 Formula <5 167 Formula 32> 168 Formula ^ 169 INTRODUCTION. Common Teems must always be employed in definitions, because a definition refers to a class of things in which each enjoys at least one property common to all the others. Each individual of a class, so defined, is called a significate. A common term does not express to the mind a distinct and adequate idea of any one of its significates, but a general notion of them all ; hence, we do not com- prehend the full scope and meaning of a definition, until we have ascertained, by careful analysis, the number, of its significates and the exact characteristics of each. Mathematics is the science which treats, primarily^ of the relations and measures of quantities; and secondarily, of the operations and processes by which these relations and measures are ascertained. QuANTriY is anything that can be increased, diminished, and measured. There are two general kinds. Number and Space, and each is subdivided into four classes. Under Number, w^e have Abstract Number, Currency, Weight, and Time; and under Space, Length, Surface, Volume, and Angular Measure. Mathematics, considered as a science of exact relation, is divided into three branches; 1. Arithmetic. 2. Geome- try. 3. Analysis. X INTRODUCTION. AKirnMETic is that branch which treats of the properties and relations of Numbers, when expressed by figures. Geoaietry treats of the properties and relations of Mag- nitudes, by reasoning directly on the magnitudes them- selves, or upon their pictorial representations. The magni- tudes considered in this branch of Mathematics, are, lines, surfaces, volumes, and angles. Analysis embraces all that portion of Mathematics in which the quantities considered are represented by letters, and the operations to be performed are indicated by signs, or conventional symbols. Its elementary branches are, Al- gebra, Analytical Geometry, and Analytical Trigonometry. In these branches, quantities of the same kind are compared by means of their unit of measure, which has a fixed value, and. is generally expressed, numerically, by the unit 1. A Variable Quantity is one which increases or dimi- nishes, according to any law, and thus passes from one state of value to another. If, in changing its value, be- tween any two limits, it passes through all the interme- diate values, it is called a continiioxis quantity. Two VALUES OF A VARIABLE QUANTITY are consccutive, when the greater cannot be diminished, according to the law of change, without becoming equal to the less. Hence, there is no intermediate value. When we say that a continuous quantity passes from one state of value to another, we mean that it either increases or diminishes; and when we speak of the next value, we mean the first value which it assumes when the change begins. These two values are consecutive. INTRODUCTION. XI If we suppose a variable quantity, denoted by x, to have a particular value, x = a^ and afterwards to assume another value, X = «', we may suppose that x changes uniformly from a to a', and assumes, in succession, all the values between its limits. These two suppositions render it impos- sible to express the change in value, either by 1, or by any of the parts of 1. For, denote the uniform change in x^hy h: then, if h could be expressed by a fiaction, however small, that frac- tion could be diminished by increasing its denominator : hence, there would be values between a and «', through which x would not pass, which is contrary to the hypoth esis. Therefore, the hypothesis that x changes uniformly, and passes through all values between x=^ a and a^ = a', renders it impossible to express the change of value by numbers. Therefore, when the change is uniform, the dif- ference between consecutive values cannot be expressed by 1, or in parts of 1. The same is also true when the change is not uniform. For, if the difference of two consecutive values could be expressed by one, or in parts of 1, it could be diminished ; hence, there would be intermediate values, which is contrary to the definition. The hypothesis, therefore, of continuous quantity, renders it impossible to express the elementary changes of value by means of numbers ; and hence, we are unable to deal with such changes by any of the methods of calculation already explained. The Differential and Integral Calculus is the name given to that branch of Mathematics which treats of the properties and relations of continuous quantities — of the laws Xn IS T li O D U C T I O N . of change to wliich tlicy may be subjected, and of the re- sults flowing from sucli changes. When we measure a quantity, great or small, the stand- ard or unit of measure, is of the same kind as the quantity measured, and the ratio of this unit to the quantity, is the result of the measurement. In the operations of the Cal- culus, the assumed unit of measure is the change which takes place in a quantity that varies uniformly. This quan- tity is called the i?idepe?ide?it variable. The quantity whose changes of value are measured, is called a function. The independent variable and function are connected by a law, either expressed or implied, and change, simultaneously, ac- cording to that law. The theory of the Calculus, therefore, rests on the fol- lowing axioms and inferences : 1. Where no law of change has been fixed, such a law may be imposed as will cause a variable to change uni- formly, and to pass through all values betn-^en any two limits. 2. The diflerence between any two consecutive values of a quantity so varying, is constant. 3. The ratio of the smallest change in the independent variable, to the corresponding change of the function, de- termines the rate of change of the function ; and the actual change is denoted by this rate into the change of the in- dependent variable. DIFFERENTIAL CALCULUS. SECTIOlSri. DEFINITIONS AND FIRST PRINCIPLES. Definitions. 1. In the Differential Calculus, as well as in Analyt- ical Geometry, the quantities considered are divided into two classes : ■*st. Consta7it quantities^ ichich preserve the same values tn the same investigation/ and, 2d. Variable quantities, which assume all possible values that will satisfy any equation ichich exjpresses the relatio?i between them. The constants are denoted by the first letters of th« alphabet, «, 5, c, &c. ; and the variables, by the final let- ters, jc, y, 0, &c. Uniform and var37ing changes. 2, There are two ways in which a variable quantity may pass from one value to another. If the variable jc, once had the particular value, a; — a, and afterwards assumed the value, x = a\ we can sup- pose : 14 DIFFERENTIAL CALCULUS. [SEC. L 1st. That during the change from a to a', x assumed, in succession, and by a uniform change, all the values between a and a', just as a body moving uniformly over a given straight line passes through all the points between its extremities; or, 2d. "We may suppose, that during the change from a to a\ X assumed all possible values between its limits, with- out the condition of a uniform change. In both cases, the quantity is said to be continuous. 3, K two variable quantities, y and x, are connected in an equation, as, for example, 2/ = ic2 + 2 ; then, to every value of a, arbitrarily assumed, there will be a corresponding value of y, dependent upon^ and result- ing from^ the value attributed to x. Thus, if we make jc = 4, we have, 2/ = 16 + 2 = 18. If we suppose x to increase from 4 to 5, we ghall have, y = 25 -h 2 = 27; thus, while x changes from 4 to 5, y changes from 18 to 27. If now we suppose x to increase from 5 to 6, y "\vill increase from 27 to 38. Thus, while jb increases unifonnly by 1, y will change its value according to a very different law. Function and variable. 4. When two variable quantities, y and a;, are con- nected in an equation, either of them may be supposed SEO. I.] FIRSTPRINCIPLES. 15 to increase or decrease iiiiifornily ; a variable, so changinsj, is calletl the independent variable^ because the law of change is arbitrary^ and independent of tlie form of equa- tion. This vaiiable is generally denoted by a?, and called simply, the variable. The change in the variable y, de- pends on the form of the equation ; hence, y is called the dependent variable, or function. When such a rela- tion exists between y and cc, it is expressed by an equa- tion of the form, y = F{x\ y = f{x) ; or, /(y, a;) = ; which is read, y a function of x. The letter i^ or y, is a mere symbol, and stands for the word, function. K y is a function of ic, that is, changes with it, x is also a function of y; hence. One quantity is a function of another^ when the two are so connected that ariy chojige of value., in either., pro- duces a corresponding change in the other. 5. If the equation connecting y and ic, is of such a form that y occurs cdone^ in the first member, y is called an explicit function of x. Thus, in the equations, y = ax -\- b of a straight line, y = 'yfl^ — aj2 . . . . of the circle, y = - ^^/A? — x^ , . . of the ellipse, y = ■y/2px of the parabola, and B y — -- ^x^ — A^ . . , of the hyj erbola, y is an explicit function of x. 1*6 DIFFERENTIAL CALCULUS. [SEC. I. But, if the equations are written under the forms, y — ax — h = ; or. A^. y) = 0, if+x^-E^ = 0; or. f{x,y) = 0, ^V ^ ^2^.2 _ ^2^2 ^ ; or. /(^, y) - 0, 2/2 _ 2j:>x = ; or. /(aj, y) = 0, ^2y2 _ jj2aj2 ^ ^2^2 3^ ; or. /(^, y) = 0, y is called an imjjlicU function of x; the nature of the relation between y and x bemg implied, but not developed in the equation. 6. It is plain, that in each of the above equations the absolute value of y, for any given value of a, will depend on the constants which enter into the equation ; this relation is expressed, by calling y an arbitrary function of the constants on which it depends. Thus, in the equa- tion of the straight line, y is an arbitrary function of a and b; in the equation of the circle, y is an arbitrary function of -R ; in the equation of the ellipse, of A and J? ; in the equation of the parabola, of 2p ; and in the equation of the hyperbola, of A and J). 7, An increasing function is one which increases when the variable increases, and decreases when the variable decreases. A decreasing function is one which decreases when the variable increases, and increases when the variable decreases. In the equation of a straight line, in which a is posi- tive, y is an increasing function of x. In the equations of the circle and ellipse, y is a decreasing function of x. In the equation of the parabola, y is an increasing func- tion of X. In the equation of the hyperbola, y is imaginary SEC. I.] FIKST PKINCIPLES. 17 for all values of x < A, and an increasing function for all 2)ositive values of ic > ^. 8, A quantity may be a function of two or more vari^ ables. If u = ax -{■ hxp-^ or w = aa;^- — hy"^ -\- cz -\- d, u will be a function of x and y, in the first equation, and of X, y, and s, in the second. These expressions may be thus written: w = /(«, y), and u = fix, y, 2). If, in the second equation, we make, in succession, the independent variables a, y, and 2, respectively equal to 0, we have, for, x=0, u= ~by^-\-cz+d =/{'(/, z), for, JB=0, and y = 0, u z=z cz + d =f('^) y and, for, x=0, y=0, and z = 0, u = d =a constant. Algebraic and Transcendeiital Functions. 9. There are two general classes of functions : Algebraic and Tra7iscendental. Algebraic functions are those in which the relation be tween the function and the variable can be expressed in the language of Algebra alone : that is, by addition, sub- traction, multiplication, division, the formation of powers denoted by constant exponents, and the extraction of roots indicated by constant indices. Transcendental functions are those in which the relation between the function and variable cannot be expressed m the language of Algebra alone. There are three kinds : 18 DIFFERENTIAL CALCULUS. [SEa I. 1. Exponential functions, in wliich the variable enters as an exponent ; as, u = a'. 2. Logarithmic functions, which involve the logarithm of the variable ; as, u — logjc. 3. Circular functions, which involve the arc of a circle, or some function of the arc; as, w = sin ic, w = cos a, u = tan x. Geometrical representation of Functions. 10. With the aid of Anal}i:ical Geometry, it is easy to trace, geometrically, the numerical relation between any function and its independent variable. Suppose we have given the equation, y = fi^)' If we attribute to x, the independent variable, in succes- sion, every value between — oo and + oo, each will give a corresponding value for y, which may be determined from the equation, y = f{x) Let be the origin of a system of rectangular co-or- dinates. From 0, lay off to the right, all the positive values of jc, and to the left all the negative values. Through the extremity of each abscissa, so determined, draw a line parallel to the axis of ordinates, and equal to the corresponding value of y; the plus valoef T y PC \ X SEC. I.] FIRST PRINCIPLES, 19 will fall above the axis of JT, and the negative values below it; then trace a curve, AMN^ through the extremities of these ordinates. The co-ordinates of this curve will indicate every relation between y and a;, expressed by the equation, y = A^)' This curve should present to the mind, not merely any particular value of x, and the corresponding value of y, but the entire series of corresponding values of these two variables. Quantities infinitely small — Differentials. 11. A quantity is infinitely S7nall, when it cannot be diminished, according to the law of change, without becoming 0. If, in the equation of the curve, y =fix) . . (1.) ic has a particular value OP, y will denote the ordinate PM, If X be increased by PQ, de- noted by A, PiKf will change to QN, which we will denote by y' \ and we shall have, y'=f{x^-K) (2.) If we subtract equation (1) from (2), we obtain, y'-y=A^ + K)-f{x) .... (3.) It is evident that each member of this equation will reduce to 0, when we make A = 0. T P ( 20 DIFFERENTIAL CALCULUS Let US suppose, -as before, that the abscis x has increased from OP to OQ, aud that the cor- responding ordinate y, has be- come 2/'. Draw through the points iV and 3/, the secant line iV^J/i If, now, we suppose the point iV to approach 3f, till it becomes consecutive with it, then, 1. The secant line will become the tangent S3IT\ 2. The abscissas OP and OQ will become consecutive; 3. The ordinates PJf, QN'^ will also be consecutive. The DIFFERENTIAL of a quantity is the difference between any two of its co7isecutive values; hence, it is indefinitely small. The differential is expressed, by writing d before the letter denoting the quantity : thus, dx denotes the differen- tial of ar, and is read, differential of jc: dy denotes the differential of y, and is read, differential of y. It is plain, that dx denotes the last value of A, in Equa- tion (3), before it becomes 0; and that dy denotes the last difference between y' aud y, as h approaches to 0. Differential Coefficient. 12. Under the preceding hypotheses, the differentials of X and y admit of geometrical interpretations. If we divide both members of Equation (3) by A, we have, y - y f{x + h) -f(x) h (4) Having drawn MR parallel to the axis of abscissas, NR will denote the difference of the two ordinates y' and y; hence, the first, and consequently the second member of Equation (4), will denote the tangent of the angle NMR^ SEC. I.] FIESTPRINCIPLES. 