ji^Tr ^- -t> IN MEMORIAM FLORIAN CAJORI TKEATISE D^FFEEEI^^TIAL AND INTEGRAL CALCULUS. BT PROFESSOR THEODORE STRONG, LL.D., M JIkmbeb of the "Ameeican PniLosopnicAL Society;" "The American Academy of Aeis and Sciences ;" and ConroKATE Membeb or "The National Academy of Sciences, U. S. A. NEW YORK: 0. A. ALVORD, PRINTER, 15 YANDEWATER STREET. 1869. Entered according to Act of Congress, in iLo year 1869, Bt THEODOKE STEONG, In the Clerk's Offic« of the Dlatrict Conrt of the Uultcd States for the District of New Jersey. CONTENTS. DIFFEEENTIAL CALCULUS. SECTION I. PAs» Definitions and First Principles . . - - 1 SECTION II. Transcendental Functions 49 SECTION ni. Yanisliing Fractions - - - - - - • 86 SECTION IV. Maxima and Minima 94 VI CONTENTS. SECTION V. Tangents and Subtangeuts, Normals and Subnormals 125 SECTION VI. Radii of Curvature, Involutes and Evolutes - - 163 SECTION VII. Multiple Points, Cusps or Points of Regression - 191 SECTION vni. Plane and Curve Surfaces 205 SECTION IX. Curvature of Surfaces, and Curves of Double Cur- vature - - - 229 mXEGEAL CALCULUS. SECTION^ L PAGB The Integral Calculus - - - .^ - - 253 SECTION 11. First Principles of the Calculus of Variations - - 316 SECTION III. Integration of Rational Functions of Single Variables, multiplied by the Ditlerential of the Variable - 350 SECTION IV. Reductions of Binomial Differentials to others of more simple forms 377 SECTION V. Integration of Differential Expressions which con^: I two or more Variables 439 VIU CONTENTS, SECTION VI. Integration of Differential Equations of the first order and degree between two Variables - - - 454 SECTION VIL Integration of Differential Equations of the first order and higher degrees, and the Singular Solutions of Differential Equations between two Variables - 484 SECTION vni. Integration of Differential Equations of the second and higher orders between two Variables - - 511 SECTION IX. Integration of Differential Equations containing three Variables 671 SECTION X. Partial Differential Equations - - - - - 583 Appendix C02 DIFFERENTIAL CALCULUS. SECTION L DEFINITIONS AND FIRST PRINCIPLES. (1.) In the Differential Calculus, numbers or quantities are considered as being constant or variable ; tbose whose values do not change during any investigation, whether they are known or not, being called constants ; while those whose values change, or are conceived to be altered, are called vari- ahles. Constants are generally represented by the first letters of the alphabet, and variables by the last letters. Thus in anJG ■\- h^ y = ax^ -\- hx -[- c ; a^h^ c are constants, and a?, y are variables. (2.) Variables that are entirely arbitrary, or arbitrary within certain limits, are called independent variables ; while those variables whose values depend on the values of one or more others that are independent of them, are Q.?i^Qdi functions of the variables, on whose values they depend. When the de pendence of a variable on one or more others is expressed or given, the variable is called an exjMcit functmi of the va- riables on whose values it depends ; but if the manner in which a variable depends on one or more others is neither expressed nor known, and is to be found from the solution of one or more equations or in any other way, the variable is called an implicit function of the variables on whose values its value depends. It may be added, that a variable 2 DIFFERENTIATION OF ALGEBRAIC EXPRESSIONS. which is expressed in variables and constants, is not consid- ered as being a function of the constants. Thus, in y = 3jj + 7, y = aa? + 5, y is an explicit function of a?, and x is an implicit of y ; and neither y nor x is considered as being a function of the figures 3, 7, or of the constants a, h. To signify in a general way, that any variable, as y, is an explicit function of another variable, as a?, we write them in such forms as y = F {x\ y = f{x), y = n 2. Let X = aj *", or Xa?"'^: 1, be proposed, in order to find n d^ the differential of X = a? "* , supposing as before m and n f, to be any positive integers. Because Xa;'" = 1, is essen- n n tially the same as the identical equation x "* a?*" = 1, it is clear that the differential of Xaj"* must equal naught, since the dif- ferential of its equivalent, 1, equals naught DIFFERENTIAL OF A POWER OR ROOT. 5 It is clear (from tlie nature of a differential), that in find- n ing tlie differential of Xx'"\ we may take tlie differential of each factor regarding the other as constant, and add the results for the whole differential ; consequently we shall have 'X.dijo''' -f x'^'dX = 0, ordX. = — ~X.x-\ix= — ^ "" dx, 771/ iiv as required. 3. If X — U « ± aj « ) ''^ we shall clearly, as before, ±£- 1 have^X = ± —-^l a'l ± x^^] x \ dx. for its differ- on q\ J ential. 4. Hence, the differential of any given power, or root of a variable or function, can be found by the following RULE. Multijjly the jpower or root hj its index, siibtract 1 or unity from the index, in the jproduct / then, multiply the result hy the differential of the variable or function, for the reqidred differential, EXAMPLES. 1. To find the differentials of x^ and {x'^y. Here we have the variable x raised to the 5th power, and the function x^ raised to the ?ith power, the indices of the powers being 5 and n ; consequently, by the rule, we shall ^s.MQhx^^dx = bx^dx and n {x^y-^dx"^ = .^.^mn-m ^ ,„^m-i^^ _ ^nnx'^^'-hlx for their differentials : noticing, that the second differential is manifestly correct, since {oif'Y = a;"""*. 2. To find the differentials of \/x = x^ and \/x^ = x^. Here J and f are the indices, and by the rule we shall have 6 EXAMPLES (continued). ds/x = dJ'=\x^''^dx = \x-}dx = 2l7aj^^ ^ %^^^' ^^^ 2 1 2 1 dj^Q? = o i—dx = Q -— <7a?; for the differentials. 3. The differentials of Ty' and 6z% are 42/flf?/and Ss"*^/^; which are obtained bj multiplying the differentials of y^ and 2", by their coefficients 7 and 5, as we clearly ought to do. 4. The differentials of ax"^ ± h and -7 a?" i ^, are max'^-'^dx and -j-x^'-hlx^ which are clearly correct, since the constants connected with the variable parts by ±, must clearly disappear when the differentials are taken, and that the differentials of «a?"* and -ya?" must evidently be a and -j times the differentials of x^ and x\ 6. The differentials of 2 \/{a^ 4- x^) ^ 2 (a^ -f- serf, and ?v/(«2 4- a;^) = I {a' + a?^)^ are 2 {a' + x^y^xdx == ^"^"^"^ and ^ (a- + a?-) ~^a?c?£c = ^— ^ — - ^ 5(^^^ + ar^)^ 6. The differentials of (cr -f- a^)~^ and (a* — aj^)~', are _ 4 (^2 ^ a!')-3izjc?ic and 6 (a^ — aj^)-'*^;^ 7. The differentials of (a^ + Sir^)-^ and (a^ - 3aj^)~^, are - 42 {a' + Sxy^xdx = - r-^'^ and — ^^. 8. The differentials of {a" + x~Y'^ and {a^—xr-y^, are 4a?"Wa? 4ar'r7aj , 4:X^dx and (a^ + x-J ~ {a'x' + If («V - If 9. The differentials of {2?/ + 'Sx-y and {2y- - Sx% are (4/ + 6x') {4.ydij + Qxdx) and 2 (23/^ - 3a^) (4yr/y - 6a?^4 PAETIAL DIFFERENTIALS AND COEFFICIENTS. 7 (5.) If X is a function of any number of variables tbat are independent of eacli other, it is customary to call the differ- ential of X taken with respect to any one of the independent variables^ a partial differential of X, and the corresponding differential coefficient is also called a partial differential co- efficient I and the algehraic sum of all the partial dfferen- tials of X, is called its total differential. If X lias two or more terms that are functions of tlie same variable, it is clear tbat we may find the differentials of such terms as before, and then take the algebraic sum of the dif- ferentials for the differential of the sum of such terms. Thus, if X is a function of x^ y, 2, &c., we shall have -y- dx^ dx -j—dy^ -j-^^) <^c., for the partial differentials of X, whose sum p-ives ^X = —7 — dx -\ — -. — dy ■\ j-dz +, &c. ; for the ^ dx dy ^ dz ^ ' complete or total differential of X ; and -y-, -^-, --y— , &c., are the partial differential coefficients. And if we have X = ^ax' — l)x-\-c^ by taking the differentials of its terms separately we shall have 6axdx and — hdx for the partial dif- ferentials, whose sum gives <:ZX = 6axdx — hdx = {6ax — h)dx for the complete or total differential of the proposed expres- sion ; and, of course, —, — = Qax == J is the corresponding differential coefficient. Eemarks. — 1. If X is a function of a single variable, its differential coefficient is sometimes indicated by writing the capital D before or to the left of X : thus, DX signifies that the differential coefficient of X is to be taken : as in 8 PARTIAL DIFFERENTIALS AND COEFFICIENTS. D (aar* —hx + c) = Saa^ — &, called the first derived function of oaj* — hx + c. And if X is a function of x^ y, &c., the partial differential coefficients -j-, -i— , &c., are sometimes expressed by the forms D^jX, DyX, &a 2. To indicate that the differential of a compound quan- tity is to be taken, we put it under a vinculum or inclose it in a parenthesis, to which we prefix d. or d (called the char- acteristic of differentials), and when the differential has been found, the quantity is said to have been differentiated. Thus d.(aP+y^ -— az) or d (s? -{■ y^ — az) indicates the differential of a^ + y^ — «2, which being taken, gives d {x^ -\- y^ — az) = 2xdx + 2ydy — adz. To make what has been done more evident, take the fol- lowing EXAMPLES. 1. To find the differential and differential coefficients of X = Za?- 5y + 93\ Here fZX = Qxdx — lOydy + 27z'dz ; and ^ = Qx, -T- = — lOy, and— i^ — = 27 z'. are the differential coefficients. dy ^' dz ' 2. Perform what is expressed hj d{ Vx^ — 2^^ + az) and ^ (.^ _aj2 + a; - 8/ - 9y^ + 7). rr.1 xdx—2ydy , , The answers are -n—o — ^.^ + adz, and V(ar^-2?/-) ^x^dx — 2xdx + dx — Vly^dy + 27y'dy ; and the partial differential coefficients are ^ ^^ and3aj-2aj+l,-12y^+27y^ 4/(ar-22^)' i/(a^-2/) RULE FOR THE DIFFERENTIAL OF A PRODUCT. 9 3. Perform wliat is indicated by By {x^ — 3/') and Djc^y {ax^—y^ + z). Ans. — 15y*, 4:ax^, and — 5(/^ ; when Dx^y is used to indi- cate the differential coefficients with reference to x and y. 4. To find the differential of the product of any number of factors, as X, Y, Z, &c.; which may (if required) be func- tions of any variables. ■ Here it is easy to perceive (from the nature of differen- tials) that d{XY) ^ XdY + YdX, d (XYZ) = TXdz -\-XZdY + YZc^X, &c., which are of like forms, are the sought answers. (6.) It clearly follows from the preceding example, that the differential of a product can be found by the following RULE. The differential of the product of any number of variables or functions, eqxials the {algebraic) sura of the differentials, that 7'esidtfrom the differential, of each factor multiplied hy the product of all the remain ing factors, EXAMPLES. 1. The differentials of xy and 3-»'y, are xdy + ydx and 3 {ardy + lyxdx) = ?>di?dy + 6yxdx. 2. The differentials of xx^ and x^t)^, equal 2x-dx + x^dx = Sx^dx, and 4:X^x\lx + Sx^x'dx = 7x^dx ; which are clearly the same as the differentials of a^ and a?^, as they ought to be. 3. The differential of {x^ + y-) {x^ — y% is {a^ 4- 2/') (2a^^^' - 2ydy) + {x^ - f-) {2xdx + 2ydy) = 4 {x^dx — y'^dy). 10 DIFFERENTIAL OF A QUOTIENT: 4. The differentials of y" {a'' + x'') x ^{a''-x-), and 2A^ are x/Ix ,, , „, Xf/X Va- + X' X -■ - + {/(a^ — x') X %rMx and 5a?^ic* <^ic + Wxdx = 8ar\7a;. 5. The differentials of <2a?yV and -ar*y~^2~^, are c ai^xy^zHz ■\- 2xz^ydy + y-z^dx) and ^ (- Zx-y-'-z-^dz - 2jrz-^if^dy + 2y--z~''xdx). X tty 6. The differentials of - —xy-^ and -, = a^'^y"^, are tX 7. 17 V'^^' — ^dy y ^ ^ J 2/2 and cZ~ = 2a??/ -^^.t? — 3^t'-?/ '^dy. Bemark. — If we put X = '-, we shall have Xy = a?, y whose differential gives X^7y + y^ZX = Joj; or, since XXX X = -, we have -- dy -\- yd ~ =^ dx] consequently, we shall J J " have d—= — 1, ~ , which is the same as found y r from xy~^. (.7.) It follows from the preceding example and the re- mark, that the differential of a fraction can be found from the following rule. IfaUiply the denom,imitor hy the differential of the nu- merator^ ajid from the product subtract th-e numerator muU WITH EXAMPLES. 11 tiplied hy the differential of the denominator^ and divide the revfiainder hy tlie square of the denominator. EXAMPLES. 1. The differentials of — and — ^r are X ar 2xHx — x^dx x^dx , T o?dx — 23rdx dx =^ =dx and —, =-^, which, are clearly coiTect ; since the form x^ ^ X , - — ■ = X, and —r = a?-\ X XT - ) and -— -, are yl ?/"* yi y" "-^ 'x\ _ ix^^-^ydx — xdy 71 1 I w I ■ — / «/ 1 I 2 ny'^x'^-'^dx — mx'^y^-^dy ___ 3. Ilie differentials of ^-=^ and4:i^;, are^^^ T . (v^a? — xdy) 4. The differentials of — and of = — ±1, are X X X X {ijdz + zdy) — yzdx ^^.^ x or -3 and -;;^^(«"±aj«)— (a"±a;")(^aj" = a'^dai"' na'^x'^-^dx na"dx ,«." ,, „. ^-iT- = ^— = - -^.TT- = ^^ = ^(«"^'^). 5. The differentials of and , are 7-- ^ a -j- X a —X {a -i- xy , adx and -f r:7. \a — xy 12 DIFFERENTIALS OF THE SECOND, FTC, ORDERS: 6. The differentials of—jr-n ^ = r and X X .= 7T —11 are ^{a'-ar) (a^^a^f dx x^dx c^dx , a^dx and 7. The differential of — -t-v 5 is '^{a- -\- or) -\- x [V'(a^+i»-) + a?P ~ a- + aj2 + aJ4/(a^ + a?)* dd'y noticing, that we shall in like manner set -5 s 7-5 sr for the differential of —,—= ^r . \/{ar + x-) — x (8.) Supposing X to be a function of x alone, taken for the independent variable ; then, since dX, from its definition, equals the sum of all the changes or variations in X, that result from the separate change or variation x'— x = dx of each x in X, it clearly follows that the differential coefficient —=- must be independent of dx; and that a double, triple, (tx &a, value of ^X must result from a double, triple, &c., value of dx, and so on ; and it is clear that the reverse is also true. It is hence evident that we may, according to custom, sup- pose dx in dX. or in dX. in the differential coefficient -j- to (Xx be unlimitedly small, and that when x is the independent variable, dx ought to be regarded as constant or invariable, for otherwise x must be regarded as a function of a variable, and of course it can not be the independent variable. dX. in the differential coefficient -y— as unlimitedly small, on WITH THE CORRESPONDING COEFFICIENTS. 13 It is further evident that for c?X = -^ dx^ "we may, if required, write dX. = -j-^ A, and regard h as being finite ; noticing, that it will generally be very convenient to regard dX. in the differential coefficient account of the minuteness oidx. Calling dX. the first differential of X, and -j~ its Jlrst dX differential coefficient; then, if tZX or -y— contains a?, and CtdG we take the differential of ^X, supposing dx constant, or a? to be the independent variable, we shall get d (<^X), which we shall represent by c^'^X, and it will be what is called the second difiexential of X ; and d -= — \- dx=^ -j^ . which is ax dx' the same as dX ~ dx^ =^ -p, , will be what is called the sec- ond differential coefficient of X. d^X In like manner, since -j-; is clearly independent of dx, if cox it contains x, we shall, as before, get d (d^X) = d^X for the third differential of X, and \ ^ -^ dx = -^ , which is dX clearly the same as d^X -i- dx^= -^--3, will clearly be what is 6? fly called the third differential coefficient of X. And we may in the same way proceed to find ^^X, d^X . . ^"X, which are called the fourth, fifth to the nth differ- ^^X ential; and the corresponding differential coefficients, -^-r, dx^ d^X d'^X — -^ -^-^. Thus, from X = a?" we get dX = nx''-^da!j 14 IMPORTANT RULE IN DEVELOPMENT. d'X = n {n-l) a;"-=<7.r', d^X = n (n-l) {n-2) x'^da?, &c, dX for the first, second, third, &c., differentials; and— p-=?id?"-\ ax ^ = n (71-1) af^-', ^ = ^ (^^-1) (^* - 2) a?"-', &c, will bo the corresponding diifferential coefficients of x\ If we put x'—x=h, or x' = X -{- h, and change a? in X into x' ; then, if the resulting value of X is expressed by X', it is manifest that X' is a function of x' or its equal x + h. If X' is developed into a series arranged according to the ascending powers of A, it is evident (from what has been done) that X and -r- A will be the first and second terms of ^ ax the series, so that we shall have X' = X + -^-A +, &c. Since x may stand for any variable, and X for any func- tion of it, it results from the preceding equation, that we can find the first difierential of the function by the following RULE. Change x in the function into a? + A, and develop the resulting function into a series arranged according to the ascending powers of A ; then the coefficient of h (the simple power of h) in the development, will equal -;- the first dif- ferential coefficient, which multiplied by dx (supposed unlim- itedly small) gives -y- dx ; which is the first differential of the function X, Thus, if we put X = ar', we get X' = {x + hf = 0^ + Sx'h + Sa-A^ + h' ; consequently, Sjt = the first difierential coefficient, and of course da^ = Sardx is the first differential EXAMPLES IN HIGHEK ORDERS OF DIFFERENTIALS. 15 m Similarly, from X = ^' " we get !L^ HI m ^-1 X! = {x -\- h)" = X'' + —x'' h +, &c., as is clear from tlie Binomial Theorem. rn, ——1 Hence, since —a?'' is the coefficient of the simple power of A, in the expansion ; it follows that we shall have dx"^ ■= — x'^ dx for the differential. 71 m Kemark. — The same differentials of x^ and a?", can be ob- tained immediately from the rule at page 5. (9.) We will now show how to find the remaining terms of J'V' the series, X' = X H — ^ A + , &c. dx Thus, by taking the differential of j— A, supposing x alone to vary, we have, according to the principles hereto- fore given, ~r^hh = jtt^^^i ^^^ twice the third term of the series. For any term in -^ hh, that results from the multi- plication of terms containing A and A taken in any order, will clearly result from the same terms when A and A are interchanged, as is manifest from the manner of obtaining -- TT /< A ; consequently, —^ — :r-^ is the third term of the series. ClX^ CCX x.Ai Similarly, from -^ ^i^ we get ~j t-^ for thrice the fourth term of the series. <^^X A^A For it is plain that any term in -^-j- z-^, that results from the multiplication of a term that contains A^ by another that 16 Taylor's theorem. contains A, will equally result in two other ways, since h^ can be formed in two other ways, by combining each h in the first h? with the remaining h ; consequently, -j-^- ■Tn'o ^ *^® fourth term of the series. It is hence easy to perceive that -77^ is the fifth term of the series, and so on. For a more full explanation of the principles used in find- ing the preceding terms, we shall refer to the solution of Example 16, at p. 56 of my Algebra, and for the common way of finding them, see p. 252 (49.), of the same work : observing, that this method is altogether more complicated than the preceding. Hence, collecting the terms, we shall have whose law of continuation is manifest : noticing, that h may be positive or negative, according to the nature of the case. Because X' is the same function of a; + A that X is of a?, it follows, if we represent X by f{x) = any function of a?, that X' will become a similar function of x + h, represented byy(a; + A) ; consequently, the series (a) becomes /(.+A)=/(.) + -^A + -^_ + .^__+,&c..(«0. {a) and {a') are different forms of what is called Taylor'^s Theorem^ which is always true when x and A are undeter- mined quantities, or when the series does not contain any fractional or negative powers of A. When particular values are assigned to x and A, the series afe also true, provided that no term becomes infinite ; but if one or more terms become maclaurin's theorem:. 17 infinite, the series are true no further than to their first in finite terms, exclusively. If we represent the particular values of X, —7-, ~rj^ &c., that correspond to a? = 0, by (X), ("7— ), \~t^\ ^^ ? then, if we change h into a?, and represent the corresponding value of X' by X, {a) will become ^ ,^, /dX\ /cPX\ x^ /d'X\ ^ which is called Maclaurin's Theorem ; in which x may be positive or negative, according to the nature of the case. Because X is supposed to be a finite function of a?, it clearly follows, if (h) gives an infinite value to any term of X, that {])) is not applicable to the expansion of X. To perceive the uses of Taylor's and Maclaurin's Theo- rems, take the following EXAMPLES. 1. To expand {x + h)\ by Taylor's Theorem. Here {x + hf and x"^ must be used for X' and X ; which . ^ , dX. . . d^X ^^ , d'X ^, d'X ^, and thence (a) becomes {x + Kf ^x'-\- 4a?Vi + 6a?Vi= + ^xh' + M 2. To expand {x + A)", according to the ascending powers of A, by Taylor's Theorem. Here X' = {x-Y h)% X = x-,'^k = nx--% ^X h^ _ n{n-l) dx" 1.2 ~ 1.2 ' 18 THESE THEOREMS ILLUSTRATED. ^ . J!_ - n(n-l)(n-2) . dx' 1.2.3" 1.2.3 '^^•' consequently, from (a) we have {x + A)" = 3. To expand X = (a + a?/, according to the ascending powei-s of X, hy Maclaurin's Theorem. Here X = (a + xf gives consequently, jutting a? = in these, we get .,.. 3 /dX\ _ , /c£^X\ . ^X . and thence, from (A) we have {a+ xf =za' + 3a'x + Sax" + x\ as required. 4. To expand ^ ^ (1 + x)-^ according to the ascend^ X -f- X ing powers of x, by Maclaurin's Theorem. Since X = (1 + x)~^, we have ^- = - (1 + .