21 which the secant line makes with the axis of ^. Denote the angle which the tangent makes willi the axis of JT by a. When the ordinates y' and y become consecutiv(\ the secant iO/ becomes tangent to the curve at the point J/, and the angle NMR becomes equal to 2/S^P; and we have, ^ = tana (5.) cly Tlie terra ~-^ is called the differential coefficient of y, regarded as a fraction of a;; hence, The differential coefficient of a fmiotion is the differen- tial of the function divided by the differential of the mdepefide?it variable. If we multiply both members of Equation (5) by dxy -^dx = tan a dx; dx but the tan a multiplied by the base dx, of the indefinitely small triangle, is equal to the perpendicular, which is the diiference between the consecutive values of y' and y, de- noted by dy; therefore, -^dx z= dy ; hence, dx The differential of a function is equal to its differential' coefficient multiplied by the differential of the variable. Limiting Ratio. 13. Let us now resume the consideration of Equation (4). h h The first member of this equation is the ratio of tlie increment A, of the independent variable, to the correspond 22 DIFFERENTIAL CALCULUS. [SEC. I. ing increment of the function, and denotes the tangent of the angle wliicli tlie secant line, drawn through the ex- tremities of y and y\ makes with the axis of abscissas. If we suppose h to decrease, the secant line "will ap- proach the tangent, and the ratio will approach the tangent of the angle which the tangent line makes with the axis of X. The tan a, is, therefore, the limit of this ratio, and since it is also the ditferential coefficient, it follows that. The differential coefficient is the limit of the ratio of the increment of the indejyendent variable to the increment of the function. A varymg ratio, of any increment of the independent variable denoted by A, to the corresponding increment of the function, denoted by y' — y, reaches its limit when h reaches its last value; and then, the values of y' and y become consecutive; therefore, the limiting ratio, is the ratio of consecutive values. Hence, if we have an expres- sion for the ratio of the increments, we pass to the limiting ratio, or differential coefficient, by making h indefinitely small. Form of the difference between two states of a function. 14. Let us resume the discussion of Equation (3). If h be made equal to 0, the first and second members will each reduce to 0. Therefore, if the second member be developed, and the like terms having contrary signs can- celled, each of the remaining terms will contain h ; else, all the terms would not reduce to 0, when A = 0. Hence, the second member of Equation (3) is divisible by h. Dividing by A, we have, y'-y _ A^+ f^^ -J'-'' . . . (4.) SEC. I.] FIRSTPRINCIPLES. 23 If, now, we pass to the limiting ratio, by making A in- definitely small, the second member will become the tan a, a quantity independent of A (see Equation 5); hence, the first term in the development of the second member of Equation (3) contains A only in the first power, and the coefficient of this terra is tan a, or the differential co- efficient. Since all the other terms become 0, when A = 0, each of them must contain A to a higher power than the first. If we designate by P, the differential coefficient of y, and by P' such a value that P'li^ shall be equal to all the terms of the development of the second member of Equation (3), after the first, that equation may be written under the form 2/'_y = PA + P'A' .... (6.) The differential coefficient, P, is independent of A, bat will, in general, contain aj; and when it does, it is a func- tion of that variable ; P', when not equal to 0, is a function of X and A. Applications of the Formula. 1. If we have an expression of the form, y =/W = «a;, we have the form or development of the second mem- ber. If we give to x an increment A, we have, ^y' =/(a; + A) = a(a; + A) =z ax +aA; hence, V' -y =A^ + h) -A^) = «>^; and y = «^ ; passing to consecutive values, ♦ ill -^ = a; and Jldx = adx. dx dx 24 DIFFEBENTIAL CALCULUS. [sEC. U 2. If we have a function of the form, y =f{^) = ax^, we again have the form or development of the second member. y' =/(x + h) = a{x + hy z= ax^ + Sax^h + 3axh^ + ah^ y' -.y=f{x + h) —f{x) = Sax^h + 3axh^ + ah^ y ^ y = 3ax2 + Saxh + ah^ ; h passing to consecutive, values, we have, ^ = dax^ ; and ^dx = Sax^dx, dx dx In the first example P = a, and P' = 0. In the second, P = Sax-, and P' = Saxh + ah^. 15. Equation (6) aifords the means of determining the differential coefficient, and the differential of any function, whose form is developed in terms of the independent variable. If we divide both members of Equation (6) by the in- crement A, and then pass to the limiting ratio, we have the differential coeflicient. If we then multiply the differential coeflicient by the differential of the independent variable, we have the differential of the function. Equal functions have equal differentials 16. If two functions, u and v, dependent on the same variable a, are equal to each other, for all possible values of ar, their differentials will also be equal. BEC. I.] F I R S T P K I N C I P L E S . 25 For, X being the independent Yariable, we have (Art. 14), u' -u = Ph-\- P'h\ v' - V = Qh+ Q'h\ in which P is the differential coefficient of w, regarded as a function of a;, and Q the differential coefficient of V, regarded as a function of x. But, since u' and v' are, by hypothesis, equal to each other, as well as u and v, we have, Fh + PVi2 = Qh-\- Q'h\ or, by dividing by h and passing to consecutive values, P = Q, . die dv , du T du y and, -rr- dx = -7- ^> that is, the differential of w is equal to the differential of V, Converse not true. 17, The converse of this proposition is not generally true ; that is, Jf tico differentials are equal to each other^ we are not at liberty to conclude that the functions from which they were derivecl, are also equal. For, let u = V ± A (1.) in which ^ is a constant, and u and v both functions of X, Giving to x an increment A, we shall nave, u' = v' ±: A^ 26 DIFFEEENTIAL CALCULUS. [SEC. I. from which subtract Equation (1), and we obtain, u' — u = v' — v^ and, by substituting for the difference between the two srtates of the function, we have, PA + r/i? = Qh -f Q'h\ Dividing by A, and passing to consecutive values, we obtain, P= Q; that IS, ^ = ^; hence, -j dx = -j-dx; or, du = dv. Hence, the differentials of u and v are equal to each other, although V may be greater or less than w, by any constant quantity A ; therefore. Every constant quantity connected with a variable by the sign plus or mi7ius, will disappear in the differen- tiation. The reason of this is apparent ; for, a constant does not increase or decrease with the variable; hence, there is no ultimate or last difference between two of its values; and this ultimate or last difference is the differential of a variable function. Hence, the differential of a constant quantity is equal to 0. 18. If we have a function of the form, u = Av, in which u and v are both functions of aj, and give to X an increment h, we shall have, u' — u = A[y' — ?;), or, Ph + rh^ = A^Qh^ Q'h% SEC. I.] FIKSTPEINCIPLES. 27 Dividing by A, and passing to the consecutive values, P = AQ, or, . Pdx = A Qdx, But, du = Pdx, and dv = Qdx ; hence, du = Adv; that is, T/ie differeiitial of the product of a constant hy a variable qxtantity^ is equal to the constant multiplied hy the differential of the variable. Signs of the diflferential coefficient. 19. If u is any function of a;, and we give to x an increment A, we have, -F = ^ + ^^; and since h is positive, the sign of the first member "will be positive when w < w' ; that is, when ic is an inci'easing function of x (Art. 7). It will be negative when w > w'; that is, when m is a decreasing function of x. Passing to consecutive values, we have, under the first supposition, -IT- = — P. under the second; hence, V dx The differential coefficie?its have the same sigii as the functions^ when the functions are increasing, and con- trary signs, when they are decreasing. If we multiply by dx, we obtain the differentials, which have the same signs as the differential coefficients. 28 DIFFEKKNTIAL CALCULUS. [SEC. I. Nature of a differential coefficient, and of a differential. 20. The method of treating the Differential Calculus, adopted in this treatise, is based on three hypotheses: 1st. That the independent variable changes uniformly: 2d. That in changing from one state of value to another, it passes through all the intermediate values; and, 3d. That any function dependent upon it, undergoes changes determined by the equation expressing the rela- tions between them ; and that such equation preserves the same general form. If the independent variable changes uniformly, and as- sumes all possible values between the limits jc = a, and X = a', we have seen that the change caimot be denoted by a number. If, then, we denote this change by dx, we mean that dx is smaller than any number ; hence, J_ _ dx ■Ox 1 dx dx^ dx^ jp But, — = — = - — = > &G. dx dx^ dx^ dx* that is, any power of dx divided by a power of dx greater by 1, is infinite; hence, any power of dx is infin- itely small, compared with the power next less. Hence, it follows : 1st. That the addition of dx to any number, can make no alteration in its value ; and therefore, when connected, with a numeral quantity by the sign db , may be omitted without error ; thus, Zax -\- dx = Zax. SEC. I.] FTUSTPRINCIPLES. 29 2d. Since dx^ ie infinitely small, compared with dx ; that is, wjinitely less than dx, we have, 5ax^dx -j- dx^ = bax^dx\ and similarly for the higher powers of dx. The quantities, dx^ dx^^ dx\ &c., are called infinitely small quantities^ or infinitesimals of the firsts second, and third orders : from their law of formation, it follows that. Every infinitely small quantity may he omitted without error when connected hy the sign ± with any of a lower order. Hate of change. 21. The measure of a quantity, great or small, is the number of times which it contains some other quantity of the same kind, regarded as a unit of measure. In the Differential Calculus, dx, the differential of the independent variable, is the unit of measure. The rate of dtf change, in the function y, is therefore expressed by -^ , and the actual change corresponding to dx, by ^-Y-dx = dy, dx ^ 22. The equation of a straight line is, y = ax + b. If we take any point, as M, whose co-ordinates are y and X, and a second point N', whose co-ordinates are y', X + h, and we have, y'-y^ah', or, y-^-JL ^ a . . (1.) 30 that is, JSTR DIFFERENTIAL CALCULUS [sec. I. MB = tanfrent NMR = a ; and, passing to the consecutive values, -~- = tanf^eut a =z a (2.) The differential coefficient -—, measures the rate of (tic increase of the ordinate y, when x receives the incre- ment dx ; and since this value is independent of x, the rate will be the same for every point of the line ; that is, the rate of asce?ision of the line from the axis of abscissas, is the same at every point. And since, dij ~ dx = dy — adx. dx the change in the value of the ordinate will be uniform^ for uniform changes in the abscissa. 23. Let us examine an equation, y =f(x) . . (1.) not of the fii-st degree. Let us suppose the curv'e AMN to be such that the abscissas and ordinates of its different points shall correspond to all possible relations between y and JC, in Equation ( 1 ). We have seen (Art. 13) that, dy dx = tan TSF = tan a ; hence. SKC. I.] F 1 R S T P K I X C T P L E S . 31 the rate of increase of tlie function, or the ascension of the curve at any point, is equal to the tangent of the angle which the tangent line makes with the axis of ab- sdssas. "We also see, that this value of the tangent of a, will vary with the position of the point M\ hence it ia a function of x\ therefore, In every equation^ not of the first degree^ the differ- ential coefficient is a function of the independeyit variable, 1. "We have seen, that when the points M and K are consecutive, the secant line, J/iV, becomes the tangent line, TMS (Art. 13). The hue MR is then denoted by dx^ and UN or RT^ (for the points N and T then coin- cide), by dy. If we give to the new abscissa, x -f dx^ an additional increment ffe, and suppose the correspond- ing ordinate, y •\- dy^ to receive the same increment as before, viz. : dy, the extremity of the last ordinate will not fall on the curve, but on the tangent line, since the triangles thus formed are similar; hence, TjT a function be supposed to increase uniformly from any assumed value, the differential coefficient will be constant, and equal to any increment of the function divided by the corresponding increment of the variable. Nature of the Differential Calculus. 24. In every operation of the Differential Calculus, one of two things u always proposed, and someti-nes both : 32 DIFFERENTIAL CALCULUS. [SEC. 1 1st. To find the rate of change in any vanable ftmc- tion when it begins to change fiom any assigned value. 2d. To find the difference between any two consecutive values of the function. This difference is the actual change in the function, produced by the smallest change which takes place in the independent variable. The use of the independent variable is to furnish a unit of measure for the increment of the fimction, tmd thus to determine its rate of change^ as it passes through all its states of value. This ratio can generally be expressed in numbers, either exactly or approximatively. 25. The increment of the function, corresponding to the smallest increment of the vanable, being the difference between any two of its consecutive values, is a quantity of the same kiiid as the function, and differs from it only in this: that it is too small to he exj^ressed bg numbers. The differential of a quantity, therefore, is merely an element of that quantity ; that is, it is the change which takes place when the quantity begins to increase or decrease, from any assumed value. When we find this element, we have the differential of the function ; and by dividing by dx, we have the differential coefficient. Hence, All the operations of the Differential Calculus comprise but two objects : 1. To find the rate of change in a function, when it passes from one state of value to another, consecutive with it. 2. To find the actual change i?i tJie function. The rate of change is the differential coefficient, and the actual change, the d'fferential. SECTION II. DIFFERENTIALS OF ALGEBRAIC FUNCTIONS. Differential of sum or difference of Functions. 26. Let r be a function of the algebraic sum of several variable quantities, of the form, in which y, 2, and w, are functions of the independent variable x. If we give 9^ to x an increment ^, we shaU have, u' — u = (y' — 2/) + (2' — 2) — {w' — w) ; hence (Art. 14), u' -ic = (Ph + rh^) + {Qh-\- Q'h') - {Lh + X'A2), or, — ^ = (P + P'h) + (C + Q'h) - (Z + iVi), and by passing to consecutive values, multiplying both members by dx^ we have, ~dx = Pdx + C^ic - Zc^o;. dx But as P, §, and X, are the differential coefficients 3i DIFFEEENTIAL CALCULUS. [SKC. n. of y, 2, and lo, each regarded as a function of x; hence, f-'cfo = f <& + f & - ^&; that i^ . ax dx ax ax The differential of the sum or difference of any numher of functions^ dependent on the same variable^ is equal to tJie sum or difference of their differentials taken separately. Differential of a product. ay. Let u and v denote any two functions, x the independent variable, and h its increment; we shall then have, u' = u + Ph + P'h\ and v' = v-\- Qh+ Q'h% and, multiplying, it'v' = (u-\- Ph-\- P'h'') (v + CA + Qli^) = uv ■\- vPh + uQh + PQh^ + &c. ; hence, -T = vP + uQ + terms containing A, h^, and hK If now we pass to consecutive values, we have, -dT = ^^+«G; therefore, d(uv) = vPdx + u Qdx = vdu + udv ; hence, The differential of the product of two functions de- pendent on the same variable, is equal to the sum of the products obtained by multiplying each by the differ* ential of the other. BEC. II.] DIFFERENTIALS OF FUNCTIONS. 35 1. If we divide by uv^ we have, that is. d(uv) __ du dv . . uv u V 27ie differential of the product of two functions, divided hy the product, is equal to the sum of the quotients obtained hy dividing the differential of each by its function. 28. "We can easily determine, from the last formula, the differential of the product of any number of functions. For, put V = ts, then, dv _ d(ts) _ dt ds V - ~^~5" - 7"^T • • • • ^^'^ and by substituting ts for v, in Equation ( 1 ), we have, djufs) _ du dt ds ^ uts '~ u t s ^ and in a similar manner we should find, d(utsr. . . .) du dt ds , dr - -^-7 -' = — + — + — -h — &G. utsr .... u t s r If, in the equation, djuts) _ du dt ds uts ~ w t s * we multiply by the denominator of the first member, we shall have, d{uts) = tsdu 4- usdt -f utds; hence, 77ie differential of the product of any nu7nber of funo tions, is equal to the sum of the products which arise 36 DIFFERENTIAL CALCULUS. [^SKC. U. by multiplying the differential of each function by tlie product of all the others. Differentials of Fractions. 29. To obtain the differential of any fraction of the form, - • V u Put, - = ty then, u = tv. Differentiating both members, we have, du = vdt 4- tdv; finding the value of dt, and substituting for t its value - , we obtain, _ du vdv dt = ^ , or, by reducing to a common denominator, vdu — udv , dt = :, ; hence, 77ie differential of a fraction is equal to the denom- inator into the differential of the numerator^ minus the numerator into the differential of the denominator^ divided by the square of the de7iominator. 1. If the denominator is constant, dv = 0, and we have, vdu du SEC. II. J DIFFERENTIALS OF FUNCTIONS. 37 2. If the numerator is constant, du = 0, and we have, ,^ udv dt = r- ; and under this supposition, ^ is a decreasing function of V (Art. y) ; hence, its differential coefficient should be negative (Art. 19). Differentials of Powers. 29.* To find the differential of any power of a function. First, take any function 7/", in which oi is a positive Avhole number. This function may be cous>idcrcd as composed of n factors, each equal to u. Hence (Art. 27), diw^) diuuuu . . . .) du du du du ~ — - = — ^^ = 1 1 H +... w" {uuuu . . . .) U U II u But as there are n equal factors in the numerator of the first member, there will be 7i equal terms m the second ;. , diiC^) ndu hence, -^ — '- = ; therefore, d{u^) = nW^^^du. 1. If 71 is fractional, denote it by — , , and make,. s r V =z w% whence, v' = u^ ; ^and since r and s are entire numbers, we shall have,. sv'-'^dv = ru^-^du'y from which we find, dv = r du = du ; • 16 88 DIFFERENTIAL CALCdI.US. [SEC. II. or, by reducing, r T 1 dv = - W dui s which is obtained directly from the function, di^w) = 7i?^«-i du, T by changing the exponent n to - • s 2. If the fractional exponent is one-half, the function becomes a radical of the second degree. "We will give a specific rule for this class of functions. Let V — w^ or, v = y^; then, dv = -u^ du = ~u ^ du = — — ; 2 2 2yw that is, The differential of a radical of the second degree^ is equal to the differential of the quantity under the sign divided hy twice the radical. 3. Finally, if n is negative, we shall have, 1 W-" = — , w» from which we have (Art. 29), 4xdx (1 - a;2)2* 42 DIFFEEEXTIAL CALClfLUS. [SEC. II. 30. u = _ yi 4- g + V 1 yi+x — y/l —X du = g + i/T-"^^)^ . Differential of a particular binomiaL 80.— 1. Let w = (a + &b")*. Put a + bx^ — y\ then, w = y" ; and (Art. 30), (7z* r= my^-'^dy. But, from the first equation, dy = w^"-^<&; substituting for y and fZy their values, we have, du = m7ib{a + Ja;")"*-^^"-^^; that is, to find the differential of a binominal 'iinction of this form, Midtiply the exponent of the parenthesis, into the ex- ponent of the variable within the parenthesis, into the co- efficient of the variable, into the bmomial raised to a power less 1, into the variable within the parenthesis raised to a power less 1, i?ito the differential of the va- riable. Rate of change of the Function. 31. AYhat is the rate of change in the area of a square, when the side is denoted by the independent variable? We have seen (Art. 2l) that the differential coefficient, — , denotes the rate of change in the function w, cor- dx SEC. II.] DIFFERENTIALS OF FUNCTIONS. 43 responding to the change dx^ in the vakie of x\ and that in all equations, except those of the first degree, this rate will be variable^ and a function of x (Art. 23). Let X denote the side of a square, and u its area ; then, u = x^, and —= 2x: dx hence, the rate of change in the area of a square is equal to twice its side ; that is, if the side of a square is denoted by 1, the rate of change in the area will be denoted by 2; if the edge is denoted by 5, the rate of change will be 10; and similarly for other numbers. 2. What is the rate of change in the volume of a cube, when its edge is the independent variable? Let X denote the edge of a cube, and u its volume ; then, u = a;', and — = dx^ ; dx hence, the rate of change in the volume, is three times the square of its edge. If the edge is denoted by 1, the rate of change in the volume will be denoted by 3 ; if the edge is denoted by 2, the rate of change will be 12; if 3, the rate will be 27 ; and similarly, when the edge is denoted by other numbers. Find the rates of change in the following functions: 3. w = 8x* — 3a;2 — ^x -\- a. A. ^2x^ — 6^ — 5. What will express the rate for x=\^ a: = 2, a; = 3? 4. u = {x^ + a) (3a;2 + b). A. 15a:* + Sx^b + Qax, Find the rate for, a: = 1, ar = 2. 44 ■ DI FFEnEXTlAL CALCULUS. [SEC, H. What is the rate for, x = 0, x = 4, x = — I? e. u =z (ax + cc2)2. A. 2 {ax + cc^) (« -|_ 2aj). What is the rate for, a = 0, jb = 1, jc = 3 ? X _^ What is the rate for, jc = 0, cc = 1 ? Hence, to find the rate of change for a given value of the variable : Find the differential coefficient^ and substi- tute the value of the varia^Je in the second member of the equation. Partial Diflferentials. 32, If we have a function of the form, u =f{x,y) (1.) the equation denotes that u is a fiinction of the two variables, x and y. If we suppose either of these, as y, to remain constant, and x to vary, we shall have, |'=/'(-'^) (^•) if we suppose x to remain constant, and y to vary, we shall have. The differential coefficients which are obtained under these suppositions, are called partial differential coefficients. SEC. n.] DIFFERENTIALS OF FUNCTIONS. 45 The first is the partial differential coefficient with respect to a*, and the second with respect to y. 33. If we multiply both members of Equation ( 2 ) by dx^ and both members of Equation ( 3 ) by c?y, we obtain, ^^ dx = f'{x, y)dx, and ^ ^^ = f'\^^ y)^y' The expressions, du - du ^ are called, pai'tial differentials; the first a partial differ- ential with respect to a, and the second a partial differ- ential with respect to y ; hence, A PARTIAL DIFFERENTIAL COEFFICIENT IS the differential coefficient of a fimction of two or more variables^ under the siipposition that only one of them has changed its value; and, A PARTIAL DIFFERENTIAL is the differential of a func- tion of two or more variables^ under the sup2)ositio?i that only one of them has changed its value. If we suppose both the variables to undergo a change at the same time, the corresponding change which takes place in u^ is called, the total differential. If we extend this definition to any number of variables, and assuma what may be rigorously proved, viz. : That the total differential of a function of any nmnher of variables is equal to the sum of the p>artial differ- entials, 4:0 DIFFERENTIAL CALCULUS. [SEC. II. we have a general formula applicable to every fimo tioii of two or more vaiiables. ^ EXAMPLES. 1. Let u = ic^ + 2/3 _ 25 then, — dx = 2xdx, 1st partial differential , uX g^y = 3yV?y, 2cl " ^dz = -dz, 3d « dz hence, du = 2xdx + 3yWy — dz. 2. Let u — xy\ then, J.?x = ydx, hence, du = ydx + xdy. 3. Let u = jc'"?/'* ; then, —-dx = mx^-'^y^dx^ — fZy = wy" - ^ ic'"(7y ; hence, cly du = maf^-'^ydx -f ny'^-^x'^dy = x^-'^y-^^mydx + 7ixdy), SEC. II.] DIFFEKEKTIALS OF FUNCTIONS. 47 4. Let hence, u = -I then, y du ^ dx — dx— — , dx y du ^ xdy - _ ydx — xdy 5. Let u ay 'X' + y = = ayix^ + 2/^) 2 ; then, du _ ayxdx hence, du _ c?t« {x^ + 2/2)^ 1 3 > .2 I -,2\2 ayxdx — ox^dy 6. Let u = xyzt\ then, e7i« = yztdx + ccs^cZy + xytdz + a-yse?^. SECTION III. INTEGRATION AND APPLICATIONS. 34. An Integral is a functional expression, either al- gebraic or transcendental, derived from a differential. Differentiation and Integration are terms denoting operations the exact converse of each other. Differentiation is the operation of finding the differ- ential timction from the primitive function. Integration is the operation of finding the primitive function from the differential function. Rules have been found for the differentiation of every form which a function can assume. Hence, in the Differ- ential Calculus, no case can occur to which a known rule is not applicable. In the Integral Calculus it is quite otherwise. In returning from a known differential to the integral from which it may have been derived, we compare the differential expression with other expressions which are known to he differentials of given functions^ and thus arrive at the form of the integral, or primitive function. The main operations, therefore, of the Integral Calculus, consist in transforming given differential expressions into others which are equivalent to them, and which are differ- entials of knoTNTi functions ; and thus deducing formulas applicable to all similar forms. The integration is indicated by placing the sign / SEC. in.] INTEGRATION. 49 before the expression to be integrated. It is equivalent to "integral of"; thus, J 2xdx = x\ is read: "Integral of 2xdx^ is equal to a^." Integration of Monomials. 35. The differential of every expression of the form, w = ic"», is du = mx'^-^dx (Art. 30), which has been found by multiplying the exponent into the variable raised to a power less one, into the differ- ential of the variable. If, then, we have a differential expression, of the form, mTS^-^dx^ or, x'^dx, we can find its integral by reversing the above rule ; that is, to find the integral of such an expression. Add 1 to the exponent of the variable, and then divide hy the new exponent into the differential of the variable.* EXAMPLES. Find the integrals *of the following differential expressions : 1. If (7w = 2xdx, I du = ^r- = x^. *J 2 X dx 2. J£ du = Zx^dx, fdu = --— = x^. ^ S X dx • M , * This rule applies to every case of a differential monomial of th« form, Ax'^dx, except that in which m is — 1 (Art. 90). 50 DIFFERENTIAL CALCULUS. [sEC III. 3. If du = oTdx^ Jdu oT+^dx ar+^ (m4- l)dx m + 1 4. Jf du = X- ^dx, I du = ^ _ _ ^ „ ' J _ 2dx 2aj2 5. If c?w = x^^/xdx^ Jdu = jx^dx = -x^V^- 36. We have seen, that the differential of the product of a constant by a variable, is equal to the constant multi- plied by the differential of the variable (Art. 18). Hence, the integral of the product of a constant hy a differ- ential, is equal to the constant 'multi2ylied hy the integral of the differential; that is, /ax^dx =z a I x'^dx = a aj" ^ ^. «/ m + 1 Hence, if the €X2yression to he hitegrated has one or more constant factors, they should, at once, he placed as factors, without the sign of the hitegral. 37. It has been shown that the differential of the sum or difference of any number of variables is equal to the sum or difference of their differentials (Ai*t. 26). Hence, if we have a differential expression of the form, du = 2ax^dx — hydy — z^dz\ we may write, J du = lajx^dx — hjydy —Jz^dz'y or, fdu = -ax^ ^^2 _^iL; that is, •/ . 3 2^ 3 ' The integral of the algehraic sum of an^f number of dif ferentials is equal to the algebraic sum of their integrals. 6KC. III.J INTEGRATION. , 51 Correction — Indefinite— Particular — and Definite Integrals. 3§. It has been shown that every constant quantity connected with a variable by the sign plus or minus, clis« appears in the differentiation (Art. 17); that is, d{a + jc*") = fZiC" = mx'^-'^dx. Hence, the same differential may have several integral functions differing from each other by a constant term. Therefore, in passing from a differential to an integral expression, we must annex to the first integral obtained, a constant term, to compensate for the constant term which may have been lost in the differentiation. For example, it has been shown in Art. (22), that, dy ^ = a, or, dy = adx, is the differential equation of every straight line which makes "wdth the axis of abscissas an angle whose tangent is a. Integrating this expression, we have, J dy z= aj dx (1.) or, y = ax', or, finally, y = ax + O (2.) If, now, the required line is to pass through the ongin of co-ordinates, we shall have, for jc = 0, 2/ = 0, and consequently, (7=0. But if it be required that the line shall intersect the 52 DIFFER EX TIAL CALCULUS. [sEC. 111. axis of Y at a distance from the origin equal to + J, we shall have, for X = 0, 2/ — + J, and consequently, (7 = + 5; and the true integral will be, y =: ax + b (3.) If, on the contrary, it were required that the right line should intersect the axis of ordinates below the origin, we should have, for X = Oj y = — J, and consequently, C = — b; and the true integral would be, y = ax — b (4.) The constant (7, which is added to the first integral, must have such a value as to render the functional equa- tion true for every possible value that may be attributed to the variable. Hence, after having found the first integral equation, and added the constant C, if ice then make the variable equal to zero, the value which the function assumes will be the true value of (7. 1. An indefijiite integral is the first integral obtained, before the value of the constant C is determined. 2. A particular integral is the integral after the value of C has been found. 3. A definite integral is the integral correspondino* to a given value of the variable. Thus, Equation (2) is an indefinite integral, because, so long as G is undetermined, it will be the equation of a SEC. III.] INTEGRATION. 53 system of parallel straight lines. Equations (3) and (4) are particular integrals, because each belongs to a par- ticular line. Origin of the Integral. 39. The origin of an integral function is its zero value. The value of the variable corresponding to the oriizin of the integral, is found by placing the second member ot the equation expressing the particular integral, equal to zero, and finding therefrom the value of the vaiiable. Thus, if in Equation (3), we make y = 0, we have, ax + b — 0^ and x ~ — -^ a "which shows that the origin of the function y (that is y =: 0), is on the side of negative abscissas, and at a dis- tance from the origin equal to — -* In Equation (4), it is at a point whose abscissa is -• a Integration between limits. 