^) -, ^=2a+x)-», -_ = _2 x3(l+»)-S and so on ; consequently, by putting .o? = in these, we have and so on. From the substitution of the preceding values in (^v), we get = l — x-\-o^ — x^-{-, &c., 1+x for the required expansion. EXAMPLES (continued). 19 5. To expand —7- -, according to the ascending powers iC ( J. Xj of a?, by Maclaurin's Theorem. Because x = reduces —zr- — r to 7- = infinity, it would w(l —x) seem that (h) is not applicable to the question. Nevertheless, since (b) gives 1 1-x we shall of course have 1 1 -\- X i- x^ + x^ +, &c., = a?-^ + l + a? + a?^ + aj^+, &c., x{l — x) as required. 6. To expand X = - = aa?^'" into a series, an-anged X according to the ascending powers of x^ bj Maclaurin's Theorem. Since -^ — — ftax-''-^ — ^ = n (n + 1) »-"-", &;c., 71 dx dx- being positive ; and since these are infinite when a? = 0, it is clear that (b) is not applicable to the question. 7. To expand X' bj (a) on Taylor s Theorem. Here, by {a) we have X' z= {x -\-Jif + (a; + A - a)-^ andX = a;^ + (a; - ay ; and putting x—-a^ these equations become X^ z= (« + Kf + A-^- = (a + A)3 + ^, and X ^a' ^- ~. It is hence clear that {a) is not applicable to the question any further than to the expansion of {a + Kf. Remarks.: — It is manifest from what has been done, that X' = {^a ■\-lif 4- A-' = a« + h-^ ■\-Za-h + ZalC- -f-A' 20 NOTICING SOME FAILING CASES is the true expansion of the proposed expression, when a? = '^-' dxdy ' dydx 2ab. Putting a? = and y = in X and these values, we get -. (f)=«.(f)=-.f=».-^ \dxdyl ~ * 26 EXAMPLES. Substituting these values in {h'\ we get X = {ax + hyf = y-^- 4- 2ahxy + y-|- =aV+2a5a?y + %=; whicli is evidently correct Kemarks. — 1. li cux + hy is eliminated from the equations -^ = 2a {ax + %) and ~^=2h{ax + hy\ dX. jdX - we get a—, 6— -=0, ay dx ^ which is called an equation of partial differences ; but, since -J- and -7- are differential coefficients, it is clearly more cor- rect to call it an equation of partial differential coefficients. 2. If X = f{ax + hy) = some function of oa? + hy^ it is easy to perceive that -^ and -j- will be of the forms dX df{ax + hy) J dX df(ax + ^y) T /v/ 7 N It is easy to perceive that the elimination oi f'{ax + hy) from these equations, gives the equation a-^ h-j- =0, the ay cix same as in 1, so that it does not depend on the form of/*, lleciprocally, if in any calculation we meet with an equa- tion of the form a-j h-^ = 0, it may clearly be supposed to have been derived from an equation of the form X =y (oa? -f &y) = an arbitrary function of oa? + Jy. DIFFERENTIATIONS OF XY. 27 3. If a =:: J, the preceding equation gives --—=-—-; con- sequently, if X —fi^x + y) = a fanction of a? + y, it follows that the jMrtial differential coefficients^ when taken sepa- rately with reference to x and y, must equal each other. For more ample details relatively to the preceding remarks, &c., we shall refer to Art. 77, &;c., p. 230, vol. 1, of the "Calcul Differentiel et Integral," of Lacroix. (11.) If X and Y represent functions of any number of independent variables, whether the variables in X and Y are the same or not ; then, we propose to show how to find any differential of the product XY. Thus, by indicating and taking the differentials of XY in succession, we get d (XY) ^ XdY + YdX, d? (XY) = X^^^Y + 2dX.dY + Yc?-X, c^^ (XY) r- Xd'Y + ZdX.d?Y + WXdY + c^XY, ... to d^ (XY) := Xd'^Y + ndXd—'Y + -^%^^ cZ=X"--Y + where it is clear that n denotes an integer, such that the nth differential of the product, indicated by (^"(XY) in the first member of the equation, is developed in the second member, and of course the equation must be considered as being an identical equation. It is clear that {c) can be obtained immediately, from the development of {dY + dXy according to the descending powers of dY and the ascending powers of g?X, by the Binomial Theorem ; being particular, in the development, to apply the exponents of the powers of o^Y and dX to the 28 DIFFERENTIATIONS OF X, Y, Z, ETC. characteristic d, and to write Y for c?"Y, and X for d^X. (See Art 91, p. 256, Vol. 1, of the "Calcul Differentiel," &c., of Lacroix.) Remarks. — 1. It is clear that the differentials of the quo- X tient ;r7u = XY~^ may be found in much the same way as before, by changing Y into Y~\ 2. If c?X, dY^ dZ, &c., stand for the differentials of any number of functions, X, Y, Z, &c. ; then the differential of the product indicated by c?"(XYZ, &c.), will be obtained from the power {dX + dY -f- c?Z -f-, &c.)", in a way similar to that of obtaining the differential indicated by c?"(YX) from {dY 4- cZX)", as explained above. For further information on what has been done, see Lacroix and "Theorie Analytique des Probabilites '^ of Laplace. To illustrate what has been done, take the following examples. 1. To find the differential of XY, indicated by dXXY). Here, by putting 3 for ?i in (c), we immediately get c^^(XY) = Xd'Y + SdXd'Y 4- ScPXdY + c^^XY : which can also be found from (c^Y + dxy = {dYf + s{dYydX + sdY{dXf + (crxy, as has been stated ; noticing that for (dYf = 1 x {dYf, we ought to write [since {dXf = 1] {clX)\dYf, and for {dXf, we must also write {dXf{dYf. For, by changing {dX)\dYf into Xd'Y, and S{dYfdX into Sd-YdX, and so on, we shall, as before, get d'{XY) = Xc^^Y + SdXcPY + Sd'XdY + d'XY. 2. To develop tf (.?ry), by means of the preceding for- mula. DIFFERENTIATIONS OF PARTICULAR EXAMPLES. 29 Bj tlie substitution of ar* for X, and if for Y, it imme- diately changes into d^ix'f) = \^;^^dxchf + Z^xyidxfdy + ^y\dx)\ since x^d{dyY = 0, on account of the supposed constancy of dy, y being regarded as an independent variable. 3. To find the second differential of -^-j or to expand c?(XY-^), when X = a?^ and Y = y. Here, since (^X^Y"^) = Xd\Y-') + 2dXd{Y-^) + d'XY-\ by putting x^ for X and y for Y, and performing the indi- cated differentiation, we get drix'y-^) =. Ix^irW - ^xy-Hxdy + ly-^dx"" Ijrdy'^ 4:xd.xdy . 2dx^ ~~ f f y ' (12.) If ^ = /'(s) and s = f'{x\ we now propose to show how to find the differential coefficient of y regarded as a function of x. Here we clearlv have dy = ' , dz, and dz = -~- dx , ^ d3 ' dx ' and consequently, by substituting the value of dz from the second m the first, we have dy r= ' -. x -^-r^ X dx\ , . , . ^y dfiz) dfix) . , which gives ~ = ' x -^^— ; as required. rt^/ CC3 CLX It is easy to perceive that if y =f{3), z = (fi (v), v = i) (a?), we shall in like manner get "^^"^ dz "" dv "^ dx '^'^' Oj. ^ ^ ¥(£) ^ (Wl ^ ^^H^) /^n , dx dz do dx and so on, to any extent. 80 EXAMPLES (continued). EXAMPLES. 1. Given y = 3^^ z = 4ji7', to find the differential of y or its differential coefficient, regarding it as a function of x, Here, from dy = Qzdz and dz = lIxHx^ we get by sub- stitution, dy = 72zardx, or -^ = 72^0^. 2. Given y = az^, z — W^ and v — cx\ to find dy, or its differential coefficient regarded as a function of x. Here, we have dy = 2azdz, dz = Shv'dv, and dv = 4:cx\lx ; consequently, by substitution, as in (d), we sball have dy = 24^ahczv'Mx, or -f- = 2'iahczv^a^. ^ ' dx 3. To simplify the differential of y = {aa? -{-har -\- cf or its differential coefficient, by putting z = aa^ + bx^ -j- Cj which reduces the proposed equation to y = z\ Here, from y = z'^ we have dy = 4:z\lZj and from z = aa^ -\- ha? + G we have dz = Saardx + 2hxdx ; conse- quently, from the substitution of dz, we have dy = 4:z^ X {Sax'^dx + 2hxdx), or ^^"^ ^^^' + ^^ + of X {Sax' + 2hx) ; which is the same result that the immediate differentiation. of the proposed equation will give. Remarks. — 1. Thus we perceive how we may often sim- plify the differentials or differential coefficients of compli- cated expressions. 2. If we have y =f{ii.z), sudh that we have u = (p(x) and 5 = (a?) ; then, we shall clearly have - df(u.z) , df(u,z) , dy = -^ ^ du + -^S — ^ dz, ^ du ^ dz ' d(l){x) dxp{x) y and du — , dx, dz = , ^ ax. itx dx EXAMPLES (continued). 31 wHcli are clearly the same as dv—-^ da + -f- dz, and da — -,- dx. dz = -j- dx, ^ da dz ' dx ' '^ dx Hence, eliminating du and dz from cZy, it will become dy du ^ dy dz , dy _ dy du dy dz dx ~ da ' dx dz ' dx' In mucli tlie same way, if we have y = f(f, V, z, &c.), t = Y{x\ V = (p{xl z = VH, &c. ; then, as before, we shall clearly have dy_ ^di dt_ dy ^ . ^ ^ + &c. . . . (^). dx dt ' dx dv ' dx dz' dx ' ' ' ' ' \ J' It is easy to perceive that we may use ( By taking the partial differential coefficients of (4) rela- tively to a?, we shall have d'^z -,( dz\ , dy dz , , (dz\ , d^ = '^[yd-a)-^^ = tx'd^^^^\d-a)--'^^'^ or, since dy dy dz , , /dz\ , drz drz -f —~f . -— and ydl-y-) -T-dx = y~J—^=y-^-^ dx dz dx ^ \dal ^ dadx ^ dxda LAPLACE AND LAGRANGE. 39 (page 22), we have cPz dy dz dz d/z r= — - . . \- y ddi? dz da dx • dxda (dy dz dy\ ,/ dz\ , '^'='^(2''£)-^'^'^ = dz dz and since (4) gives -j- ^= y -j--, this is easily reduced to dx CLCL '^ dx^ ^V daJ ' ^^ da Differentiating the members of this equation relatively to iB, we have ^. -(43 -¥'£} ,M4J dadx dxda dx -7- day on account of the independence of a and a?, and the differen- tiations relatively to them. It is easy to perceive that may, as before, be re- duced toe? 1 2/^-^1 -f- o?a, which gives and proceeding with this, as before, we have d^_ dx'~ which, as before, gives (^2 Pi): dx da"", S^^K^'S^^"''^''^'''^'^ 40 THEOREMS OP If the values of -i— and y, that result from putting a? = in them, are represented by -^ and y\ we shall have dz' ,dz' d^-z' ^y'db) , Hence, from the substitution of these values in (3), we get "="+ 1^' rf^) *+ -^^- 1:2 + —d^- 1:2:3 +• *^ ■ (A); which clearly holds good, when any like functions of z and z' are put for z and z' in it ; noticing, that (A), thus generalized, is called the Theorem of Laplace / and if we put 1 for a?, in (A), it will become what is called the Theorem of La Grange. To perceive some of the uses of (/i), take the following EXAMPLES. 1. Given 'bz'^ — cz -\- d^=^ 0, to find 2 in a series of the known quantities. The equation is readily changed to the form consequently, for in (1) we put 1, or unity, and y = s" ; also, a = - and x = -. By putting a? = we get z' =^ a and thence -^ = 1 ; also, y = z'"" gives y' = s'" =a", and thence y -5— = a". LAPLACE AND LAGRANGE. 41 Because y = a" and ^ = 1' ^ (2/'' ^) -^ ^* becomes \ ■ = 2;2a2"-' also, ^^ (2/^^ ^) -r- ^ct^ = 6^' (a^'O ^ ^a^ = 8/1 (3/i - 1) a^"-^ and so on. Hence, collecting the results, we shall get z = a-\-a^x + 2na^'^-^ ~ + 3?i(3?i - l)a^"-^ j^ +, &c. If 7^ = 3, 5 = 1, c rrr 3, and d— — 1, tlie proposed equation becomes ^'^ — 3^ — 1 = ; whicli gives a = — ^ , and a? = -^ . Hence, from tbe preceding series, we shall have _ _ 1 _ jL 1 £_ _ ^~ 3 81 729 19683 ' - ^^^^ ,&c. = - 0.3172, 19683 which is one of the roots of the equation s'^ — 3^ — 1 = ; correctly found in all its figures. 2. Given hz— cz"" -\- d— 0, to develop s in a series. Since the equation is equivalent to d o o SO that n = 3, a = 77 and a? = ^ in this equation. If - is put for - a and x in the series for ^, it becomes ^-e'-ey-."'^ = 0.3333 + 0.23112 + 0.053416 -f, ifec. =3 0.61787 +, &c. ; and hence v = I/3 = 1^0.61787 = 0.85173, whose first two decimal places are correct 3. Given As"+ 'Bz'"+ Cz^^-h . . . . + N = 0, to find 2. Since the equation is equivalent to 2*^ = - ^ - -^- (B3"' + C2^'' +, &c.) = a-\-xy; N 1 we have a = — v- and i»y = r- (B3"' + C3"''4-, &c.); and -A. -A. we may evidently put x = r- and y = Bz^'-^- 03"'' + , &c. From what precedes, we get z = {a + xi/Y, which corre- sponds to <^ (a + xi/) in (1), p. 37 ; which, by putting x = 0, gives z' = a^^j which gives LAPLACE \ AND LAGEANGE. I — M chJ 1 1_ - a« n da n 43 and y' = B^''^' + Qz"'"+, kc. = Ba'^ + Ca" +, &c. Hence, from (h) we get l-n 1 £' r»^' ^^ n s = a" + (Ba" + Ca''*" +, «&c.) -— x + nf n" \ — n d [(Ba"^ + Ca» +, &c.y a"^~ ] x" da 1.2n^ ' v' n" \ — n cZ -^[(Ba"^+Ca"+,&c.ya "] ^ , ^ ~' da? 1.2.3 7. "^' ^''• To illustrate ttis formula, we shall take the equation ^3 _ 3^ _ 1 ^ 0, under the form z" — z-^ — 3 = 0. Hence, A = 1, B = -1, C =: 0, D = 0, &c., N = -3, a^-^^Z,x=.-\ n=2, n'^ -1, n" = 0, «&c. From the formula, we get 2?iV. "" consequently, by putting 3 for a and 2 for n, and giving the square roots the ambiguous sign ± , we get 2 = ± V3 + g± ^ +,&c. =: ± 1.7320 + 0.1666 T 0.0138 +, &c. Hence, we have 1.88 + and —1.54— for approximate values of two of the roots of the proposed equation, cor- rectly found to two places of figures in each. Eemarks. — 1. It is sometimes necessary to distingiiisli 44 DIFFERENT METHODS. between total and partial differential coefficients. Thus, if - du du dp du dq du dr dx ^ dp dx dq dx dr dx ' we call the first member of the equation the total differential coeffeieiit^ and the terms that compose its right member are its parts, or what are called the partial differential coeffi- cients. 2. li p = a?, it is clear that the equation will be reduced du _ du du dq du dr dx dx dq dx dr dx'' where it will be perceived that the total coefficient -7- in the first member of the equation, is apparently the same as the partial quotient in the second member; consequently, for distinction's sake, we inclose the partial quotient in a parenthesis, thus (-j-)- Hence, the preceding equation will be written in the form, du _ ldu\ du dq du dr ^ dx ~~ \dxl dq dx dr dx ' and we may clearly proceed in like manner in all analogous cases. (17.) It may not be improper, in concluding this section, to notice some of the different methods that have been used by different authors in treating the Differential Calculus. 1. Leibnitz and Newton, the illustrious founders of the Calcalus under different forms, respectively used the infin- itesimal method^ and that of the liiniting ratio. Thus, to find the differential of x^ ; we change x into x-\-h and thence get {x -\- hf — ar^ = Sx^h + Sxh^ + A^, for what is generally called the difference of x^ ; noticing, that it is some- DIFFERENT METHODS. 45 times called tlie increment or decrement of x"^ accordingly as it is positive or negative. If K is finite, the difference being evidently finite, is called 2^ finite difference / and is often denoted by writing the Greek letter J (delta), called the cJiaracteristio of finite differences^ before or to the left of x^ ; and since h^=x + h — x^ we write Ax for h ; consequently, for {x + hj— x^= dx'/i + Sxh? + h% we may write Jx'^ = Sx"Jx + 3x{Ax)- + {Jxf : noticing, that x^ or (more generally) x"^ -^ c, c being constant, is often called the integral of Jx^ or of its equivalent, Sx^Ax -\- SxJj^ + Ax\ If h is unlimitedly small, or an infinitesimal, it is clear that Bx^h + Sxh^ + h^ will also be unlimitedly small, or an infinitesimal ; and if infinitesimal differences, sometimes called dvfferentials^ are distinguished from finite differences by writing d for J, tJien^ according to the method of Leibnitz, the equation {x + hy — ar' = 3a?-A + Bxh^ + h^ be- comes dd(^ = Sardx + Sxdx^ + dx^ ; for which, on account of the comparative minuteness of Zxdx^ and da^^ toe may evi- dently write dd^ = Sx^dx, which is of the same form, that our rule at p. 5 will give for the differential of a^ : noticing that a?^ + c is called the general integral of dx\ or of its equivalent, Sx^dx. To signify that the integral of any finite difference is to be taken, the Greek letter ^ (sigma) is generally written be- fore or to the left of the difference, inclosed in a parenthesis, if necessary. Thus, ^Jx" = I \Zx\Ax) + Zx^dxJ + {AxJ'], which clearly equals SIx^Jos -f S2:x{Axy + ^Axf^ is used to denote that the integral of Ax^, or of its equivalent, 3x%Ax) + dx{Jxf + {Axf , is to be taken ; and since (x + h)^ — x"^ — {iv 4- hf -\- G — x^ — c= A{x^ + c\ c — const: the most general form of the indicated integral is a?^ + a 46 DIFFERENT METHODS. In much the same way we indicate the integral of any proposed differential, by writing /, called the sign of integra- tion, or the characteristic of integrals^ to the left or before the diflereflllial, as before. Thus, we have in which c = coast, =:f'dx-d:c = Zfj^ddn = a?' + c: noticing, that the constant c is used for generality, or to make the integral applicable to any case tliat may be required. Again, resuming {x -f hf — x^ = 3xVi + Sxh^^ + A^, and dividing its members by A, it will become /I which clearly shows if h is diminished indefinitely, the right member has Sx- for its limit Hence, according to the common method of taking the limit, by putting h = 0, the equation is reduced to the form .0 dx^ - z=Sx'\ or since for - we ought evidently to write ~ we d^ have -7- = Sx'dx ; see my Algebra, pages 256 and 257. Since {Sx"" h + ^x/i^ +A-^) -^h=Sa^ -\-Zxli' +A^ tliis quotient is often (with great impropriety) called the ratio of the incre- ment or decrement of y? to the corresponding increment or decrement of the independent variable x ; and 3.2?^, the limit of the quotient, is often improperly called the limit of the ratio when h is infinitesimal. The preceding process in substantially the same as Newton's method of limits. Because [a (x + A)" + c — {ax'' + c)] = a\nx''-'h+ ^^^^— ^ X "-» h' + , &c.] 1. Ji DIFFEKENT METHODS. 47 is Tinder the form of an exact difference, if h is finite, the equation (agreeably to what has been done) can be expressed by the form A{(ix^ + c) = a \nx''-^Ax + ^^^V^— ^"~' (^•^)' + » ^^.J ; 1 . Z hnt if h is infinitesimal, the equation is equivalent to diax"^ 4- c) = nax'^~'^dx. Similarly, because {x -f h) (]/ + k) —xy^= xh + yh + Kh is under the form of an exact diJfference, if h and k are finite, the equation may be expressed by the form A {xy -\- c) — xAy-^yAx + AxAy; but if h and k are infinitesimals, the equation becomes d (xy + 6') = xdy + ydx ; by rejecting dxdy on account of its comparative minuteness. It is manifest from these examples, that in order to find the integral of any finite difierence or differential, it must be exact, or be reducible to a difference or differential which is either exact, or differs insensibly from an exact difference or differential. 2. Eesuming the equation {x + hf -0^ = ^xVi + Sx/r + h^ , and putting dx for k in the first term SjcP/i of the difference of ./, it will become Sx^dx. If the operation to be per- formed on x^, in order to obtain Sx'^dx from it, is denoted by d. x^ or dx^, we shall have d.x^ = dx^ — 3x'dx ; which indi- cates and expresses the differential of ar', obtained by defi- nition, accordlnfj to the method proposed hy the celebrated Lagrange. Supposing X to be any function of a?, and that X becomes X' when x is changed into x ±,h\ then, supposing x and h 48 DIFFERENT METHODS. to be undetermined, Lagrange proved that X' may be ex- pressed by tbe form X ± X, A + Xo 7-^ ± X3 — — + , &c.; in which, he called Xi, X2, X3, &c., the first, second, third, &a, derived functions of X ; and it is easy to perceive that the series is the same as Taylor's Theorem. 3. The difficulties and unsatisfactoriness that have attended the treatment of the first principles of the Differential Calcu- lus, appear to us to have arisen from the circumstance, that it has been thought necessary to convert X' into a series of the form X + Ah + A^A'^ -f- AJi^ -f , &c., and then to reduce the difference X' —X = A^ + A/r + A.A^ + , &c., to its first term AA, in order to get dX. = Kdx , or the differential of X. For this process has evidently introduced the infinitesimals of Leibnitz, and the limiting ratios of Newton and others, into the Calculus, as furnishing reasons why the terms A/i-, AgA^, &c., must be rejected, in comparison to A A. Whereas, the true reason for the omission of these terms, is that so long as x and h are indetermi nates, the term A A rep- resents the sum of all the changes of X that result from the separate change x' — oj = A of each x contained in X . And it is evident from the reasoning in (9) at p. 15, that we may consider the terms that follow the second term -7-^ h in Taylor's Theorem, as deducible from it when x and h are regarded as being inde terminates, in a way very analogous to that of finding the terms that follow the second term from it, in the investigation of the Binomial Theorem: see Ex. 16, p. 56, of my Algebra. SECTION IL TRANSCENDENTAL FUNCTIONS. (1.) "When a function is sncli tliat it can not be expressed by means of its variable and constants in a finite number of algebraic terms, it is called a transcendental function. Thus, log a?, a^, sin a?, cos a?, &c., are transcendental functions : the first being a hgaritJmiic function, tbe second an exponential function, and the third and fourth are circular functions. (2.) Any number or quantity may he expressed in a transcendental form. For if a represents any number or quantity, it is clear that for a we may write / a^ a"" a^ , \ I a"- a^ a^ o \^ 1 / a"- a^ a^ c \' 1 + r-T+T-T+'M 1:2:3+'^'- . , . , a^ a' a' m which ^~T"*""3 X+'^^-' is called the hyperbolic or Napierian logarithm of 1 + a. c^ a^ a^ Hence, if we put a 9- + "o" — j- +, &c. = A, we shall A" A^ clearly have l4-a = l + A + — - + — --— +, &c. ; and in 52 l^ ^4 like manner, if 5 — — + — — +, &c., is represented by B, we shall have l + 5=::l + B + ^+ -^-^l +' ^^ 3 50 TRANSCENDENTAL FUNCTIONS. (3.) The product of the correspondinff members of these equations will be of a similar form. For we shall clearly have (1 + a){l + h) = l + a + h + ah = l + (A + B)+ (A + B)^ j^ + (A + Byji^+,&c. If a-i-h + ah — ^ ^ + 3 ^— , &c., is represented by C, it is clear from what has been done, that the preceding equation is equivalent to ^ + ^ + r2 + ui+'^ = H-A + B + (A + B)=j?2+(A + B)»jl3+,&a; which clearly gives C = A -f- B. Because A and B are the hyperbolic logarithms of 1 + a and 1 + h, and that C is the hyperbolic logarithm of their product, it results from the preceding equation, that the hyperbolic logarithm of a product equals the sum of the logarithms of its factors. • If the members of C = A + B are multiplied by the arbitrary multiplier m^ called the rnodulus ; it is clear that its properties will not be changed, and we shall get mC = mA + 77?B; such, that mA, w-B, and wiC may be called logarithms of 1 + a, 1 + ^, and of their product. Hence, in any system of logarithms, the logarithm of a product equals the sum of the logarithms of its factors / reciprocally, tfte logarithm of a quotient equals the logarithmt of the dividiind, minus that of the divisor. Hence, too, tJie logarithm of a power equals the logarithm of its root multiplied hy the index of the power ; and re- LOGARITHMIC FORMULJB. 