40. Having found the indefinite integral, and the par- ticular integral, the next step is to find the definite in- tegral; and then, the definite integral between given limits of the variable. Let us take the particular integral found in Equation (3),, y — ax ■\- h. If it is required to find the value of the function y, for- a given value of the variable sc, as, x = x\ y will be- come a constant for this value, and w^e shall have, y' = ax' ■\-h (5.) which is a definite integral.- 54 DIFFEKEXTIAL CALCULUS, [sec. III. ^^' M f- a y p « If we wish the value of the function corresponding to a second abscissa, x = x'\ we shall have, y" = ax" + 5 (6.) If we subtract Equation ( 5 ) from Equation ( 6 ), we have, y" -y' = a{x"-x') .... (7.) which is the definite integral of y, taken between the lim- its, X = x\ and x = x" . If; x' = OF, and x" = OQ; then, y' = PM, and y'' = QJST; hence, y" -y' = a{x" - x') = NB ; Therefore : The integral of a func- tion, taken between tioo limits, indi- cated by given values of x, is equal to the difference of the definite integrals corresponding to those limits. Let us now explain the language employed to express these relations. The modified form of Equation ( 1 ), J(dy)^^^ = ajdx, is read: "Integral of y, when x is equal to jb';" and J(dy)^^Tr = ajdx, is read: "Integral of y, when x is equal to a";" and 05" j(dy) = afdx, IS read : Integral of the diiferential of y, taken between the limits, x' and a;"; the least limit, or the limit correspond- ing to the subtractive integral, being placed below. SBC. in.] INTEGRATION, EXAMPLE. 1. What is the integral of du = Qx^dx, between the limits a? = 1, and cc = 3, if in the primitive function u reduces to 81, when aj = 0. Jdu = fdx^dx = dx^ + C ; hence, fdu = 3aj3 + C. But from the primitive function, w = 81, when aj = 0; hence, (7 = 81, and, fdu =3x3+81 (1.) /W«,-i = 3 + 81 = 84 . . (2.) /(^w)«,-8 = 81 +81 = 162 . . (3.) s f(du) = 162 - 84 = 78 . . (4.) What is the value of the variable corresponding to the origin of the integral (Art. 39) ? Making the second member of Equation ( 1 ) equal 0, 3a;3 + 81 = 0, or, x = — 3. Integration of particular binomials. 41, To integrate a differential of the form (Art. 30), du = (a + bar)V'dx (1.) 56 DIFTERENTIAL CALCULUS. [SEC. m. The characteristic of this form is, that the exponent of the variable icithout the parenthesis is less by 1 than the exponent of the variable within. Put, {a + bx"") = z ; then, (a + bx^'Y = g*" ; and, dz nbx'^-'^dx = dz j whence, a^-Wa; = — : ; hence, 7ib Jdu =f{a + bx'^Yx^'-^dx =f-^ ^dz 2'"+! 7ib (m + l)nb' and consequently, _ (c^ + bx-Y + ^ Hence, to find the integral of the above form, 1. If there is a constant factor^ j^/ace it without the sign of the integral^ and omit the pmcer of the variable loithout the parenthesis' and the differential: 2. Augment the exponent of the parenthesis by 1, and then divide this quantity^ loith its exponent so increased^ by the exponent of the parenthesis^ into the exponent of the variable within the parenthesis, into the coefficient of the variable. EXAMPLES. 1. f(a 4- Sxyxdx = ^-^~ + (7; and 2. fm{a + bx'^yxdx = ~(a + ^^)^ + 0. mn{a - icx'Yx^dx = - —(a - 4«c*)^ + C. SEC. ni.] INTEGRA r ION. 57 Integration by Series. 42. The approidmate integral of any function of the form, du z= JCdx, may be found, when ^ is such a function of x, that it can be developed into a series. Having made the development of the function -X", in the powers of x, by the Binomial Formula, we multiply each term by dx, and then integrate the terms separately. When the series is converging, we readily find the approximate value of the fimction for any assumed value of the variable. EXAMPLE. 1. Find the approximate integral of, in which, X = (1 - a;2)"^ Developing, (1 — x^) ^, by the binomial formula,! (1 - a:^^ = 1 + T'"^ + Y- T'^* + y4 4*' + *"•' multiplying by dx, and integrating, we obtain, * Bourdon, Art. 166. University, Art. 32. f Bourdon, Art. 135. University, Art. 104. 58 DIFFEBBNTIAL CALCULUS. [SEC. m, from which we obtain an approximate value of w, cor- responding to any value we may give to x. APPLICATIONS TO GE0:METEICAL MAGNITUDES. Equations of Tangents and Normals. 43. We have seen, that if a? and y denote the co-ordinates of every point of a curve, —- will denote the tangent of the angle which the tangent line makes with the axis of abscissas (Art. 13). This value of -^ ^ ' dx was found under the supposition that the second secant pomt became consecutive with the first; hence. Any two consecutive points^ must, at the same time, he in the chord, the curve, and the tangent. Denote the co-ordinates of the point of tangency, in any curve, by x" and y" . If through this point we draw any secant line, its equation will be of the fonn, 1/ - 2/" = a{x - x'y* If the second point of secancy becomes consecutive with the first, we shall have (Art. 13), dy" hence, the equation of the tangent line is, y - 2/" -&==-*") • • • • ^'-^ * Bk. I. Art. 20. BBC. in.] TANGENTS AND NORMALS. 59 If, in the equation of any curve, we find the value of -^-f, , and substitute that value in Equation ( 1 ), the equa- tion will then denote the tangent to that curve. 1. By differentiating the equation of the circle, {b2 + 2/2 _ 7^2^ o'r, a;"2 _^ y'n ^ j^^ we have, dx" ~ y"' hence, V - y" = - ^,{x-x")', or, by reducing, yy" + xx" = B"^, 2. By differentiating the equation of the ellipse, we have, dx" "~ ^y'* 3. By differentiating the equation of the parabola, we have, dx" ~ y"'^ 4. By differentiating the equation of the hyperbola, we have, dy" _ B'^x" dx" ~ Ahj"' Substituting these values, in succession, in Equation (1), and reducing, we shall find the equation of the tangent line to each curve. • Bk. II. Art. §. \ Bk. III. Art. 14. % ^k. IV. Art. §. 60 DIFFEKEXTIAL CALCULUS. [sEC. lH. 44. The equation of the normal is of the form, y-y" = a'{x-x") .... (i.) But since the normal is perpendicular to the tangent, at the point of contact, a 1 + aa' — 0,* or, hence, the equation of the normal is, rJx" y-y dy T,(^-^") dx" dy"' . (2.) By differentiating the equation of the circle, the ellipse, the parabola, and the hyperbola, finding in each differ- dx" cntial equation the value of ^— ^, substituting that value in Equation ( 2 ), and reducing, we shall find the equation of the normal line to each curve. Value of tangent, sub-tangent, normal, and sub-normal. 45. Let P be any point of a curve; TP the tangent, TR the sub-tangent, PoV the normal, and MN" the sub-normal. Then, in the right-angled tri- angle TPB, PR = TRx tan PTR = TR x <]y. dx' hence. TR = PR dy dx dx y-j- = Sub-tangent. ♦ Bk. I. Art. 23. SEC. III.] TANGENTS AND NORMALS. 61 46. The tangent TP is equal to the square root of the sura of the squares of TR and PIt\ hence, TP = yy^l +^ = Tangent. 47. Since TPN is a right angle, PPJST is the com plement of TPR\ it is therefore equal to PTJR^ and con- sequently its tangent is y- ; hence, PN = v-^- = Sub-normal. ^ dx 48. The normal PN is equal to the square root of the sum of the squares of PP and PN'\ hence. PiVr = y^l + g = Normal. 49. Apply these formulas to lines of the second order, of which the general equation is, 2/2 — f)2X -f nx'^,* Differentiating, we have, dy m -\- 2nx m + 2nx ^^ 2y 2y/mx + nx^' substituting this value, we find, m =y'^ = ■'Sl^-^ = Sul>tangent. ^ dy m + 2rvx ^ -^ V ^ dy^ V ^ \m-{-2nx/ * Bk. V. Art. 42. DIFFERENTIAL CALCULUS, SEC. UL BK = y-f — — = Sub-normal. ^ dx 2 PN = y \/l + -J^ = y^^ + nx'^+ -(m + 27ixy. By attributing proper values to m and w, the aboTe formulas will become applicable to each of the conic sections. In the case of the parabola, w = 0, and we have. TR = 2x, BR = m TP =z -y/mx + 4a;2, PN -^' mx -f -rn^. Asymptotes. 50. An asymptote of a curve is a line which continually approaches the curve, and becomes tangent to it at an infinite distance from the origin of co-ordinates. Let AX and AY he. the co-ordinate axes, and y - y dy' dx r,{^- ^"), the equation of any tangent Une, as TP. If, in the equation of the tangent, we make, in succes- sion, y =z Oj X = 0, we shall find, . dx" dv" &EC, III.] ASYMPTOTES. ' 63 If the curve CPI^ has an asymptote i?^, it is plain that the tangent I'T will approach the asymptote IiJ5Jj when the point of contact P, is moved along the curve from the origin of co-ordinates, and T and J) will also approach thc; points Ji and !FJ and will coincide with them when the co-ordinates of the point of tangency are infinite. In order, therefore, to determine if a curve have asymp- totes, we substitute in the values of ^2^ and AD, the co-ordinates of the point which is at an infinite distance from the origin of co-ordinates. If either of the dis- tances AT, AD, becomes finite, the curre will have an asymptote. If both the values are finite, the asymptote will be- inclined to both the co-ordinate axes ; if one of the dis- tances becomes finite and the other infinite, the asymptote will be parallel to one of the co-ordinate axes; and if they both become 0, the asymptote will pass through the origin of co-ordinates. In the last case, we shall know but one pomt of the asymptote, but its direction may be determined by finding the value of -^ , under the sup- position that the co-ordinates are infinite. 51. Let us now examine the equation, of lines of the second order, and see if these lines have asymptotes. We find, 2^/2 _ ^x AT = X- m 4- 2nx m + 2nx^ 64 AD DIFFERENTIAL CALCULUS mx 4- 2?ix'^ [sec. m. mx 2y 2^/m^ 4- 7ix'^' which may be put under the forms, AT = — 4-2/1 X AD = m and making a; = oo, we have, AR AE m 2v^' If now we make w = 0, the curve becomes a parabola, and both the limits, AE, AE, become infinite ; hence, the parabola has no rectilinear asymptote. If we make n negative, the curve becomes an ellipse, and AE becomes imaginary ; hence, the ellipse has no asymp- tote. But if we make n positive, the equation becomes that of the hyperbola, and both the values, AR, AE, become 2B^ finite. If we substitute for m its value, -7-, and fof n its value -^, we shall have, AR = - A, and AE = ± B. Hence, of the lines of the second order, the hyperbola alone has asymptotes. «EC. III.] EECTIFICATION OP CUKVES. 65 ^ Differential of an arc. 52. "We have seen that, when the points which limit any arc of a curve become consecutive, the chord, the arc, and tangent become equal (Art. 43) ; therefore, the differential of an arc is the hypothenxise of a right-angled triangle of which the base is dx^ and the perpendicular dy. Hence, if we denote any arc, referred to rectangular co-ordinates, by z, we have, dz •v/^M-e?j/2. . (1.) or, z = f ^dx' + dy'' . . {2.) Rectification of a plane curve. 53. The rectification of a curve is the operation of finding its length; and when its length can be exactly expressed in terms of a linear unit, the curve is said to be rectifiahle. To rectify a curve, given by its equation: Differentiate the equation of the curve and find the value of dy"^ in terms of x and dx ; or of dx^ in terms of y and dy^ and substitute the value so found in the differential Equation (2). TJie second memher will then contain but one variable and its differential; the integral will express the length of the arc in terms of that variable. EXAMPLES. 1. Find the length of the arc of a circle in terms of the radius. The equation of a circle whose radius is 1, referred to rectangular axes, when the origin is at the centre, is, a;2 -f- y2 _ 1^ Q^ DIFPEEBNTIAL CALCULUS. [SEdH. Denoting the arc by 2, we have, dz = ^dx^ 4- dy\ or, z = f-^dx^ + dy\ From the equation of the circle, we have, xdx + ydy — ; hence, dy"^ — ; 1 ■"" X r /77~ x^dx^ r dx p^ _i dx. Developing the binomial factor into a series, by the binomial formula,* multiplying by dx, and integrating, we have (Art. 42), _1 la^ . l.SicS 1.3.5a;7 (l_x^) V^ = . + ^ + _-+__ + &c. + C K we suppose the origin of the integral to be at JEJ, the correspond- ing value of X will be zero, and (7=0. If now we integrate between the limits a; = 0, and a; = ^, we shall obtain the value of the cor- responding arc in terms of the radius 1. But X, or FM, is the sine of the arc JEJP^ denoted by z ; and when jb = J, 2 = 30° ; hence, 30° =f(l - x^f'^dx = 1 + 2^3 + ^. + ^<^'^ •Bourdon, Art. 135. University, Art. 104. SEC. III.] RECTIFICATION OF CUEVES. 67 hence, „^o . ./I . l-l-l . 1-3.1.1 1.3.5.1.1 , . \ and by taking the first ten terms of the series, we find, It = 3.1415926. . . , a result true to the last decimal figure. We have thus found the semi-circumference of a circle whose radius is 1, or the circumference of a circle whose diameter is 1. 2. Find the length of the arc of a parabola, whose equation is, 2/2 = 2jtXB. Differentiating and dividing by 2, we have, ydy = pdx, and consequently, «&»= J\ill occupy every part of the rectangle OC, SEC. in.] QUADRATURES. 71 Since* the equation of the line ^ (7 is, y = K we shall have, for the differential of the surface, ds = hdx. Integrating between the limits jc = 0, and a; = 5, and observing that (7 = 0, when a; = 0, we have, h fds = I hdx = hx = hb; that is, 27ie area of a rectangle is equal to tJie product of its base hy its altitude. Area of a triangle. 58, Let ABC be a right-angled triangle, and C the origin of co- ordinates. Denote the base AB by 5, and the altitude GB by h. De- note any line parallel to the base by y, and the corresponding altitude by X, If we suppose the base AB to be moved towards the vertex of the triangle, along CB as a directrix, and so to change its value, that. h : h :: 2/ : ic. or. y hx h' it is plain that it wiU generate the surface of the triangle. If we denote the surface by 5, we have, ds = ydx\ 72 DIPFEEENTIAL CALCULUS. [SEC. UL substituting for y its value, and integrating between the limits a; = 0, and a; = A, we have, d, = j^Jxdx = ^- = -; that is, The area of a triangle is equal to half the product of the base hy the altitude. Area of the parabola. 59. Find the area of any portion of the common para- bola whose equation is, 2/2 ::;: ^yx\ whcuce, y = \/2px. This value of y being substituted in the differential equa* tion (Art. 55), gives (Art. 36), fds = f^/2^dx = ^fx^dx = ?^a;5 4. C; or, s = -^ = -xy + C. If we estimate the area from the principal vertex, where a; = 0, and y = 0, we have, (7=0, and denoting the particular integral by s\ we shall have, 2 s' = -xy\ that is, o The area of any portion of the parabola^ estimated 2 from the vertex^ is equal to - of the rectangle of the abscissa and ordinate of the extreme point. The curve w, therefore, quadkablk. SEC. III.] QUADRATURES. 73 1. To find the area of a parabola from the vertex to the double ordinate through the focus. We have, for these limits, aj = and x = ip. Denoting the integral by s", ip we have, J ds = s" = ip^, which denotes the area bounded by the curve, the axis, and the ordinate ; hence, if we double it, we shall have the required area; or, 2." = ip== 1^2 =1(2^)2. That is. The area is equal to one-sixth of the square described on the parameter of the axis. 2. If the area be estimated from the ordinate through the focus, where x = ip^ and y =i?, G must have such a value as to reduce the first member to 0: for, this is the origin of the integral. We have, J ^^ — f^ + ^> and for the particular case of the focus, fds = ^xipxp + C= \p^ + (7; hence, 1^,2+ (7=0; or, {7=-Ji?2. Hence, the integral from x = \p Xo any value of x is, fds = -xy - -p\ Area of the circle. 60. The equation of the circle referred to its centre and rectangular axes is, y2 _ 7.2 _ a;2 . Qj.^ y ^ y?^— x^ ; 74 DIFPBBENTIAL CALCULITS. [SEC. IH hence, the differential equation of the area (Art. 57) is, ds = (V^- aJ^^^ .... (1.) in which the origin of the area is at the secondary dia- meter, where a; = 0. From Formula S^^ page 189, we have, But, by Formula (13), Art. 99, we have, /(r»-.)-^.^=/-^_ = sin-'?+(7; x" whence, by substitution, we have, 8 = lx{r^ - x^y + Ir^ sin"* - + C , (2.) Estimatmg the area from the secondary diameter, where a; = 0, we have, (7=0. If we integrate between the limits of jc = 0, and jB = r, we shall have one quarter of the area of the circle. When we make a; = r, in Equation ( 2 ), the first term in the second member becomes ; and in the X second term, - becomes 1, and the arc whose sine is 1, r is 90°, which is denoted by — , to the radius 1 ; hence, 2 r Jds = ^r2 sin-i 1 = 2^' X ^; or, Area of the circle = ^(o^^ ^ -) = r^'K, SEC. m.] QUADRATURES. T5 Area of the ellipse. 61. The equation of the ellipse, referred to its centre and axes is, ^V ^ ^2a;2 = ^2jj2 . iience, B and the differential equation of the area is, ds = ~(A' - x^ydx. The second member of this equation differs from the second member of Equation ( 1 ), of the last Article, only in the constant coefficient -r , and the constant A"^ for A r^, within the parenthesis ; hence, the integral of that expression becomes the integral of this, by multiplying it by -J , and changing r into A ; that is, Area of ellipse = • — - — = A.B.ir; that is. The area of an ellipse is equal to the product of its ""semi-axes multiplied by -r. 1. Let Q denote the area of a circle described on the transverse axis, and Q' the area of a circle described on the conjugate axis ; then, ^2^ = §, and i?V = Q'; hence, 76 DIFFERENTIAL CALCULUS. [SEC. III. ^2^2 ^2 ^ QQ^^ and AB-JT = y^Q x Q'; that is, The area of an ellipse is a mean proportional between the two circles described on its axes. A C D E B X QUADRATURE OF SURFACES OF REVOLUTION. 62. Let oacdeb be a plane curve, 0J5 the axis of abscis- sas, and Oo, Aa, Cc, &c., con- secutive ordinates; then, oa, ac, ed, &c., -R-ill be elementary arcs. The surface described by either of these arcs, while the curve revolves around the axis OB, will be an element of the, surface. We have seen, that when the ordinates are con- secutive, the chord, the arc, and the tangent, are equal (Art. 43) ; hence, the surface described by any arc, as «c, is equal to that described by the chord; that is, equal to the surface of the frustum of a cone, the radii of whose bases are Aa = y, Cc = y -\- dy, and of which the slant height ac = ^dx^ + dy^. Hence, if we denote the surface by s, we have,* ds = -^(22/ + 2y + 2dy) X i^dx^ -}- dy^; or, omitting 2dy (Art. 20), ds = ^ry^dx"^ + dy'^'j that is. The differential of a surface of revolution is eq^ial to the circumference of a circle perpendicxdar to the axis., int6 t/ie differential of the arc of the meridian curve. SEC. ni.] SURFAOES OP REVOLUTION. 77 Therefore, to find the measure of any surface of revo- lution : Find the values of y and dy^ from the equation of the meridian curve^ in terms of x and dx ; then mhstitute these values^ in the differential equation^ and integrate between the proper limits of x. Surface of a cylinder. 63. If the rectangle -4(7 be revolved around the side AB^ DC will generate the surface of a cylinder. Since the generatrix is parallel to the axis AB^ its equation will be, y = b, and hence, dy — 0. Substituting these values in the differential equation of the surface, we have, Jds = f2iryx/dx'^ + dy^ = f^^bdx = lifbx -h C. If we suppose A to be the origin of co-ordinates, 'C = 0, and integrating between the limits x — and X = h^ we have, 8 = 2bir7i; that is, The measure of the surface of a cylitider is equal to the circumference of its base into the altitude. 78 . DIFFERENTIAL CALCULUS. [SEC. HI. Surface of the cone. 64. If the right-angled triangle CBA be revolved around the axis AC^ GB will generate the convex surface of a cone. If we suppose C to be the origin ^ of co-ordinates, the equation oi BG will be, y =1 ax, and dJy = adx. Substituting these values in the differential equation of the surface, we have, fds = Jl'Tiaxy/dx^ 4- aHM = J2'iraxdx^ + a^ + (7, (Art. 35) = -R-aa^VT-f a^ + G, Estimating the surface from the vertex, where a; = 0, we have, (7=0, and s = tax' ViT If we make x = h = AG, and BA = b, we have, a = - , and consequently, n that is, The convex surface of a cone is equal to the circumference of the base into half the slant height. «EC. in.] 8URFACE0F THE SPHERE. 79 Siuface of the sphere. 65. To find the surface of a sphere. The equation of the meridian curve, referred to the centre, is, a;2 + y2 = i22. differentiating, we have, xdx + ydy = ; 56. _ xdx , _ , dy = , and dy^ = Substituting for dy^ its value, in the differential of the surface, which is. ds = 2'jry^dx'^ + dy^, we have, J*ds = J'liryd dx^ + ^<^a;2 r= J^'r^Edx = 2ri?a; + O. if If we estimate the surface from the plane passing through the centre, and perpendicular to the axis of JC, we shall have, s = 0, for X = 0, and consequently, (7=0. To find the entire surface of the sphere, we must inte- grate between the limits x = -\- H, and x = — M^ ^and then take the sum of the integrals, without reference to their algebraic signs; for, these signs only indicate the position of the parts of the surface with respect to the plane passing through the centre. Integrating between the limits, ic = 0, and a; = 4- i?, so DIFFEREXTIAL CALCULUS. [SEC. m. we find, * = 2'ri22; and integrating between the limits x = 0, and a; = — i2, there results, 8 z= - 2'S'i22 ; hence, Surface = 4crjR2 = 2^i2 X 2i2 ; that is, Equal to four great circles^ or equal to the curved surface of tJie circumscribing cylinder. 1. The two equal integrals, s =z 2'rri22, and « = — 2*i22^ ' indicate that the surface is divided into two equal parta by the plane passing through the centre. Surface of the paraboloid. 66. To find the surface of the paraboloid of revolution. Take the equation of the meridian curve, y2 = 2ixc, which being difl^erentiated, gives, dx^y^y, and ax^ = P" Substituting this value of dx in the differential of the surface, (Art. 62), we have, ds = '+p'f-P'i Surface of the ellipsoid. 67. To find the sui-face of an ellipsoid described by- revolving an ellipse about the transverse axis. The equation of the meridian curve is, whence, B^ xdx B xdx ^y =" - T^-TT = - ^ A^ y A y^— x^' 82 DIPFEEENTIAL CALCULUl [sEa ni. substituting the square of dy in the differential of the surface, and for y its value, |,/3?^r^. we have, di = iv^^dxy/A* - {A^- S^)3? (1.) hence, fds = I^^-^VA^ - B^fdx^J^-^ - x*. Put, 21'-— yGi^ — B^ — D^ Q. constant quantity; and A* A^ - B^ = I^, also a constant. and we have. fds = Dfdx^^ - x' E A B J> With C, the centre of the meridian curve, and the radius jK, describe a semi-circle. Then, / dx y/^ — a;2, is a circular segment of which the abscissa is ic, and radius R. If, then, we estimate the surface of the ellipsoid from the plane passing through the centre, and estimate the area of the circular segment from the same plane, any portion of the surface of the ellipsoid will be equal to the corresponding portion of the circle, multiplied by the constant D, Hence, if we integrate the expression, SEC. hl] cubatuee op volxtmbs. 83 JdXy/m - X\ between the limits a; = 0, and x = A^ we shall have the area of the segment CGFB^ which denote by B\ Hence, \ surface ellipsoid = Z> X Z)'; and Surface - 2D x J)'. 1. If we make ^ = ^, in Equation (1), the ellipsoid becomes a sphere, and we have. 8 = J^ntEdx = liiRx 4- G. If we estimate the surface from the plane passing through the centre, (7=0, and integrate between the limits jc = 0, and jb = i?, we have, \ surface of sphere = 2-^7^ ; hence, Surface = 4-^722. CUBATUEE OF VOLUMES OF REVOLUTION. 68. Cubatuee is the operation of finding the measure of a volume. When this measure can be found in exact terms of the measuring cube, the volume is said to be cahdble. 69. A volume of revolution is a volume generated by the revolution of a plane figure about a fixed line, called the ojxis. If the plane figure OoacdebB, be revolved about the axis of X, it will generate a volume of revolution. 8i DIFFERENTIAL CALCULUS. [sec. IU. A C D E B X Let us suppose the ordinates Aa^ Cc^ Dd^ &c., to be consecutive. During the revo- lution, any element of the sur- face, as, AacC^ "will generate the frustum of a cone, of which the radii of the bases are Aa = 2/, Cc = y -\- dy^ and the altitude, AC z= dx. This frustum will be an element of the volume, and will have for its measure,* lb/' + (y + dyy + y{y + dy)\dx. If we denote the volume by Fi develop the terms within the parenthesis, multiply by dx^ and then reject all the terms containing the infinitely small quantities of the second order (Art. 20), we shall have, dV xy'^dx. The area of a circle described by any ordinate y, is t!y'^\ ; hence. The differential of a volume of revolution is equal to the area of a circle perpendicular to the axis into the differential of the axis. The differential of a volume generated by the revolution of a plane figure about the axis of Y^ is rrx^dy, 70. To find the value of V for any given volume : Find the value of y'^ in terms of a, fro7n the equation of the meridian curve; substitute this value i?i the differ- ential equatio?i, and th^n integrate between the required limits of X. * Leg., Bk. VIII. P. 6. f Leg., Bk. V. Prop. 16. mas. in. CUBATURK OP VOLUMES. 85 KX.VMPLES. 1. Find the volume of a right cylinder with a circular base, whose altitude is h and the radius of whose base is r. We have for the differential of the volume, dV = 'jry^dx; and since y = r, we have, fdV =f-.f~dx', integrating between the limits a; = 0, and cc = A, h fdV = V= 'jrr^x = irr^h; that is, The measure of the volume of a cylmder is equal to the area of its base multijylied by the altitude,^ 2. Find the volume of a right cone with a circular base,, whose altitude is A, and the radius of the base, r. If we suppose the vertex of the cone to be at the origin: of co-ordinates, and the axis to coincide with the axis of" abscissas, we shall have, y = ax, or, y = |aj, and y2 _ ^a;2. substituting tliis value of y"^, we have. * liCgcndre, Bk. YIII. Prop. 2 J. V 86 DIFFERENTIAL CALCULUS. [SEC. HI. Integrating between the values ic = 0, and aj = A, h fdV= r= cj J = ^r^ X ^'; that is, The measure of the volume of a co9ie is equal to th6 area of the base into one-third of the altitude.* 3. To find the volume of a prolate spheroid, f The equation of the meridian curve is, ^2 and dV = '^^2^^^ ~ x^)dx ; hence, r- 'Si-- -%)*"• = 3(3^''"' - «') + (» - 2)aic— 3; and for the fourth. t^ic^ = w(/i - 1) {k 2) Cw — 3)aaj'»-*, SEC. IV.] SUCCESSITE DIFFERENTIALS. £3 It is plain, that when n is a positive integral nuii'btr, the function u = aa", will have n differential coefficients. For, when n differ- entiations have been made, the exponent of x m fJie second member will be ; hence, the nth. differentia? co- efficient will be a constant, and the succeeding one? ff-ill be 0. Thus, ■^ = n{n-'l){n-2){n-^) a. 1 d'^ + hi Sign of the first differential coefficient- »2. If we have a curve whose equation is, y = A^)y and give to X any increment h, we have (Art. 13), y^-y _ fix + h) - fix) h - h ' and passing to the consecutive values, <^y -~- = tan a. ax If we so place the origin of co-ordinates that the curve shall lie within the first angle, 7i will be positive, and y' — y will be positive at all points where the curve 94 DIFFIiPwENTIAL CALCULUS. [sEC. IV, recedes fiora the axis of X, and negative where' it ap- proaches the axis; and this is true for consecutive as well as for other values. Hence, the curve will recede from the axis of JC when the first dijferential coefficient is positive^ and approach the axis when that coefficient is negative. The general proposition for all the angles and every possible relation of y and x, is this : The curve will recede from the axis of X when the ordinate' and first differential coefficient have the same sign^ and approach it when they have different signs. 1. To determine whether a given curve, as ABC^ recedes from, or approaches to the axis of X, at any point, as G : Find, from the equation of the curve, the first differential co- efficient, and see whether it is positive or negative. 2. If the tangent becomes parallel to the axis of JT at any point, as JB, -^ = tan a = ; hence, a = 0. dx If the tangent becomes perpendicular to the axis of X^ at any point, as A dy dx = tan a = CO ; hence, a = 90*= SBC. IV.] SUCCESSIVE DIFFERENTIALS, 95 Eign of the second differential coefficient. 73. A curve is convex towards the axis of abscissas when it lies between the chord and the axis; and con- cavCf when the chord lies between the curve and the axis. 1. Figures ( 1 ) and ( 2 ) denote two curves, the one con- vex and the other concave towards the axis of JC. Let J?M be any ordinate of either curve, P'M' an ordinate consecutive with it, and P" M" an ordinate con- secutive with F'M\ If we designate the ordinate PM by y, P' Q' will be denoted by dy (Art. 21), and we shall have, FM' = y + dy, and since P"M" is consecutive with P'M\ P"M" = y + c?y + % + dy) = y + Idy + d'y. MP + P"M" Since, hence, and MM' = M'M" = dx, QM' = QM'= y-±y-±^^^y^,,j^^^ QM'-FM' = QP d^ ~2~ 96 DIFFERENTIAL CALCULUS. [sec. it. In the case of convexity^ QM' > F'M\ and then, d^y is positive. In the case of concavity, QM' < jPJf' , and then, d^y is negative ; and since dx^ is always positive, the second differential coefficient will have the same sign as the second differential of y. If w^e take the case in which the ordinates are nega- tive, the second differential coefficient will stiU have the same sign as the ordinate, when the cm've is convex, and a different sign when it is concave. Hence, The second differential coefficient will have the same sign as the ordinate when the curve is convex toioards the axis of abscissas, and a contrary sign when it is concave. 1. The second differential of y is derived from dy in the same way that dy is derived from y (Art. 72) ; viz.: by producing the chord PP', and finding the difference of the consecutive values of F"Q" and SQ'\ w^hich is JP"S, The co-ordinates x and y determine a single point of the curve, as P ; these, in connection with dx and dy, determine a second point, P', consecutive with the first ; and these two sets of values, in connection with the sec- ond differential of y, determine a third point, P", con- secutive with P. SEC. IV.] SUCCESSIVE DIFFERENTIALS. 9^ Hence, the co-ordinates x and y, and the first and second differential coefficients, always determine three consecutive points of a curve. 2. When the curve is convex towards the axis of abscissas, the tangent of the angle which the tangent line makes with the axis of iZj is an increasing function of X ; hence, its difierential coefficient, that is, the second differential of the function, ought to be, as we have found H, positive (Art. 19). When the curve is concave, the first differential coeffi- cient is a decreasing function of the abscissas; hence, the second differential coefficient should be negative (Art. 19). Applications. 