51 ciprocallj, the logarithm of a jpower^ divided hy its index^ equcds the logariihm of its root. If the logarithm of a number or quantity, whose modulus is m, is indicated by writing log before or to the left of it (inclosed in a parenthesis when necessary), we shall clearly have log (1 + a) = m /a — ^ + -|- — ^ + ? &c. j (a) ; which we shall call the Logarithmic Theorem. It is evident from what has been done, that we shall have (l + a)' = l + A,«+(^J^+^^J+,&c (J); which is called the Eicponential Theorem^ in which A and Kx are the hyperbolic logarithms of 1 + a and (1 + a)^. If A = 1, Q>) becomes .*.... (^0; which gives . l + «=l+l + ^ + j^ +,&c. =2.7182318284 +,&c. which is generally expressed by ^, and is called the hase of hyperholic logarithms^ since its hyperbolic logarithm is sup- posed to be unity or 1 ; consequently, putting 6 for 1 + « in {l)'\ it becomes ,« = ! + , + i5+j|_+,&o. r): which shows, if we put e^ = N, that we shall have x = the hyperbolic logarithm of N, since that of ^ = 1. If we write log before a number or quantity (inclosed in a parenthesis if necessary) to denote its hyperbolic logarithm, it is clear that log (1 + d)^ = Ax ; and as 62 LOGARITHMIC FORMULA. log (1 4- ay = 7n log (1 + af^ we get log (1 + ^y = rnAx. If we assume m A = 1^ or m = -r-, the preceding equation becomes log (1 + ay = x, and of course log (1 + a) = 1 ; consequently, 1 -f a represents the base of the logarithms denoted by log. Hence, assuming (1 + ay = N, we have log (1 + ay = log N = a? ; since 1 + a is supposed to be taken for the base of the logarithms represented by log.- Because log N — m log N = — ^ — , it results that vje shall get log N, hy dividing the hyperholic logarithm of N hy the hyperbolic logarithm of the hase^ or hy inAiltijplying it hy the modulus (-r)'-, reciprocally, log N, multiplied hy the hyperholic logarithm of the hase^ or divided hy the modulus^ gives the hyperholic logarithm of N". Thus, if the base 1 + a = 10 = the base of common logarithms, the tables of hyperbolic logarithms give log 10 = 2.3025850929, and thence the modulus (m) = \ = ^— =0.4342944819. ^ ^ A log 10 Again, from the tables we have log 2 = 0.6931471, and thence we get log 2 = the common logarithm of 2, equals ^'?^^^itJ^ = 0.6931471 X 0.4342944 = 0.3010299, log 10 which agrees with the common logarithm of 2, as given by the logarithmic tables. Eeciprocally, log. 2 X log 10 = log 2 -- 0.4342944 = 0.6931471, equals the hyperbolic logarithm of 2. It follows from what has been done, that the calculation LOGARITHMIC FORMULA. 63 of a table of logarithms to any base may be considered as being reduced to the calculation of hyperbolic logarithms. For examples in illustration of the calculation and use of logarithms in the solution of problems, the reader is referred to p. 527, &c., of my Algebra. . Eesuming e'^^l+ajH-— + —— +, &c., from {})"\ p. 51, and changing x successively into xV—1 and — xV—i^ we get the equations aj2 a^yHT _ _ ^ _^ x' ~ ^ 1.2 + 1.2.3.4 1.2.3.4.5.6 +' ^^•' "•- 12:3-^ 12:3x5 -'H^-^> ^ ~ 1.2 ^ 1.2.3.4 1.2.3.4.5.6 ^' ' (^ - r2j + 1:2:8.4:5 -'H"^-'' By taking the half sum and half difference of these equa- tions, we get (^v^+ ,-x»'^) ^2 = 1- ^ + j^^ -, &c, and (.^--- .---) ^ 2 1/^ = 1 - j^ 4- ^^ -, &o. ; which (in trigonometry) are called the cosine and .s{??e of a?. Denoting the sine and cosine by writing sin and cos for them, the preceding equations may be written in the forms sm X — ^=^- , and cos x = . . (c). 54 DIFFERENTIATING LOGARITHMS. By adding the squares of (c'), we get sin- x + cos' a? = 1 ; wliicli is also evident from sin a? = a? — +, &c., and cos a? =? 1 — r-^ +, &c. We are now prepared to show how to find the differentials of logarithmic, exponential, and circular functions. (4.) To show how to find the differentials of logarithmic and exponential functions. We will show how to find the differential of a variable or function represented by log x. From (a), given at p. 51, if we put a? for 1 + a^ we must clearly put a? — 1 for a, and we shall have , ^ (aj-l)2 (x--rf (x-Vf „ , in which m is the modulus; consequently, by taking the differential of this, m being constant, we shall have d([ogx) = m\l-{x-l) 4- {x-Vf -{x-lj +,kQ.'\ x dx mdx 7ndx ~ l'V{x - 1) ~ ~^ ' dx and when 7n = 1, we have d (log x) = — . X Hence the differential of the logarithm of a variable or function can be found by the following BULK 1. Divide the differential of the variable or function by the variable or function, and the quotient, multiplied by the modulus, gives the differential. 2. If the modulus is unity, or the logarithm hj^perbolic, then divide the differential by the variable or function, for the differential EXAMPLES. 65 Eemabks. — 1. Wlien it is possible, hyperbolic logaritbins ought always to be used in finding differentials, because their differentials are of the most simple forms. 2. It clearly results from the rule that the differential of a variable or function equals the differential of its hyperbolic logarithm multiplied by the variable or function. EXAMPLES. 1. The differentials of log (a + x) and log ax= log a -f log x. mdx , 7n.dx are and . a + X X cc ' 2. The differentials of log {x -f y) and log - = log x — log y, dx + dy , dx dy ydx — xdy are and - = . X -\- y X y xy 3. The differentials of log {a? + x") and log (rt^ — a?^) = log (a + x) + log {a — x\ 27nxdx T dx dx 2xdx are -7- — 5 and — ; = ^ ^ . a' -\- x"" a -\- X a — X w — xr 4. The differentials of log (ar — a^) = log (a? + a) -f- log (a? — a) 2 and log - — log 2 — log a?, a? d\c dx 2xdx are h X + a X — a a^ — (T^ which is the same as to divide d (a>^ — a-) by (a^ — a-), and dx 0. The differentials of log |/ (a^ + a?')"*' == log {a^ + ar*) ^ and log[^+V(^±a^)]are^;^, V{^±a^' 66 DIFFERENTIATING EXPONENTIALS. 6. The differential of log ^jfl "^ ^ -"^, is xdx /I 1 \ _ ^^^ ^{a^ + a?-) \ |/(tt* + ar^) — a ~ |/(a» + ar) + a/ ~ a?|/(a' + aj^) 7. The differential of ax"" is aa?"* x = maa?"* - Wa?. a? 8. The differential of xy is xy X d log ajj^ = a^ycZ (log a; + log y) = a?y ( h —1 = ydx -f xdy. 9. The differential of ^, is?(^ - ^) = IH^ZI^. y' y\x yJ v" 10. The differential of a^ is a^ X tZ(loga^) = a^d(\oga x x) = a^ loga x dx ; which can be also found from assuming y = a*, or diJ log y = x log a, or — = log adx, or dy=^y log adx=a^ log ac?a?, as before. It is hence evident, that when the ' exponent of an expo- nential is alone variable, we can find the differential of the exponential by the following RULE. Multiply the hyperbolic logarithm of the base or root of the exponential by the exponential, and the product by the differential of the variable exponent Remark. — If the base of the exponential is also variable, then we must add the differential, regarding the base as alone variable to the preceding differential ; and the result will be the complete differential, when the base and exponent of the exponential are vaiiable. DIFFERENTIATING EXPONENTIALS. 57 EXAMPLES. 1. The differentials of 2^ and 3*^, are 2^ x log 2 x dx^ and 3^ log S X dy: noticing that 2 and 3 are the constant roots of the exponentials, whose variable exponents are x and y. 2. The differentials of e"^ and e~% are e'^dx £ind — e~^dx] since log e = l. 3. The differentials of Ja^ and c'''^, are ha^ log a x dxj and c'^^logc X ao?a7. 4. The differentials of e^°s-^ and a^°sx^ are ^iogx_^ and a'^^^logax — . X ° X 5. The differentials of ay"^ and 2/-^, are xay'^-^dy + ay"^ log y x dx and — xy~^~^dy — y~^ log 2/<^a?, as is clear from the rule and remark. 6. The differentials of a^' and e"', are a*' log ae^c^a; and e°' log a X a^dx ; noticing that e'^ and a^ are variable expo- nents of a and 6, and that e stands for the hyperbolic base. 7. The differentials of z""' and (log x) '''^ ^, are s"" log 2 X {yx'-^ -^ dx -\- x^ log ajc/^/) + x^z" -^ 6/5;, and (log a;)^"«^ x log (log x) \-\ogx x (log a?)^°e^-^ x — = [logxy^^ ^ X (1 + log^^') — : X noticing, that the notation log" x is used for log (log a?), and we may also represent log [log (log a?)] by writing log^ x ; and so on, to any extent. 8. The differentials of e v(« -x') and e ''^^'* , are 3* 68 CIRCULAR FUNCTIONS. and e'°«*~-^ . X log a; 9. The differentials of e^"^-^ and ^-^ ^^ , are ^xi^^T^ |/Zri,y;c and e-^ "^^^ y. — \^ — V dx . 10. The differentials of a-^ ^~^ and a"^ ^'-^ , are ^i v^ 1 iQg ^ X ^/^ 1^—1, and rt-'' ^"-^ log « X — dx V — 1. (5.) TFi? i^e7^ noz^ 5/io>w; Aoi AB , if X represents the angle B, it is clear that we must have X + a.- -^3- +,&c. ^(X + .)(l--^^-y3^-%,&c.)>.--^4-,&a X = X -\- x' = A RIGHT ANGLE. 67 for the proper representation of the inequality; consequently, AC being expressed in terms of X in a way similar to the representations of the other sides in terms of their opposite angles, it clearly follows that AC must be the sine of X = sin ABC. From AC = xi-x'- ^^jj^ +, &c., - X- j^ + , &c, we must have X or the angle ABC equal to a? + x\ the sum of the angles A and C . Hence (see figure), if from the right angle Bf we draw the right line BD meeting AC in D, so as to make the angle CBD = the angle C, we shall have the anoxic ABD = the ansjle A . Hence, the lines AD, DB, and DC, are equal, and the points A, B, C, lie in the circumference of a circle whose center is D and radius DB . If the angles A and C equal each other, it is clear that AB = BC, and of course AC' = AB^ + BC^ or 4AD^'=2AB= or AB- =. 2AD- = AD^- + BD^; con- sequently, in the triangle ADB the angle D equals the sum of the remaining angles of the triangle. But, since the triangles ADB and CDB are clearly identical, it results that their angles at D must equal each other, and of course from the well-known definition of a right angle, each of them is a right angle. Hence, the angles at A and B in the trian- gle ADB are together equivalent to a right angle ; and in the triangle CDB, the sum of the angles at C and B is equiva- lent to a right angle. Hence, the sum of the angles of the triangle ABC is equivalent to two right angles, and because the angle B equals the sum of the angles A and C, it is clear that B is a right angle, and that the sum of the angles A and C is equal to a right angle ; and because the angles A and C make the same sum, whether they are equal or unequal, it clearly follows that their sum is always a right angle. 68 THE ANGLE IN A SEMICIRCLE IS RIGHT. Also, because the angle B is always in a semicircle whose center is D and diameter AC, it follows that the angle inscribed in a semicircle is always a right angle. Eeciprocally, if one angle of a triangle is right, the sum of the other two angles is right, and the square of the numer- ical value of the side opposite to the right angle equals the sum of the squares of the numerical values of the other two sides. For ABC (see fig.) being the triangle, a circle de- scribed on AC as a diameter, must evidently pass through the right angle, and the triangle coincides with one of the triangles that have been considered ; and thence the truth of the proposition is manifest. It may be added that the sum of the three angles of any rectilineal triangle is easily shown to be equal to two right angles. (8.) We now propose to ^how how to find the numerical values of angles. Eesuming the right triangle ABC from p. 65, we have, according to what is there supposed, AC to represent any arbitrary unit of length, while the angles A and C are represented by a? and x\ and CB = sin a?, AB == sin x' = cos X. If from A as a center, with AC as ^a radius, the arc CGr is described meeting AB produced in G, it will represent the value of sin x. By taldng the differentials of X — THE LENGTH OF THE ARC GC. 69 CB = sin X and AB = cos x^ we sliall (as at p. 61) have d. CB = cos xdx and d. AB = — sin xdx^ which give (as at p. 58) \\{d. CB)"^ + (t/. AB)-] = cfe, supposing x and sin x to increase while cos x decreases. If from B toward A, BH is set off to represent d , AB == — sin xdx and HE drawn parallel to CB, meeting the tangent to the arc CGr at C in E ; and if through C, CD is drawn parallel to AB, meeting HE in D ; then, EC represents dx^ and ED =: o^.CB = cos xdx. For the right triangles ACB and ECD give the proportions AC or 1 : EC :: cos x : ED, and 1 : EG :: sin a; : CD = BH, which give DE = EC x cos a?, and BH = EC x sin x. Since (neglecting the signs) BH = sin xdx^ the second of these equations gives EC x sin a? = sin xdx^ or dx = EC ; consequently, the first becomes DE = cos xdx, as it ought to be. Because the arc GC and the angle x commence to- gether at G, and increase together from G toward C, and that the increase of the arc at any point is clearly in the direction of the tangent (at the point), CE evidently repre- sents the differential of the arc GC ; consequently, since dx — GE, it follows that dx represents the differential of the arc GC, and, of course, x equals GC ; agreeably to what has been supposed. Eemarks. — 1. It is easy to perceive that we may proceed in much the same way as above, to find the differential of any proposed arc of any plane curve, by expressing it, in terms of the differentials of its rectangular co-ordinates, like AB and CB ; that is, . by taking the square root of the sum of the squares of their differentials at any point of the curve, for the differential of the curve at the same point. 2. In our reasonings we have, and shall, generally, take it 70 FINDING THE ARC X IN TERMS OF t. for granted that the reader is familiar with the definitions and leading principles of Geometry and Trigonometry. Thus, in the figure, supposed to be constructed, AC, CB, AB, are called the radius, sine, and cosine of the arc GC ; also, AG, GF, and AF, are called the radius, tangent, and secant of the same arc. 3. AC being represented by 1, since the equiangular triangles ABC and AGF give the proportion AB : BC :: AG : GF = t, or cos x : sin x\\\\t, or its equivalent, sin x =■ t cos aj, in which t = the tangent of x. Since sin a; = a? - ^-^-g, &c., cos a? = 1 -^ -f- ^^a ~' ^'^' ' consequently, the preceding equation may be written in the form, di? m^ i 0? x^ \ "^ - 12:3 + risAS -'*"• = n^ - 1.2 + ixsii - *°) ' which clearly shows that x can be expressed in a series of the odd integral powers of t. For a simple inspection of the terms shows t to be the first term of the series ; and to get the second term, we put t + A^' for aj, and thence have ^ + ^^'~ m +' &c. = 25 - ^ +, &c. ; consequently, if we determine A, on the supposition that the terms involving f destroy each other, we shall have A/3 ^' _ ^ . _ 1 1 _ 1 ^^~iM-~i:%' ""^ ^-"1:2 + 1:23- "a: If ^ — - + Af is put for X, we shall in like manner get A = ^ ; o O and so on. Hence, we shall have FINDING THE CIRCUMFERENCE OF A CIRCLE. 71 «;=<-f + |-y+,&c (.); whicli is a very useful formula for finding the circumference of a circle. l^lius, if X is the numerical value of half a right angle, since X ■=. x\ we have sin 3? sm X = cos X, and, oi course, t = = 1 ; ' ' cos iC consequently (since ^ expresses the numerical value of a right angle), by putting 1 for t and - for x, we shall have 77 , 1 1 1 Again, if x is one-third of a right angle, we shall have x' == 2x; and cos x = sin x' = sin 2a; = 2 sin x cos Xj or sin a? = -, and thence cos x = -^'y L it consequently, from t = we get ^ = -— cos X yo From the substitution of this value of t in (e), we clearly . ^ i /i 11 1 1 P \ ^^' 6 ^ ^3 1^ - O + 5:3^- 773^3 + 9:3^-' ^^-j' which will enable us to find the numerical value of rr to any required degree of exactness. The value of tt to eight decimal places is easily found to be 3.14159265 ; which is clearly the numerical value of two right angles, or the semicircumference of a circle whose radius is the unit of length ; consequently, the product of TT and R, the radius of any other circle, gives Rtt for the length of the semicircumference of the circle whose radius is B. 72 IMPLICIT FUNCTIONS OF For series of more rapid convergency than the above, the student is referred to page 70, volume 1, of Lacroix's " Calcul Ditfei-entiel," and to page 797 of Eutherford's edition of " Button's Mathematics." (9.) We will now show how to find the differential of an arc regarded as a function of its sine, cosine, etc.; which are sometimes called inverse functions. 1. If sin 2 = y and cos s = aj, we get from what is done at page 58, cos zdz = dy and sin zdz = — dx, or / • • c o -.N 7 dy , , dx (smce sm- z + cos- z=i)dz= — ~z=:r and dz = ■ ; Vl-f VT^' and in like manner, if we put tan z = t and cot z = t' { we get, from page 59, — — = dt and . „ = — dt' \ or dz = co^zdt and dz= — &\v^zdt', which are equivalent to dt , , dt' dz = ■ :z and dz 1 + 2? 1-^t"' Also, if sec s — 5 and cosec z = s\ we get, from what is shown at page 61, tan z sec zdz = ds and cot z cosec zdz = — ds' ; which, from tan z = Vs^ — 1 and cot z = Vs'^ — 1, are reducible to , ds , , ds^ dz = — — and dz In much the same way, from page 61, if we put versin z = l—- cos z = v and coversin z =1 — sinz = v\ we get sin zdz = dv and cos zdz = —dv\ ARCS DIFFERENTIATED. 73 , dv 1 , dv' or dz = — — and dz= — : sin 3 cos s consequently, since cos s = 1 — -y and sin s = 1 — -y', we get , dv T . dv dz = J and dz= — V2v' - v"' 2. It is manifest that the radius of tlie arc in the preceding formula is 1, or unity, which may easily, from the principles of homogeneity in the members of the equations, be reduced to an arc whose radius is r, after the following manner: Thus, for J/- and x^ in the first two equations, write ~ and — 2- and they become , dy , , dx dz = — ; — and dz = ; — , which are easily reduced to , rdy , , rdx dz = — ; — - — and dz — and in like manner the remaining equations become , r'dt , 7^1' dz = —7, -: , dz= — , T^ds , T^ds' dz = — — , dz= — dz = rdv - _ rdv' which are adapted to the arc z whose radius is r. Eemarks. — 1. Diiferentials that are not of the preceding forms, can often be reduced to thena. Thus ^ :i d Z is equivalent to /25 - Ux' ■m x" 7^ REDUCTIONS OF FORMS. •which, is the differential of a circular arc whose radius is J and sin = a; divided by 5. In like manner the differen- , -dv tial -s — i-r-r is reducible to — ; t:^ — r ; which is the dif- t^-S") ferential of a circular arc whose radius = 1 and tangent = - -y, divided by db. 2. In like manner, differentials can often be reduced to those of known logarithmic forms. Thus the differential -g — -^ is reducible to the known logarithmic differentials dx — dx T 2adx H , and a -\- X a — X XT — a* . , , , dx dx IS equivalent to , ^ X — a X -\- a which are differentials of well-known logarithmic forms. (10.) We will conclude this section by noticing some of the more important properties of the expressions e^^-^ — cosa?+ sin a? V~l and e~'^^~'^ = cos;??— sina? V — 1, or their equivalents cos X = -„ and sm x = tc-- ; 2 Sv^TZTi see page 53. It is manifest that for the fii-st two of these forms, we may take e^'^^~'^ — cos a? db sin a? V — 1 ; by using the upper signs (in the ambiguous signs) for the first, and the lower signs for the second. DE moivre's formula. 75 If mx is put for x , we sliall have or, because ± m X V _ 1 — (fi ±x ^— 1 \ w (^±x ^-ij'^z^ (cos aj± sin a? 4^ — 1)"*, we stall get (cos a? db sin a? I^^l)'" = cos mx ± sin ma? 4^—1 . . . {f) ; wliicli is called De Moivre^s Formulce. Expanding tlie first member of this equation according to the ascending powers of ± sin a? -/ — 1 by the Binomial Theorem, and equating the real and imaginary parts of the members of the resulting equation, separately, we readily get cos mx = cos"* X — — ^— — - cos'" ~ ^ x sin^ x _j ^ ^v _^.v / cos"*-'*aJsm*aj— ,&c., JL . Z . O . 4: and • / m 1 (?/? — l)(m — 2) ^ o . , sm maj = m smaj I cos"* ~^ a? ^r^ cos"* ~ ^ sm'^ x , {m - l)(m --^ ^){m - 3)(m - 4) ^ . • . o \ + ^^ ^^^^ 7~r-i-^~^ X cosaj"*-^ sm^a?— , &c. 2. 3. 4. ' / If in these equations we successively put m = 2^m = 3, &c., we get cos 2x = cos- X — sin- x = cos^ x — (1— cos^ a?) = 2 cos^ a? —1, sin 2a7 == 2 sin a? cos a?, cos 3a7 = cos'^ a;— 3 cos x sin^ x = cos'^ a? — 3 cos a? (1— cos^ a?) = 4 COS'^ 37—3 COS X, sin 3aj = 3 sin a? cos^ x — sin'' a? = 3 sin a? — 4 sin^a?, and so on. 76 IMPORTANT FORMULiE. If in the expressions for cos x and cos ma?, sin x and sin inx^ we put o'^ ~'^ = y, and of course e~''*^-~^ = -, then 1 1 we get 2 cos x = y -\ — ^ , 2 cos mx = y"" + -^.