74L. The equation of the circle, referred to its centre and rectangular axes, is, a;2 -}- 2/2 _ 7^2 . hence, ~ = • dx y X Placing = 0, we have, a; = 0. Substituting this value of x in the equation of the circle, we have, y = ±i?; hence, the tangent is parallel to the axis of abscissas at the two points where the axis of ordinates intersects the circumference. K we make, -^ = — ? = oo , we have, y = ; substituting this value in the equation of the circle, SB =s :t: i2 ; licnce, 98 DIFFERENTIAL CALCULUS. [SEC. TV. the tangent is perpendicular to the axis of abscissas at the points where the axis intersects the circumference. 1. For the second differential coefficient, we find, ^ _ _ ^ which will be negative when y is positive, and positive when y is negative. Hence, the circumference of the circle is concave towards the axis of abscissas. 2. If we apply the same process to the equation of the ellipse, of the parabola, and of the hyperbola, we shall find that the tangents, at the principal vertices, are parallel to the axes of ordinates ; that the second differ- ential coefficient and ordinate, in all the cases, except that of the opposite hyperbolas, have contrary signs ; and hence, aU these curves, except the conjugate hyperbolas, are cor^ cave toicards the axis of abscissas. maclaurin's tiieoeem. T5. Maclauein's Theorem explains the method of de- veloping into a series any function of a single variable. Let u denote any fimction of a, as, for example, u = (a + x)'" (1.) It is required to develop this, or any other function of Xy into a series of the form, u = A + Bx + Cx^-h Dx^ + ^E^ + &c. . . (2.) in which A, B, C, i>, &c., are independent of x, and arbitrary functions of the constants which enter into the SEC. IV.] MACLAURIN'S THEOKEM. 9? second member of Equation ( 1 ). When these coeffi- cieAts are found, the form of the series will be known. Since the coefficients, A^ B, C, + 2.3.4^ + &c. &c., &c. ; A = («),_o ~" Wa;/aj-o _ 1 p«\ - 1.2\(fe'/»,_o 1 lcPu\ ~ 1.2.3\c&Vb_o Ac. &c. ; 100 DIFFERENTIAL CALCULUS. [SEC. IV. hence, whicli IS Maclaurin's Formula. In applying the formula, we omit the expressions jc = 0, although the coefficients are always found under this hypothesis, EXAMPLES. 1. Develop (a + aj)*", by Maclaurin's Formula, A = a*", B = { — \ = m{a + a;)'"-^ = ma*"-*, ^ - 2U^^/ ~ 1.2 ^""^"^^ - 1.2 "* ' _ _l_/^\ _ m(m-l) (m-2) , ^ - 1.2.3Waj3J - 1 2 3 ^""^^^ _ m (m- 1) {m - 2) , ^12 3 ' &c., &c., &c. Substituting these values in Equation (2), we have, (a + xY = a" + ma'^-^x + ^ ^^ "7 ^^ a'^-^ar^ the same result as found by the Binomial Formula. SEC. IV.] maclaurin's thkokem. 101 2. If the function is of the form, u = — i- = (« + x)--^ = a-4l + ^)~'. a + X ^ \ at we find, ^ = -, a du\ w . V « 1 1 ji = m ^ ^i^a^x)--= -7-1— ^ \dxJx^O V -r ; (a + ic) _ l/^w\ _ — 1 X — 2((nr +jC)2f _ J^ a^ ~ 2.3W/« = 0~ 2.3 ~ , &c., are independent of y, but functions of a, and arbitrary func- tions of all the constants which enter the primitive func- tion. It is now required to find such values for the ex- ponents «, Z>, c, &c., and for the coefficients A^ B^ (7, Z>, &c., as shall render the development true for all possible values that may be attributed to x and y. In the first place, there can be no negative exponents. For, if any term were of the form. it might be written. By--, B and making y = 0, this term would become infinite, and we should have, which is absm'd, since the function of ic, which is inde- pendent of y, does not necessarily become infinite when y = 0. The first term A, of the development, is the value which the primitive function u' assumes when we mako V = 0. SEC. IV.J TAYLOR'S TnEOKEM. 105 If we designate this value by u, we shall have, u = f(x). If we differentiate Equation ( 1 ), under the supposition that X varies, the partial differential coefficient is, du' dA dB ^ dC , dB and if we differentiate, regarding y as a variable, the partial differential coefficient is, du' ■^ = a^y«-i + bCy^-' + cBy<^-'^ + &c. . . (3.) But these differential coefficients are equal to each other (Art. 78) ; hence, the second members of Equations ( 2 ) and ( 3 ) are equal. Since the coefficients are inde- pendent of 2/, and the equality exists whatever be the value of y, it follows that the corresponding terms in each series will contain like powers of y, and that the coefficients of y in these terms will be equal * Hence, a — 1=0, b — 1 = a, c — 1=J, &c., and consequently, u\ and u > ic" ; the curve therefore ascends just before the ordinate reaches a maximum value, and descends immediately afterwards ; hence, at the point of maximum, it is concave towards the axis of abscissas (Art. 73). Since the curve ascends just before the ordinate reaches the maximum value, the first differential coefiicient will be positive ; and since it then descends^ the first differ- ential coefficient will be negative immediately after the 108 DIFFERENTIAL CALCULUS. [sec. maximum value (Art. 72). Hence, at the point of maximum value of the ordmate, the first differential co- efiicient will change its sign, and therefore passes through 0. Since the curve is concave towards the axis of abscissas, the second differential coefficient is negative (Art. V3) ; hence, the conditions of a maximum value of u are, du dx = 0, and y^, negative. 82. Denoting the consecutive or- dinates, as before, by u\ u, u'\ if u is a minimum. u < u\ and w < w'' ; the cui-ve, therefore, descends just before the ordinate reaches a minimum, and ascends immediately afterwards; hence, at the point of minimum, it is convex towards the axis of abscissas. Since the curve descends just before the ordinate reaches the minimum value, the first differential coefficient wUl be negative; and since it then ascends^ the first differential coefficient will be positive immediately after the minimum value (Art. 72). Hence, at the point of minimum, value of the ordinate, the first differential coefficient will change its sign, and therefore passes through 0. Since the curve is convex towards the axis of abscissas, the second differential coefficient is positive (Art. 73) ; hence, the conditions of a minimum value of «, are, du d?u dx^ and ^^ , positive. SEC. v.] maszima and minima. 109 83, Hence, to find the maximum or minimum value of a function of a single variable: 1. Find the first differential coefficient of the function^ place it equal to 0, and determine the roots of the equation, 2. Find the seco7id differential coefficieiit^ and substi- tute each real root, in succession, for the variable in tJie second member of the equation; each root which gives a negative result, will correspond to a maximum value of the function, and each which gives a positive result will coT' respond to a minimum value. Point of inflection. §4. A POINT OF INFLECTION is a poiut at which a curve changes its curvature with respect to the axis of ab- scissas. , "When a curve is concave towards the axis of abscissas, its second difi*er- ential coefficient is negative (Art. "72) ; when it is convex, the second differ- ential coefficient is positive (Art. 72) : therefore, at the point where the curve changes its curvature, the second differential coefficient changes its sign, and consequently passes through zero. In the first figure, the second differential coefficient, at the point M, changes from negative to positive ; in the second, from positive to negative. At the point M, in both figures, the first differential coefficient is equal to 0, and the tangent line separates the two branches 110 DIFFEREXTIAL CALCULUS. [SEC. V. of the ciiiTe. When the second differential coefficient is 0, the ordinate at the point has neither a maximum nor a ndnimmn. There are three consecutive points of the curve which coincide with the tangent, at the point of inflection. This is shown by the equality of the co-ordinates of the point M (in the curve and tangent), and of the first and sec- ond differentials. EXAMPLES. 1. To find the value of x which will render the func- tion y a maximum or minimum in the equation of the circle, y +x - M. ^- ^, making, — - = 0, gives, x = 0. The second differential coefficient is, f^ = - ^L±^'. When, cc = 0, y = R\ dx^ y^ 5 > ./ > hence, ^ = " i' which being negative, y is a maximum for M positive. 2. Find the values of x which render the function y a maximum or minimum in the equation, y = a ^ hx -^ x"^. Differentiating, o» t=-6 + 2«, and ^ = 2; dx SEC. V.J MAXIMA AND MINIMA. Ill and making, — 5 + 2a; = 0, gives, X = 2 Since the second differential coefficient is positive, this value of X will render y a minimum. The minimum value of y is found by substituting the value of jc, in the primitive equation. It is, y = a--- 3. Find the value of x which will render the function u a maximum or minimum in the equation, w = a* + 5^ic — c^x^. J = J3_2c^^, hence, « = ^, and ^3 = -2«^ hence, the function is a maximum, and the maximum value is, 4^2 ^ = «* + zr2 4. Let us take the function, u = ^a^x^ — ¥x + c^. We find, ^ = 9a2a;2 _ 54 and jc = ± — The second differential coefficient is. 112 DIFFERENTIAL CALCULUS. [SEC. V. Substituting the jjIus root of £C, we have, which gives a minimum, and substituting the negative root, we have, £ = -.«.., which gives a maximum. The minimum value of the function is, u = c^ — — - ; and the maximum value. 9a 5. Find the values of cc, which make u a maximum or minimum in the equation. ^ jc = 1, a maximum. u = ic^ — 5x* 4- 525^ — 1. X = a; = 3, a minimum. 6. Find the values of x, which make u a maximum or minimum in the equation, u =z x^ — 9ic2 -f 15a; — 3. ( aj = + 1, a maximum. Ans, ] [x = + 5, a minimum. SBC. v.] MAXIMA AND MINIMA. 113 1. Find the values of x, which make u a maximmn or minunum in the equation, u z= x^ — Sx^ -{- Sx -{- 7. Ans. There is no such value of a;, since the second differ- ential coefficient reduces to 0, for a; = 1 ; hence, only- one condition of a maximum or minimum is fulfilled.* 85* Notes. 1. In applying the preceding rules to practical examples, we first find an expression for the function which is to be made a maximum or minimum. 2. If in such expression, a constant quantity is found as a factor, it may be omitted in the operation; for the product will be a maximum or a minimum when the variable factor is a maximum or minimum. 3. Any value of the independent variable which renders a function a . maximum or a minimum, will render any power or root of that function, a maximum or minimum ; hence, we may square both members of an equation to free it of radicals, before differentiating. 8. To find the maximum rectangle which can be in- scribed in a given triangle. Let b denote the base of the triangle, h the altitude, y the base of the rectangle, and x its altitude. Then, u = XT/ = the area of the rectangle. But, b : h : : y : k — x; bh ^ bx hence, y = ^ , * We have limited the discussion to a single class of maxima and minima, viz. : that in •which the first differential coefficient of the func- tion is 0, and the second negative or positive. lU DIFFERENTIAL CALCULUS [sec. V. and consequently, hJvx — hx^ b,\ ,v u = . = j-(hx - X'); and omitting the constant factor — , we may write, u' = hx — x^ ; for, the value of x, which makes u' a maximum, will make u a maximum (Art. 85) ; hence. dx = A — 2a;, or, A 2' therefore, the altitude of the, rectangle is equal to half the altitude of the triangle; and since, the area is a maximum (Art. 81 ). 9. What is the altitude of a cylinder inscribed in a given cone, when the volume of the cylinder is a maxi- mum? Suppose the cylinder to be in- scribed, as in the figure, and let ABz^a, BC=h, AD = x, ED = y\ then, BD = a — x = altitude of the cylinder, and ify^{a — x)* = volume = v . . (1.) From the similar triangles A£JI> and ACB, we have, * Legendre, Bk. VIII. Prop. 2. BEC. v.] MAXIMA AND MINIMA, 115 X : y : : a : b; whence, y = bx' Substituting this value in Equation ( 1 ), we have, Omitting the constant factor — -, we may write, v' — x^{a — X) ; for, the conditions which will make v' a maximum, will also make v a maximum (Art. 85). By differentiating, we have, c = dx Placing, lax - 3a;2 = 0, we have, 2 a; = 0, and x = -a. 3 But, dx'' = 2a — 6x = — 2a. Hence, the cylinder is a maximum, when its altitude is one-third the altitude of the cone. 10. What is the altitude of a cone inscribed in a given sphere, when the volume is a maximum? Denote the radius of the given sphere by r, and the centre by C, Let A be the vertex of the re- quired cone, J3D the radius of its base, which denote by y, and denote the altitude AB by X, Then, 116 DIFFEEENTl A.L CALCULUS. [sec. 2/2 — 2ra; ,2 . * and if we denote the volume of the cone by v, V = ^'rrx{2rx - jc2) = i .... (2.) Subtracting Equation ( 1 ) from ( 2 ), member from mem- ber, we have, w' — w = a'a^ — a' — a'(a^ — 1) ; whence, = a* — 1 (3.) Put, a = 1 + 5, and develop by the binomial formula; we then have. 120 DIFFERENTIAL CALCULUS. [SEC. VL Substituting this value of a*, in Equation ( 3 ), and dividing by A, we have, If we now pass to consecutive values, by making h numerically equal to 0, we have, du I «>' . h^ h*- h^ . and putting for h its value, a — 1, we have. Denoting the second member of Equation (4) by A:, we have, —-T- = Ar, or, du = da' = a'kdx . . (5.) that is, the differential of a function of the fonn a*, is equal to the function, iiito a constant quantity Jc^ de- pendent on a, into the differential of the exponent. Relation between a and 1c, 87. The relation between a and h is very peculiar, and may be determined by Maclaurin's Formula, » = «- = W + (S> + riS)- + ryS)- + &c (6.) SEC. VI.] EXPONENTIAL FUNCTIONS. 121 First, if we make aj = 0, the function a* = I = luy The successive differential coefficients are found from Equation ( 5 ) ; viz. : S = "'*> ^* O = *' ^~j = -z- = da'k = a^'k'^dx; hence, £ = -^' -^ &) = '- = a'^, and (g) = ^', &c^ &c., &c. Substituting these values in Equation ( 6 ), "we have, If we make a = 7 , we shall have, I 111 ^ 1 ^ 1.2 ^ 1.2.3 ^ ' designating the second member of the equation by c, and jemploying twelve terms of the series, we find, e = 2.7182818. . ..; 1 hence, a* = e, therefore, a = e* . . (7.) Equation ( 7 ) expresses the relation between a and k. 122 DIFFERENTIAL CALCULUS. [sEC. VI. A system of logarithms, called the Naperian system, has been constructed, whose base is, e = 2.7182818.... This, and the common system, whose base is 10, are the only systems in use. The logarithms, in the Naperkn system, are denoted by Z, and in the common system by log. "We see from Equation (7), that k is the Naperian logarithm of the number a. If we take the common logarithms of both members of Equation (7), we shall have, log a = k log e (8.) The common logarithm of e = log 2.7182818 .... = .434284482 . . . . , is called the modulus of the common system, and is denoted by M. Hence, if we have the Naperian logarithm of a number, we can find the com- mon logarithm of the same niimher hy multiplying hy t/ie modulus. I^ in Equation (8), we make a = 10, we have, 1 = k\oge\ or, ^ = log e = M\ that is, the modulus of the common system is also equal to 1, divided hy the Naperian logarithm of the common base. S8. From Equation (5), we have, du da* y , — = — = /cdx. u a* If we make a = 10, the base of the common system, X = log M, and f7u 1 du ^^ XI k u SEC. VI.] LOGARITHMIC FUNCTIONS. 123 that is, the differential of a common logarithm of a quantity is equal to the differential of the quantity di- vided by the quantity into the modulus. 89. If we make a =^ e, the base of the Naperiau system, x becomes the Naperian logarithm of w, and k becomes 1 : see Equation (7) ; hence, M = I ; and , du dx = — ; a'' that is, the differential of a Naperian logarithm of a quantity is equal to the differential of the quantity di- vided hy the quantity ; and in this system^ the modulus is 1. ^ 90. Having found that Jc is the Naperian logarithm of a, we have from Equation ( 5 ), ' du = a'ladx; that is, the differential of a function of the form a*, is equal to the function^ into the Naperian logarithm, of the base «, into the differential of the exponent. EXAMPLES. 1. Find the differential of w = a*. du = a'ladx, 2. Find the differential oi u = Ix. du = — = aj-Waj. X Note. This case would seem to admit of integration by the rule of Art. 35; but that nile applies to alge- 32i 11IFFEKENTIAL CALCULUS. [sEC. VI. braic ftmctions only, and this form is derived from a transcendental function. S. Find the differential of w = y*. lu = xly, hence, — = jc— + lydx', hence, u y "^ ' by clearing of fractions, and reducing, du = xy' - '^dy + y*ly dx\ that is, equal to the sum of the partial differentials (Art. 32). 4. Find by logarithms the differential of w = Qty, lu z= Ix -\- ly-y* hence, — = 1 — -; and by reducing, i4> tij y du — ydx 4- xdy (Art. ay). 5. Find by logarithms the differential of w = - . lu — Ix — ly \\ hence, by differentiating, du dx dy ^ . ^ . — = ~ ; and by reducmg, ^^^yJx-_^ (Art. 29). * Bourdon, Art. 230. University, Art. 1§5. f Bourdon, Art. 231. University, Art. 185. SEC. VI.] LOGARITHMIC FUNCTIONS 125 6. Find the differential of u = l(-^-]. , ladx du = ' a^ — x^ r. Find the differential of w = l(-^=:^=] V-i/oM- xy du jc(a2 + x^) 8. Find the differential of u = {a' + 1)\ du = 2a^{a' + l)ladx, d^ I 9. Find the differential of w = • a' 4- 1 , la'ladx ■ du =: • (a- + ir 10. Find the differential of u = — = (-)'• X' \x/ DijQferenfial forms which have known integrals. 91. If we have a differential in a fractional fonn/ in which the numerator is the differential of the denomina- tor, we know that the integral is the Naperian logaritlim of the denominator (Art. §9). It frequently happens, however, that we have to deal with fractional differen- tials which are not of this form, but which, by certaui algebraic artifices, may be reduced to it. We shall give a few examples of such reductions. J 26 DIFFERENTIAL CALCULUS. [SEC. VI. dx Form 1. , /: Put jc^ _j_ ^2 _ -y2 . then, xdx = vdv. Add vc?a5 to both members; then, xdx + vdx = vdx + vdv ; hence, (a; + tj)(7ic = v{dx + ^y) ; whence, dx + (Zy f7a; dx « + y V ViC^ rt a2 ; hence, /dx -\- dv _ r dx X + V ~ J .^^T^jT^ Bnt in the first member, the numerator is the differen- tial of the denominator ; hence, /-^= = l{x + v) = l(x + v^^^^). Form 2. /- (fa; Put Va;2 ± ^ax = v ; then, x"^ ± 2ax = v\ Adding a^ to both members, and extracting the square root, vdv X ± a = y/v^ + d?- ; hence, dx and ya2~± 2oa; Vu2 + a2 SEC. VI.] LOGARITHMIC FUNCTIONS. 127 But from the first form, Substituting for v its value, and for V^+ ^S ^^^ value, r^_ dx _ _ i{x ± a ■\' ^/tP- ± 2ajc). „ 2(2cfa; 2adx Form 3. -r r; or, -^^ 5 ^2 _ jg2 » a;2 — a2 2ac?a; 2ac?a; - + C. ver-sin-'- + C a = tan-i- + C. a 136 rUFFERENTIAL CALCULUS. fsEC. TI. Applications. 100. We may readily find the relation between the diameter and the cii*cumference of a circle from either of the first four equations of Art. 97. 1. To find this ratio from Equation (5), which is, Developing by the Binomial Formula, we have, ^ = 1 + 22/^ + 2-:^y* + 2-;^^^ + &c.; whence, 1 1 Q 1 ^ *» dz =dy+ -y^y + ^ y*dy + —^ y^dy + 1. 104. If we suppose the base of the system of loga- rithms to be greater than 1, the logarithms of all numbers less than 1 will be negative;* therefore, the values of y, corresponding to all abiseissas between the limits of a; = 0, and X -= AE = 1, will be negative; honce, these ordi- nates are laid off below the axis of JT. AMien x ■= 0, y = — 00. Therefore, when tlie base is greater than 1, the corresponding curve is QPEK' . The curve cannot extend to the left of the axis of I", since negative numbers have no real logarithms.f Base < 1. ^ 105. If the base of the system is less than 1, the log- aritlims of all numbers greater than 1 are negative; and of all numbers less than 1, positive. Under this supposi- tion, the curve assumes the position Q'I'EK. The parts * Bourdon, Art. ^J>5. University, Art. 1 gC \ Bourdon, Art, 33(3. University, Art. 1S6 142 DIFFEKEXTIAL CALCULUS. [SEC. VU. of the curves EPQ^ EP Q\ are concave towards the aids of abscissas; the parts EK^ EK\ are convex; and both curves, throughout their whole extent, are convex towards the axis of Y. Asymptote. 106. Let us resume the equation of the curve, y — logic. If we denote the modulus of a system of logarithms by 3f, and differentiate, we have (Art. 88), dx^^ dy M dy = -Jf; or, ^^ = -• But, -^ denotes the tans^ent of the an He which the dx tangent line makes with the axis of abscissas; hence, the tangent will be parallel to the axis of abscissas when a; = 00, and perpendicular to it, when x = 0. But, when x = 0, y = — cc; hence, the axis of ordinates is an asymptote to the curve. The tangent which is par- allel to the axis of -Z", is not an asymptote; for, when jc = 00 , we also have, y = co (Art. 50). Sub-tangent. 107. The most remarkable property of this cui-ve, is the value of its sub-tangent T'Ii\ estimated on the axis of logarithms. We have found, for the sub-tangent, on the axis of X (Art. 45), -^ = %"■■ BBC. VII.] TRANSCENDENTAL CUKVKS. 143 and by simply changing the axis, we have, T'R' = -j^x = M (Art. 106); hence, ax ^ ' The sub-tangent^ taken on the axis of logarithms, is equal to the modulus of the s'gstem from which the curve is constructed. In the Napeiian system, M = \\ hence, the sub-tangent is equal to 1, equal to AE. In the common system, it is denoted by the number, .434284482 . . . 108. If a circle KFG be rolled along a straight line, AL, any point of the circumference, as P, will describe a curve, which is called a cycloid. The circle NPG is called the generating circle, and 1\ the generating point. Since each succeeding revolution of the generating circle will describe an equal curve, it will only be necessary to examine the properties of the curve APJ^L, described in one revolution. We shall, therefore, refer only to this part, when speaking of the cycloid. If we suppose the point P to be on the line AL, at A, it will be found at some point, as X, after all the points of the circumference shall have been brought in contact with the line AL. The line AL will be equal to the circumference of the generatmg circle, and is called the vu DIFFERENTIAL CALCULUS. [sec. vn. brfse of the cycloid. The line -Z?J/', drawn perpendicular to the base, at tlie middle point, is called the axis of the cycloid^ and is equal to the diameter of the generating circle. Transcendental Equation of the Cycloid. 109. Let CN be the radius of the generating circle. Assume any point, as A, for the origin of co-ordinates. Let us suppose that when the generating point has de- scribed any arc of the cycloid, as ^-P, that the point in which the circle touches the base has reached the point N, Through N^ drnw the diameter KG^ of the generating circle : it will be perpendicular to the base AL. Through P. draw PR perpendicular to the base, and PQ parallel to it. Then, PP — KQ will be the versed sine, and PQ the sine of the arc NP to the radius CN". Pat, CR = r, AP = X, PP = JSTQ = y, we shall then have, PQ = -v/2ry-2/% ^ =A]Sr-PN'= arc NP-PQ-, hence, the transcendental equation is, ver-sin~^y — ^/2ry — y"^. 8K0. Vn.] TEANSCENDENTAL CUKVES. 145 Differential Equation. 110. The properties of the cycloid are most easily deduced from its differential equation. This is found by differentiating both members of the transcendental equation. We have (Art. 97), Tcly w, pg^ and then draw through P the parallels PN^ PG\ and PN will be a normal, and PG a tangent to the cycloid at the point P. Position of Tangent. 112. The differential equation of the curve, dx v^y - y^ SEC. VII.] TRANSCENDENTAL CURVES. 147 may be put under the form, ^ ^ V2ry - y^ _ /5r __ ^ dx y ~~ V y If we make y = 0, we shall have, dx ' and if we make y = 2r, we shall have, dx hence, the tangent lines drawn to the cycloid at the pointa where the curve meets the base, are perpendicular to tlie base ; and the tangent drawn through the extremity of the greatest ordinate, is parallel to the base. Curve Concave. 113. If we diflerentiate the equation, dx = ——= _ , V2ry — 2/2 regarding dx as constant, we obtain, V2ry - y2 or, by reducing and dividing by y, = (2ry — y^)d^y + rdy\ whence we obtain, rdy^ ^ ^^- 2^-y^' and hence, the curve is concave towards the axis of abscissas (Art. ya). us DIFFERENTIAL CALCULUS. [SEC. VU, Area of the Cycloid. 114. The area of the cycloid may be found in a very simple manner, by constructing the rectangle AFBM^ and considering the portion AFB, If we regard F as an origin of co-ordinates, FB as a line of ab- scissas, and take any ordinate, as, KH z = 2r y> we shall have, Rut, d{AHKF) = zdx. zdx = — — _y M^ _ ^y^2i'y — y2; V2ry — y2 whence, AHKF =fdy^/^— y^ + (7. But this integral expresses the area of the segment of a circle, whose radius is r, and versed-siiie y (Art. 99), that is, of the segment MIGE. If now, we estimate the area of the segment from J/, where y = 0, and the area AFXH from AF^ in which case the area AFKH — 0, for 2/ = 0, we shall have, AFKR = MIGE; and taking the integral between the limits y = and y = 2r, we have, AFB = semi-circle MIGB, and consequently, area AIIBJSI = AFBM - MIGB. SEC. VII.] T R A N S C E N D E X TA L CURVES. 319 But the base of the rectangle AFBM is equal to the semi-circumference of the generalhig circle, and the alti- tude is equal to the diameter ; hence, its area is equal to four times the area of the semi-circle 3IIGB ; there- fore, area AHBM = 33fIGB ; hence. The area AHBL is equal to three times the area of the generating circle. Surface described by tho Cycloid. 115. To find the surface described by the arc of a cycloid when revolved about its base. The differential equation of the cycloid is, 6x= y^y Substituting this value of dx in the differential equation of the surface (Art. 62), it becomes, Applying Formula (E), (Art. 170), we have, L 3 3 y ^2rg - yd But, 23 150 DIFFERENTIAL CALCULUS. [sEC. VH. heiKie, 8 = 2;rV2r[- ?2/^v^ -y'- lr{2r - 2/)^] 4- C. If we estimate the sui-fiice from the plane passing through the centre, we have C = 0, since at this point 5 = 0, and 7/ z= 2r. If vre then integrate between the limits y = 2r, and y = 0, we have, s = - surface = — — cr-; hence, 2 o s = surface = — — cr"'*, 3 ' that is, the surfiice described by the cycloid, when it is revolved around the base, is equal to 64 thirds of the generating cii'cle. The minus sign should appear before the integral, since the surface is a decreasing function of the variable y (Art. 19). Volume generated by the area of the Cycloid. 116. If n cycloid be revolved about its base, it is re- quired to find the measure of the volume which the area ^11 generate. The differential equation of the cycloid is, -v/27^- y2 If we denote the volume by T", we have (^Art. 60), '^fifcly dV = V^ri/ — y* SEC. Vn.] TEANSCENDENTAL CURVES 151 If we apply Formula (E) (Art. 170), we shall find, aftoT three reductions, that the integral will depend on that of dy ^2ry - 2/2 But the integral of this expression is the arc whose vereed sine is - (Art. 99). Making the substitutions and reduo- tions, we find the volume equal to five-eighths of the circumscribing cylinder. Spirals. 117. A Spiral^ or Polar Line, is a curve described by a point which moves along a right line, according to any law whatever, the line having at the same time a uniform angular motion. Let AJBG be a straight line which is to be turned uniformly around the point A. When the motion of the line begins, let us sup- pose a point to move from A along the line, in the direction AB C. When the line takes the posi- tion ADE, the point will have moved along it, to some point, as Z>, and will h.nvc described the arc AaD, of the spiral. When the line takes the position AD'E\ the point will have described the curve AaBD\ and when the line shall have complet- ed an entire revolution, the point will have described the curve AaBD'B. If the revolutions of the radius-vector be continued, the 152 DIFFEnENTIAL CALCULUS. [sEC. VII. generating point will describe an indefinite spiral. The point A, about which the right line revolves, is called the pole ; the distances AD, AD\ AJS, are called ra- dius-vectors or radii-vectores ; and the parts AaDD'B, BFF' C, described in each revolution, are called spires. If, with the pole as a centre, and AB, the distance passed over by the generating point in the direction of the radius-vector, during the first revolution, as a radius, we describe the circumference BEE\ the angular motion of the radius-vector about the pole A, may be measured by the arcs of this circle, estimated from B. If we designate the radius-vector by u, and the meas- nring arc, estimated from J5, by f, the relation between u and t, may be expressed by the equation, u = f[t), or u = ar, in which 7i depends on the law according to which the generating point moves along the radius-vector, and a on the relation which exists between a giv€?i value of w, and the corresponding value of t. General Properties. lis. When n is positive, the spirals represented by the equation, u = at^y will pass through the pole A. For, if we make ^ = 0, we shall have, u = 0. But if 71 is negative, the equation will become, a or, « = -, SEC. VII.] TRANSCENDENTAL CUPwVES, 153 from which we shall have, for, ^ = 0, w = 00, and for, ^=00, w = ; hence, in this class of spirals, the first position of the generating point is at an infinite distance from the pole: the point will then approach the pole as the radius-vector revolves, and will only reach it after an infinite number of revolutions. Spiral of Archimedes. 119. If we make t* = 1, the equation of the spiral becomes, u =z at. If we designate two different radii-vector es by u' and m", and the corresponding arcs by t' and t'\ we shall have, u' = at\ and w" = at", and consequently, w' : u" : : t' : t" ; that is, The radii-vectores are proportional to the measuring arcs, estimated from the initial point. This spiral is called the spiral of Archimedes. If we denote by 1, the distance which the generating point moves along the radius-vector, during one revolu- tion, the equation. will become. 1 = at; or, 1 x - = f . ' a 154 DIFFEKEXTIAL CALCULUS. [SEC. TTI. But since t is the circumference of a circle whose radius is 1, we shall have, - = 2ty and consequently, a = — • Parabolic SpiraL 120. If we make n = -, and a = v^> w® have, for the general equation, u = v^ X t^ ', or, u^ = 2pt, which is the equation of the parabolic spiral. If ^ = 0, w = ; hence, this spiral passes through the pole. Hyperbolic Sj^raL 121, If we make w = — 1, the general equation of ipirals becomes, u = at-^; or, ut = a. This spiral is called the hyperbolic spiral^ because of the analogy which its equation bears to that of the hy- perbola, when referred to its asymptotes. If, in this equation, we make, successively, < = 1, « = 2' * = 3' ^ ^ V ^°" we shall have the corresponding values, u = a, u = 2a, w = 3a, u = 4a, &c. SBC. Vn.] TBANSCENDENTAL CUEVES, 155 Logarithmic Spiral. 