^ 2 sin X V— 1 = y , and 2 sin mx V—\ = v"' -. Supposing m to be a positive integer; by raising the members of 2 cos x =^ y + - io the mAh power, and uniting the first and last terms, the second and last but one terms, and so on ; we shall evidently have 2» cos" a, = (y- + A) + m (y"-= + -~) . m (m — 1) / ^ . 1 \ If in is an odd number, since y"" -\ — - = 2 cos 7?ix, iT'"^ H ;;r:2 ~ ^ cos {m — 2) a;, and so on, we readily get 2 cos mx = cos mx + m cos (m — 2) a? + cos (tti — 4),'» +, &c., until the number of terms = 1.2 ?yi + 1 2 When 7?i is an even number, we have 2'"~* cost's? = , _.. mim—X) cos mx-{-m cos [m — 2) a? + — j— — ^ cos (w— 4) a? +, &c., - terms containing cosines ; to ^ m(m-l)x...x(y +1^ untn there are — terms containing cosines ; to which must be added the term 1-2X xy IMPOKTANT FORMULAE. 77 If in these formulae we put 1, 2, 3, &c., successively, for 7??, we readily get the following TABLE. 1. COS a; = cosaj; 2. 2 cos^ a? = cos 23? + 1 ; 3. 4 cos^ X = cos 3a? -f 3 cos x ; 4. 8 cos"* X = cos 4;r +4 cos 2;c+3 ; 5. 16 cos^ a? = cos 5a? + 5 cos 307 -1-10 cos a?; 6. 32 cos^ a? = cos Ga? + 6 cos 4a? + 15 cos 2.r 4- 10 ; 7. 64 cos^a? = cos 7a? + 7 cos 5a? + 21 cos3a?-h35 cos x; 8. 128 cos^ X = cos 8a? + 8 cos 6a? +28 cos 4a? +56 cos 2a?+35 ; and so on, to any extent that may be desired. If m is an even number, and the members of 2 sin a? V~l =^ y are raised to the mth power, then, by proceeding as before, we shall clearly have ± 2"*-^ sin*" a? = cos ma? — m cos (m — 2) a? . 7n {771 — 1) H -:r-^ — - COS {m — 4:)x — , &c. ; 1 . Ji noticing, that + must be used for ± , in the first member of the equation, when m is exactly divisible by 4, and that — must be used when it is divisible by 2, or not divisible by 4. It may be added, that there will here be -^ terms containing cosines ; together with the term ± ^m{m.- l)x....x (y + V 1 . 2x X -TT- 78 IMPORTANT FORMULJE. in whicli 4- must be used for ± when rn is divisible by 4 ; and wben in is not divisible by 4, we must use — . When m is an odd number, by proceeding as before, we shall have ± 2"*~^ sin"* X = sin mx — m sin (m — 2) x H Y~2 — ^^^ ^^^ — 4) a? — , &c., until the number of terms equals — ^ — ; noticing, that + must be used for ± in the first member of the equation, when m — 1 is divisible by 4 ; and that — must be used in the contrary case. If 1, 2, 3, 4, 5, &;c., are successively put for in in the pre- ceding formulae, we readily get the following TABLE. 1. sin X = sin a? ; 2. — 2 sin^a? = cos2a?— 1; 3. —4 sin' a? = sin 3a?— 3 sin x ; 4. 8 sin* X = cos 4,r — 4 cos 2a? + 3 ; 5. 16 sin^a? = sin 5a;— 5 sin 3aj-f lOsina;; 6. —32 sin'' x = cos 6x—6 cos 4a; + 15 cos 2a;— 10 ; 7. —64 sin^a; = sin 7a?— 7 sin 5a; + 21 sin 3a;— 35 sin x; 8. 128 sin^ a; = cos 8a;— 8 cos 6a? +28 cos 4a; +56 cos 2a? + 35; and so on, to any required extent. Resuming the simultaneous equations 2 cos x = y -\ — , and 2 cos mx — if ^ — -^, from p. 76 ; it is easy to per- ceive thjvt they are equivalent to the equations y^ — 2y cos a; + 1 = 0, and 'y^'^ — 2y'" cos ma; + 1 = 0. FACTORS OF y"'"' — 2y"' COS + 1 = 0. 79 Because these equations are coexistent, it is clear that the first is a quadratic factor of the second. If we have an equation of the form yim _ 2y« COS 6> + 1 = y'^"' — 22/"^ cos {6 + ^nz) + 1 = 0, since cos 6 = cos {0 + 2nn\ n being an integer ; then we shall have y^ — 22/ cos I ) + ^' for the general representative of its quadratic factors. Pat- ting successively, 0, 1, 2, 3, &c., to /i = m — 1 for n in the quadratic factor, we clearly get 2^2^/1 _ 2y^ cos + 1 == \if — 2?/ cos — + l) X {y'- 2y cos ^-^ + l) x {y' - 2y cos ^^— +l), &c., to m factors. It is evident that these factors are different from each other, and that they are the only quadratic fac- tors which the equation can have ; since t^ == r/z, ?i = m + 1, n^=.m ■\-% kc, will merely give repetitions of the factors found. Thus, the quadratic factors of since cos := ^ or 6 = 60°, will easily be found to be y' - 1.8793852 . y -f 1, y' - 1.5320888 . y + 1, and 2/^-0.3472964 y + 1; and in the same way, since gives cos = — ^ , we readily get 6? == 120°, and thence we 80 FACTORS OF y-'" — ly"^ COS + 1 = 0. shall have f - 1.5320888 .y + 1, y- - 0.3472964 -2^+1, and y^ — 1.8793852 . y + 1, for the quadratic factors. If we have an equation of the form y'"* — 2ay"* -}- 1 = 0, in which a is numerically not greater than unity, it is clear that it may in like manner be resolved into quadratic fac- tors. Consequently, if each quadratic factor is resolved into its two simple factors, the roots of the proposed equation will be known. If a = 1, the equation becomes 1 . .» r. 2n7r having ^ — zy cos h 1 for its general quadratic factor, since cos Inn = 1. Putting 0, 1, 2, 3, . . . . to m — 1, inclusively for n, the particular quadratic factors will be found to be * rt 47r , . c, rt 2(m— l)7r . if — 2y cos h 1 . . . . to v" — 2y cos -^ — h 1, for the last factor. Because 2(m — l)7r ^ 27T __!^ _^_ = 27r , 7/2/ m it is clear that {2m — 1) 27r cos ^^ ^n = cos — , 7 a m and, in like manner, 2 (m — 2) TT 47r cos — ^ — = cos — , m m and so on ; consequently, for „ ^ 2(m — l)7r . j^ _ 2y cos -^———- + 1, . ^ 27r , we may write y- — 2y cos 1- 1 ; PACTOKS OF if"^ — 2?/"" COS -f 1 = 0. 81 for y^ — 2y cos — ^ ^ h 1, we may write y^ — 2y cos — — h Ij and so on. Hence, we shall have (^- - ly = (y _ 1)= . (y^ - 22/cos ?^ + l)' X {y'- 2y cos ^ + 1 ) , &c., to — jz — factors, when 7n is an even number ; and to — ^ — factors, when m is an odd number. Consequently, extract ing the square roots of these equal products, we shall have 2/"'-l=(y-l). (y^-2y cos ^ + l) . (y» - 22/cos J + l) &c., to factors when m is even, and to factors when 7)1 is an odd number. Thus the factors of y^— 1 = 0, are y — 1, y'^—2y cosy + 1, y'^ — 2y. cos y + 1, and y + 1 ; and those of 2/^ — 1 = 0, are 2/ — 1 , y" — 2y cos y +1, and y^ — 2y cos y + 1- In like manner, if a = — 1 , our equation becomes y2m^2y-4-l==(y"^ + l)'=0; whose general quadratic factor is 2 , _ 2717T+7r y'+2y.cos —— + 1, since cos i^n-n + tt) = — 1. 4* 82 FACTORS OF xf"" — %f^ COS ._im = 2/" + l"N or y^'+l = PA' x PB' x PC x , &c.: noticing, that the equations ± {y^^—l) = PA x PB x PC x , &c., ^''^ -i- 1 = PA' X PB' X PC X , &c., are called Cotes's Properties of the Circle; see pp. 32 and 33 of Young's " Differential Calculus." SINGULAR PROPERTIES OF THE CIRCLE. 85 ."Remarks. — There are one or two singular properties of circular functions that it may not be improper to notice in this conuection. Thus, resuming the equation e^^~^ = cos x + sin a? l/— 1, from p. 53, and putting a? = -, we have e^'^'^'^V-l, or e~'^=:{V^)''~'; which, expanded according to the ascending powers of x, by {b"\ given at p. 51, gives for one of the properties. And by taking the hyperbolic logarithms of the members _7r _ -| of e 2"— (|/_- !)*'-!, we have — - = V^l x ^ log — 1, or TT =: — 4/ — 1 log — 1, for the other property : noticing, that TT = the semicircumference of a circle whose radius =: 1, and that e stands for the base of hyperbolic logarithms. See pp. 33 and U of Young's "Differential Calculus." SECTION HL VANISHING FRACTIONS. (1.) When tlie numerator and denominator of a fractional expression are each reduced to naught or vanish, bj giving a particular value to a common variable, the expression is called a vanishing fraction. Thus, —TT 2\ ^ ^ vanishing fraction : since, by putting a (fl/ — a ) a for X, it is reduced to . ,, ■ .,, = t:. It is clear, from ' a{a^—a^) aj"— a^ (x — a) (x^-\- xa + a^) a (x^— a^) a {x — a) {x -\- a) that it is reduced to the form ^, bj putting a for a? ; since the factor x — a (which is common to the numerator and denominator) becomes a — a = 0. It is hence evident, that vanishing fractions result from the vanishing of factors that are common to their numer- ators and denominators. (2.) Because the quotient arising from any division is man- ifestly independent of any factors that are common to the dividend and divisor, it is clear that by erasing such factors from the dividend and divisor (or dividing them by their greatest common divisor) before the particular value is put for the variable, and then putting the particular value in the result, we shall get the true valua ILLUSTRATIONS. 87 Tlius, since —,-. ^ = ^^— — ^— — ,^ -—-^ a {x^— a"") a {x — a) {x + a) is reduced to 7 — by erasing tlie factor x—a from a{x -}- a) -^ ^. its numerator and denominator ; then, by putting a for x in - — 7 — - — r— , we get, after a slight reduction, ^ for the true value of the proposed fraction, when a is put for x in it. (3.)^ If for generality, we use ^~ to stand for any vanish- Jj X ing fractional form, which becomes - when a is put for x ; then, if A denotes the true value, we shall have - = A . ¥x To find A, we may clearly put .j^^t— = A, or ¥x=A x F'a? ; Jd X then to eliminate the vanishing factor, when it has neither a negative nor fractional exponent, we may differentiate the members of Fx = A x F'a? on the supposition of the con- d¥x stancy of A, which will give A = 'jr?T- ? ^^^ if the right member of this for x= a is reduced to 7: , we may evidently, as before, put A = -wn^y- , and so on, until a fractional form Ct't: X will finally be obtained, in which both the numerator and denominator will not vanish when a is put for x ; which will clearly be the true value of the proposed fraction. x^ — 3.'» + 2 Thus, to find the true value of the fraction when X = 1] which reduces it to the form Sx'- 6x^ + S' 0* 88 ILLUSTRATIONS. Here, Fa?, F'a?, and a, are represented by ajs _ 3aj + 2, Sa^ — 6u;» + 3, and 1 ; consequently, from d ix" - 3a; + 2) = (3£r - 3) dx and d{Zx^ - Qs? + ^) = (123?^ — 12aj) dx, we have A == — r^ — --^^ = - when xz= 1, V2a^—12x Hence we have d^{a^-Sx+ 2) _ d{3x' -S) _ 6a? - ^ c^ (3a;^ - ear' + 3 '^ d (ISa,-^ - 12a!) ~ 36a!2 - 12 ' /» -j which becomes ^ r-^ = r , when 1 is put for a?, which is oO — 1^ 4: the true value of A, that of the proposed fraction, when 1 Ls put for x in it. Fa? (4.) Still using ^^ to represent a fractional form that be- Jj X comes ^, when a is put for x ; then, the vanishing factor that is common to the numerator and denon;inator, whatever may be its nature, can be eliminated from the fraction after the following manner : Thus, put a + h for x in Fa? and F'a?, and expand these functions by Taylor's Theorem, or in any other way, accord- ing to the ascending powers of h ; and they (by omitting the vanishing terms) will evidently be reduced to the forms Ah" -f BA* + , &c., A'A^' -f B'A*' + , &c. Hence, we shall have Fa? _Y{a-{-h) _ AA^ +BA^-f,&c. F^a! "" F\a + h) ~ A!h^' ^- BVi^^ +, &c. ' and hence it is clear that -r-> A"-''', when a is put for a?, ex- presses the value of the proposed fraction. Thus, if a is ILLUSTRATIONS. 