122. Since the relation between u and t is entirely arbitrary, we may, if we please, make, t — log w. The spiral described by the extremity of the radius- vector, xmder this supposition, is called the logarithmic spiral. Direction of the measuring arc. 123. The arc, which measures the angular motion of the radius-vector, has been estimated from right to left, and the value of t regarded as positive. If we revolve the radius-vector in a contrary direction, the measuring arc will be estimated from left to right, the sign of t will be changed to negative, and a similar spiral will bo described. Sub-tangent in Polar Curves. 124. The Sub-tangent, in spirals, or in any curvCj referred to polar co-ordinates, 18 the projection of the tangent on a line dra\\Ti through the pole, and perpendicular to the padius-vector passing through the poiut of contact. Let A be the pole, AR = 1, the radius of the measuring arc, P any point of the curve, TP a tangent at P, and Al\ 156 DIFFEPvENTlAL CALCULUS. [SEC. VH. perpendicular to AP, the sub-tangent. Let AP' be a radius-vector, consecutive with AJP, and P§, an arc de- scribed from the centre A. Then, JViV"' = dt, and QP — duy and, since JPQ is parallel to iViV', we have, P§ = itdL But the arc PQ coincides with its chord (Art. 43), and since Q ia a right angle, the triangles PQP' and 7L1P are similar; hence, AT : AP :: PQ : QP' ; therefore, Sub-tangent AT : u : : udt : du. Whence, Sub-tangent AT = "^ = -t--^\ du n 125. In the spiral of Archimedes, we have, 1 n = 1, and a = ~; hence, AT = --^ If we make t = 2^, circumference of the measuring circle, we shall have, AT = 2^, circumference of the measuring circle. After m revolutions, we shall have t = 2mie, and consequently, AT = 2mV = m.27n'!r-j that is, The 8iib-tanr/eJit, after in revQlutions^ is equal to m times SEC. VII.] TRANSCENDENTAL CUEVES. 157 the circumference of the circle whose radius is the radiuS' vector. This property was discovered by Archimedes. 126. In the hyperbolic spiral, ?z = — 1, and the value of the sub-tangent becomes AT =z — a ; that is, The sub-tangent is constant in the hyperbolic spiral. Angle of the Tangent and Radius-Vector. 127. We see that, AT _ icdt AP ~ du ' denotes the tangent of the angle which the tangent line makes with the radius-vector. In the logarithmic spiral, of which the equation is we have, hence, -j-p = -y- = M; that is, In the logarithmic sjyiral, the angle formed by the tangent and the radius-vector passing through the point of contact^ is constant ; and the tangent of the angle is equal to the modulus of the system of logarithms. If t is the Naperian logaritlim of w, JLT is 1 (Art. 89), and tlie angle will be equal to 45°. t = log u, dt = du —M; AT AP = udt da ~~ 158 DIFFERENTIAL CALCULUS. ] SEC. VII, Value of the TangenL 128. The value of the tangent, in a curve referred to polar co-ordinates, is, Dififerential of the Arc. 129. To find the differential of the arc, which we de- note by 2, we have, or, by substituting for PP\ QP\ and P§, their Values, when P and P' are consecutive, we have, Differential of the Area. 130. The differential of the area ADP, when referred to polar co-ordinates, is not an elementary rectangle, as when referred to rectangular axes, but is the elementary sector APP'. The area of this triangle is equal to AP X PQ ^ j^ ^^ denote the differential by ds, we have, AP'x QP (u -f du)udt ^^ = 2 = 2 * or, omitting the infinitely small quantity of the second order,, ududt (Art. 20), u^dt ds = -^, SEC. VII.] TKANSCENDEXTAL CURVES. 159 which is the differential of the area of any segment of a polar line. • Areas of Spirals. 131. If we denote by 5, the area described by the ra- dius-vector, we have (Art. 130), ds = — ; and placing for u its value, at"" (Art. 11 t), ds z= — - — , and s = + C. If ?* is positive, C = 0, since the area is 0, when t = 0. After one revolution of the radius-vector, t = 2if, and we have, _ g" (2r)^'» + ^ ^ "" ~4fi + 2 ' which is the area included within the first spire. 132. In the spiral of Archimedes, (Art. 119), 1 a = -—, and n = 1 : hence, for this spiral we have, ^ - 2172' which becomes - , after one revolution of the radius-veo- 3 If tor ; the unit of the number - , being a square whose o side is 1. Hence, the area included by the first sjvre^ is equal to one-third of the area of the circle whose radius is the radius-vector^ after the first revolution. ICO D I F F E i: E N T I A L CALCULUS. [SEC. VH. Ill the second revolution, the radius-vector describes a second time, the area described in the first revohition ; and in any succeeding revohition, it will pass over, or re- describe, all the area before generated. Hence, to find the area, at the end of the mth revolution, we must in- tegrate between the limits, t = {m — 1)27^, and t = m.2'!f, which gives,' If it be required to find the area between any two spires, as between the ?nth and the (m + l)th, we have for the whole area to the (m + ])th spire, 3 . ' and subtracting the area to the mth. spire, gives, (m 4- 1)^ — 2m3 + hn — If s= ^^ -'Tf = 2m^, o for the area between the mth and {m + l)th spires. If we make m = I, we shall have the area between the first and second spires equal to 2':f ; hence, the area between the mth and {m -h l)th spires, is equal to m times the area between the first and second. 133. In the hyperbolic spiral, ?i = — 1, and. we have, ds = -^^^^ a° ^^ have, hence we have. 4y2 _[_ ^2 X — a = ; rt 4^3 T 2?/2 m — p = -^, and £C — a = — -^^ — — ; substituting for y its value in the equation of the involute, we obtain, y = 7n-a;% 3 -13 = — ; x-a= -2aj--; and by eliminating jc, we have, 10/ 1 \3 P2 1^ / 1 \ 27m\ 2 / ' vhich is the equation of the evolutc. If we make (3 zzz 0, we have, « = Im; 176 DIFFEREXTIAL CALCULUS. [SEC. VD. an4> University, Art. 1§0* INTEGRATION OF FRACTIONS. 181 Substituting these values for A and JB, we obtain, 2adx _ dx dx x^ — a^ ~ X — a X + a^ integrating, we find (Art. 89), adx /ddx — = l{x ^ a) — l{x + a) + (7; consequently, 2. Find the integral of, 3a; — 5 dx. aj2 — 6a; + 8 Resolving the denominator into its two binomial factors, (x — 2), and {x — 4), we have, 3a; — 5 A , B "i — a^ . Q = ::: a + z rJ lience. a;2 — 6a; + 8 a; - 3a; — 5 ^a; — 4^ + -i^a; — 2^ 2 A/y. _L » ' 3-2 __ 6a; + 8 a;-^ — 6a; + 8 by comparing the coefficients of a;, we have, - 5 = - 4^ ~ 2^, S = A + B, 7 1 which gives, -^ = 2 ' ^ ~ ~ 2 ' Bubstitutmg these values, we have, f 3a — 5 _ I f dx ^ 1 f dx J x^-Qx 'Vf'^ - ~ 2^ ^^^ "^ 2V ^3-4 + ^ = Jlog(a;-4) _llog(a;-2)+ G. 25 •182 INTEGRAL CALCULUS. Hence, for the integration of rational fractions: 1st. Hesolve the fraction into partial fractio?is, of which the 7mmerators shall he constajits^ and the denominators factors of the denominator of the given fra^ction, 2d. Fi7id the values of the numerators of the partt(H fra€tio7is, and midtipli/ each hy dx. 3d. Integrate each partial fraction separately, and the sum of the integrals thus found will be the integral sought, INTEGKATION BY PARTS. 162. The integration of differentials is often effected by resolving them into two pails, of which one has ^ known integral. We have seen (Art. 27), that, d{tiv) =z udv + vduy whence, by integrating, uv = J udv + / vdu, and, consequently, J udv = uv —Jvdu. Hence, if we have a differential of the form JTdx, which can be decomposed into two factors P and Qdx, of which one of them, Qdx, can be integrated, we shall have, by making / Qdx = v, and JP = u, , fxdx =fpQdx = J udv = uv - Jvdu . (].) in which it is only required to integrate the term J vdu. INTEGRATION BY PARTS. 183 EXAMPLES. 1. Integrate the expression, v?dxy/a?- + 7?, This may be divided into the two factors, £c2j and xdx^o?- + a;^, of which the second is integrable (Art. 41). Put, u = JB^, and c?y = xdxy/d?- + x^\ then, . Substituting these values in Formula (1), and finally. 2. Integrate the expression. (a2 - iB2)* _ s The two factors are, aj, and xdx{a^ — jb^) ^ — - 1 w = a; ; c?y = xdx(a^ — a;^) - ; v = - •/(a2 — a2) / w^y = ^— + / , ; whence, /X'^dX X , . , a , A _x N 1 = ^^ . + sm-1 - (Art. 99). (a2 - a;2)2 -/^^ - a;2 a 1S4: INTEGRAL CALCULUS, INTEGRATION OF BINOMIAL DIFFEEENTIALS. t Form of BinomiaL 163. Every binomial differential may be placed imder the form, in "which m and n are whole numbers, and n positive ; and in which j9 is entire or fractional, positive or nega- tive. 1. For, if m and n are fractional, the binomial takes the fopm, i 1 x^dx{a + bx^)P* If we make x = 2^, that is, if we substitute for x, an- other variable, 2, with an exponent equal to the least common multiple of the denominators of the eiqponents of X, we shall have, x^dx{a + bx^)P = ez^dz{a + bz^)P, in which the exponents of the variable are entire. 2. If 71 is negative, we have, x'^-'^dx{a + bx-")p, and by making a; = -, we obtain, - z-'^^^dz{a + bz'')Py in which n is positive. 3. If X enters into both terms of the binomial, giving the form, x^- dx{ax'' -\- bx'')Py BINOMIAL DIFFERENTIALS. 185 in which the lowest power of x is -vmtten in the first term, we divide the binomial within the parenthesis by ic'", and multiply the factor without by cc''^; this gives, which is of the required fomi when the exponent *w + j^'-i, is a whole number, and may easily be reduced to it, when that exponent is fractional. When a Binomial can be integrated. 164, — 1. If ^ is entire and positive, it is plain that the binomial pan be integrated. For, when the binomial is raised to the indicated power, there will be a finite nmn- ber of terms, each of which, after being multiplied by iC'"~Wa;, may be integrated (Art. 35). 2. If m = w, the binomial can be integrated (Art. 41) 3. K jo is entire, and negative, the binomial "will take the form, (a + hx'^y^ which is a rational fraction. • Formula ^. For diminishing the exponent of the v^ujabla without the pa* renthesis. 165. Let us resume the difiercntw^ bmomial, 186 INTEGRAL CALCULTTS. If we multiply by the two factors, jc" and aj-", the value will not be changed, and we obtain, Now, the factor cc" -if7ic(a + bx'')p is integrable, what- ever be the value of p (Axt. 41). Denoting the first factor, af-" by w, and the second by dv, we have, du = (m — 7i)a;'"-"-Wa;, and v = -^ — . ,, ^ ; and, consequenUy, /a^-.&(a + te»). = But, /*af" - « - 1 Ja;(a + hx'^yp "^ - = Jrffn-^-'^dx{a + ^ic«)?(a + ^"3 = a f^-'^-^dxia + 5a;")p + h fx'^-'^ dx{a + ^"j^; substituting this last value in the preceding equation, and collecting the terms containing, I x"^-"^ dx{a + hx^y, af-»(a + 5a;")P + i — a(m — n) j x'^-^-'^dx{a + ^")^ " {p + l)^i6 ' BINOMIAL DIFFEKENTTALS. 187 whence, (^) J*x'^-'^dx(a + bx'*)p = b(2^n + 771) This formula reduces the differential binomial, fx'^-'^dx{a + bx"")?, to fx'^-"-'^ dx{a + 5x")i'; and by a similar operation, we should find, I ^m-n~i ^^^^ _j_ Ja;n)p^ ^q depend on, / £c'"-2"-ic7ic(a + 5af*)^ ; consequently, eacA operation dimiiiishes the exponent of the variable without the parenthesis by the exponent of the variable within. After the second integration, the factor m — n^ of the second term, becomes m — 2n\ and after the third, m — 3w, &c. If m is a multij^le of n, the factor 7n — w, m — 2/1, m — 3w, &c., will finally become equal to 0, and then the differential into Avhich it is multiplied will disap- pear, and the given differential can be integrated. Hence, a differential binomial can be integrated^ lohen the ex- ponent of the variable without the parenthesis plus 1, ia a multiple of the exponent within. APPLICATIONS. 166. We have frequent occasion to integrate differential binomials of the form, - -y==z = cC" dx{a^ — x^) ^ . 188 INTEGRAL CALCULUS. The differential binomial x'^-'^ dx{a -{- Ix^^y will assume this form, if we substitute, tor m. m + ] L; (( a, . a2; (( b\ . -i; (C n, . 2; (C P^ . -i. Making these substitutions in Formula ^, we have, /af^dx y/a^ — x^ ftn — l m ya^ -x^ -\ ^^ '- 1 . ; so that the given binomial differential depends on, /x^-'^dx '^a? — ^ and in a similar manner this is found to depend upon, x"^-^ dx and so on, each operation diminishing the exponent of a? by 2. If m is an even number, the integral will depend, after m operations, on that of. /-- —^^ ,,, sin-i - (Art. 99). J Va" - x^ « binomial dipfeeentials. 189 Formula 2S. For diminishing the exponent of the parenthesis. 167. By changing the form of the given differential binomial, we have, fx'^-'^dx(a + hx'')P-'^ (a + bx"") = a fx'^-'^ dx{a + 6a;")?-i + b fx'^+''-'^dx{a + 5a;")i'-i. Applying Formula ^ to the second term, and observing that m is changed to m + w, and ^ to j9 — 1, we have, af(a 4- bx'^)P — am / xf^-^ dx{a + Ja;")^-^ ^(/??i + m) Substitutmg this value in the last equation, we have, (IB) Jx"^-^ dx{a + bx'')P z= af»(a + bx'^y -h pna I x"^-"^ dx{a + bx")?-"^ pn + m ' in which the exponent of the parenthesis is diminished by ], for each operation. APPLICATIONS. 1 . Integrate the expression dxla^ -f x^Y . The differential binomial x'^-'^dx{f:i + bx'')p will assume 190 INTEGRAL CALCULUS. this form, if we make m = 1, a = d^^ 6 = 1, n = 2, and jt> = f . Substituting these values in the formula, we have, jdx{a?' + x'Y = -^ Applying the formula a second time, we have, 7? But we have found (Art. 9l), *^ yd' + x^ ^^"^^' . /(/^.(a^ + x^)i = — — ^ — '— + ^dx^ — ^ + — 'l\x ^ a/« + «V + ^^ 2. Integrate the expression, dx^^r^ — x^. The first member of the equation will assume this form, if we make, m = 1, a = r^, 6 = — 1, w = 2, and jt) = i. Substituting these values in the formula, we have, whence, by substitution (Art. 99), ^^J^ binomial diffeeentials. 191 Formula . For diminishicg the exponent of the parenthesis when it is negative. 169. It is evident that Formula ^I^ will only diminish p, the exponent of the parenthesis, when p is positive. We are now to determine a formula for diminishing this exponent when p is negative. We find, fi-om Formula 2S, pna ' T\Titing for 2h — i^ + Ij "^^ have, (2>) fx^-hJx(a + hx^)-P ^ x'"{a. + bx'')-P+^ — (m + ;i — ;?;3) fx'^-hlxia + 5a;«)-i'-»-i 7ia{2) — I) When p z=z 1, ^> — 1 =z ; the second member be- comes infinite, and the given expression becomes a ra- tional fraction. APPUCATTOXS. 1. Integrate the expression, / dx{2 — x^) ^. ^ BINOMIAL DIFFE The first member of Equation ^ if we make m = 1, a = 2, I pz= —^, Substituting these values, since the coefficient of the second term, in the formrJa, becomes zero. Returning, then, to the example under the last formula, we have, J x-^dx{2 -x^) " = ^—- '— + -^—Y' ^' 2. By means of Formula 3J>, we are able to mtegrate the expression, (^-^ = <^^-' + ^^)-''. when p is a whole number. The general formula will assume this foim, if we make m = 1, X = z, a = a"^, b = 1, n = 2. Each application of the formula will reduce the expo- nent — p, by 1, until the integral will finally depend on that of _^^ = itan-^- + G (Art. 99). a^ + z^ a a ' Formula 2^. When the variable enters into both terms of the binomiaL lyo. Let it be required to integrate the expression, xidx ^ , „.-4 = xidx{2ax — x^) ^. 'yjlax — AL CALCULUS. nay be placed under the form, 1 . -^' ^) t)y making, we shall then have, fx~^dx{2a - x)~^ = X "( 2a - «)^ , 2a(q - ^) p f-f , ,„ ,-^ V L_ _| \i J ^ ^dx(2a — x) ^ 1 If we observe that, jc ^ = a; a", and x ^ = x x ^, and pass the fractional powers of x within the parentheses, we shall have, x^dx (^) /-7. \/2ax — x^ x9-'^y2ax — jb2 ^ (2q — l)a P xi- ^dx + 9. 9. "^ ^/2ax — x^ Each application of this formula diminishes the expo- nent of the variable without the parenthesis by 1. If 3' is a positive and entire nimiber, we shall have, after q reductions, f—J^=r= = ver-sin-^ - + <7 (Art. 99). *^ ^/2ax - a;2 « •tr^ jr-v: . ' J^T/' ''^\^'r,, .,y ■ - A '.^■^TfVM v9!fW'r,'W1f^(fr.r-rrjiffjni(pxr'., ■?»»