89 greater tlian a\ it is clear that the value of the iraction equals 0, since -^ ti"-''' = 0, when A = ; when a -- a\ the v^alue of the fraction is -^, , since a — a' = ^ reduces A''-"' to A*' = 1 ; and when a' is greater than «, the value A^ ~ A'A^ A A ^, A'*-"' = -TTiT^n::^ = infinity when A = 0, on account of the infinitesimal divisor A'*'"" in A'A«'-« Hence, a fraction whose numerator and denominator are reduced to naught by a particular value (a) of the variable, may be found by the following RULE. 1. Divide the differential or differential coefficient of the numerator, by the differential or differential coefficient of the denominator, and substitute the particular value of the vari- able in the result ; then if the numerator and denominator of the fraction thus obtained are not both reduced to naught, it will be the value of the vanishing fraction. If, however, the numerator and denominator of this frac- tion vanish ; then we must proceed with the second differen- tials or differential coefficients of the numerator and denomi- nator in the same way as before ; and so on, until a fraction is obtained whose numerator and denominator do not both vanish for the particular value of the variable ; which will, of course, be the correct value of the vanishing fraction. 2. If in the preceding process any differential coefficient becomes infinite, for the particular value {a) of the variable ; then, we must, as at p. 88, change the variable into « + /?, in the numerator and denominator of the proposed fraction, and 90 EXAMPLES. develop, by particular processes, the numerator and denomi- nator into the forms AA".+ BA* + , &c., and A7i''' + B7/-'' + , &c., arranged according to the ascending powers of A ; then, as at p. 88, the true value of the vanishing fraction will be A A "^ expressed by -r^ A*'-'*', when A = ; which equals 0, -7-7 » ^^ ^ A. infinity, axxiordingly as a — «' is positive, naught, or negative. Eemark. — Examples that do not fall immediately under this rule can often be reduced to it, and thence their values found. EXAMPLES. QT* ___ Q^nfl /yj* _l_ lyfl 1. The value of y^ j^ = s , when 1 is put for a?, . f]^—a^ , - , , Za^ — lax—x" , 13 vg — T-^ ; and the value of — — — , when x = a, — 0' ox — Oct " " "^' __ 2. To find the value of -=== — , when x — a. Vx^ - a' Put a ~\- h for x, and the expression is immediately reducible *° ^ Su^ltuT+Zy ' ^^'''^' ^y P''"'°S ^ = 0, gives /I for the answer. Otherwisa Kepresenting the sought value by A, w^e easily get a^ — Sax + 2ar = A^{x^ — a^), which gives 2x — a = A" (ar -\- xa + x^\ by erasing the factor x — a from its members ; consequently, putting a for a?, the answer is A =— =^. r 3a 3. To find , — , when 1 is put for x. EXAMPLES. 91 Putting 1 4- A for a?, the expression reducesto (3A + 2A^)^ _ U + 2A^)^ _ / 27A-^ +, &c. \^ _ (3A + 3A^ + A^)* ~ (3 A + 3A^ + A^)^ ~ V ^^'' +, &c./ ~" (3 A ±, &c.)S wliicii, by putting A = 0, gives naught for the true value of the proposed fraction, when 1 is put for x. 4. To find the value of -^ ^ -. , when x = a. or — ar a — X When X = a^ the dividend and divisor are evidently un- limitedly great, instead of being infinitesimals, as in the pre- ceding examples. Performing the division before putting a for «, we get 1 ^ _1_ _ 1 . or — x^ ' a — X ~ a -\- X ^ consequently, putting a for x, the answer is — . X 1 5. To find the value of the difference it — , , X — 1 log X when X = 1', the logarithm being hyperbolic. Reducing the terms of the proposed expression to a com- mon denominator, skives the fraction '^' , ,— ; which {x— 1) log X is under the form of a vanishing fraction. Dividing the second differential coefficient of the numera- 1 tor of this fraction by that of its denominator gives - — '- X vy for the quotient ; which, by putting .1 for a?, gives -^ for the answer. 92 EXAMPLES. n 6. To find the value of the product (a? — 1) tan ^ a?, when 1 is put for x. When a? — 1 = 0, tan - x becomes tan - = infinity ; con- sequently, one of the factors equals 0, while the other is infinite. Since tan ^ x= , the product becomes , 2 nx ' ^ nx^ cot^ cot^ which is a vanishing fraction ; since its numerator and denom- inator both vanish when x = l. Consequently, dividing the differential coefficient of the numerator of this fraction by that of its denominator, we 2 X sm'* ^ X get , which, by putting 1 for a?, since sin h — 1> o gives for the answer ; and in much the same way, the tan - X value of , when a? = 0, is infinite. e^ 7. To find the value of -. , when x = 0. a? — sm a? From (h") page 51, we have Cu sm X e' = l-\-x -^ — -r +,&c.,and ^o- » consequently, (,x _ ^sin.>^ -^ (a, ^ sin a>) = 1 + ^t.^-^ +, &c., which, by putting a; = 0, gives 1 for the answer. o mu 1 ^ 2a; — sin a? , af — a? , 8. The values of ■ and -^ , when x = X 1 — x and 1, are 1 and 0. EXAMPLES. 9. To nnd the value of -. r , when a is w* — a^ put for X. Put a + A for X, and the answer will be found to . Ida ^ 1 be -J- , or ba more nearly. 10. The values of :: — and — — -. r , when 1 x—1 ax~^ — a~^ 3 and a are put for a?, are 1 and — 11. The value of —. ^ — r. r , when a? = a, is a^ — 2a^x + 2ax^ — a?^ ' ' unlimitedlj great ; and that of when a? = a, is {^ay, 12. The values of ajn+i_fl^" + i a? — bax ■\- 4:0? and a;" — cC" ^x" — lax + 4a« ' 71 + 1 when « is put for x, are a, and 3. 13. The value of ^-- , when cc = 0, is ^ . For most of the preceding examples the reader may be referred to pages 60 and 61 of Young's " Differential Cal- culus." SECTION IV. MAXIMA AND MINIMA. (1.) A VALUE of a function greater than the immediately preceding and following values is called a maximum^ while a value less than those values is called a minimum. Thus, since three successive values of a function of any variable, as x, may clearly be expressed by the forms ¥ (x — h), Yx, and ¥ {x + h); ¥x will be a maximum or mininum, accordingly as it is greater or less than each of the other values, from any finite value of h (however small), to A = 0. (2.) Hence, supposing the functions ¥{x — A) and ¥{x-\- h) to be converted into series arranged according to the ascend- ing powers of A, they may clearly be expressed by the forms ¥x + A {- hy -\- B {- hf +, &c., and Faj + A(A)«4-B(Af +, &c., in which A, B, &c., are supposed to be independent of A, while the index a is less than h, h less than c, and so on ; these series (like the functions they represent) being each less or greater than ¥x from a very small value of A to A = 0. It is clear that these expansions may be written in the forms F(aj-A) = F^+(-A)''[A + B(-A)*-«+C(-A)''-« + ,&c.], and F {x + A) = ¥x + h' [A + BA''-'^ + Ch'-'' +, &c.] ; in which the indices h — a^c — a^ kc, are clearly all positive. DEDUCTION OF FORMULAS. 95 If A is different from 0, it is clear tliat so small a finite value may be given to A, that A shall be greater than the sum of all the other terms within the braces, in the expansions ; consequently, when F.» is a maximum or minimum, the terms A( — A)" and AA"" must accordingly, each be negative or positive. Hence, a must evidently be either an even in- teger, or a vulgar fraction which (in its lowest terms) has an eveii integer for numerator and an odd integer for its de- nominator; and A must be negative or positive, accordingly as '¥x is a maximum or minimum. (3.) Regarding x and h as indeterminates, we may, by Taylor's Theorem for the above formulas, write ,, ^ d{^x)j dX¥x)h'' d^(Fx) A' and To reduce these expansions to the preceding conditions, we must put the coefficient of h equal to naught, or assume the equation ' ^ = 0; and the expansions will be re- duced to F(. - A = F. + -^-^ -- - -~^J j-^^^ +, &c., and F(.. + h) ^ F^ + ^-^ S + ^^^'^^A +' ^^^ which are clearly of the requisite forms, since h? is the lowest power of h^ in them. When a function is a maximum or minimum, any con- stant factor or divisor of it may be omitted, and vice versa. Also, any positive power or root of a igiaximum or mini- mum, must also be a maximum or minimum. And the re- 96 RULE FOR A MAXIMUM OR MINIMUM. ciprocal of a maximum is a minimum ; and that of a mini- mum is a maximum. (4.) It is manifest that the maxima and minima of a func- tion of a single variable may be found by the following RULE. 1. To find when y^ a function of a?, is a maximum or minimum ; put the first differential coefficient ~~ = 0^ and find the real roots of the equation. Substitute each real root in ~, y|, &c., until the first which does not vanish is obtained; then, if it is of an odd degree, it can not corre- spond to a maximum or minimum of y ; while if it is of an even degree, it will correspond to a maximum or minimum of y, accordingly as its sign is negative or positive. 2. To find other maxima, and maxima that may result from the unlimited increase of -^, we put -j- = infinity; or, which comes to the same, we assume its reciprocal dx -J- = ; and find the real roots of this equation. Then, the roots which, put for x in ?/, make it greater than its adjacent values, will give maxima ; while those which make y less than its adjacent values, give minima: noticing, that those roots which do not make y a maximum or minimum, can not correspond to the maxima and minima of the question. 3. If any real root of -^ = 0, w^ien substituted as di- dx rected in 1, makes jhe first differential coefficient, which does not vanish, infinite, then the true value of the term must EXAMPLES. 97 be found bj tbe ordinary processes of algebra, and thence the corresponding maximum or minimum may be determined. EXAMPLES. 1. To find tbe maximum and minimum of yz=:z^^—^x''+ l^x — 7. Here -# = becomes ar* — 3a? + 2 = ; whose roots are ax (Px x = l and x = 2. Substituting a? = 1 in -7^ = 2aj — 3, it be- comes — 1, which being negative shows that if we put 1 for X in y, we shall get its maximum. Also, putting 2 for x in -^ = 2x — 3, it becomes — = 1 ; which, being positive, shows if we put 2 for x in y, we shall get its minimum value. 2. To find the minimum value of y = a^ — {a + h)x + ab. Here -f- = ^ becomes 2x — {a-\-h) — 0, which gives X = — ^ — , and -T3 = 2 ; consequently, putting — -— for x m 2/, we have y— — I — « — ) ? a mmimum. 3. The minima values of y — g? — 2ax -\- o? -\- 1 = {x — af + 5, and y = {x — «)*, are evidently y = h, y = ; while y = (x — of, admits of neither a maximum nor minimum. 4. To divide 2rfi into two parts, whose product shall be a maximum. Because m-\-x and 7n — x when added equal 2m, they may clearly stand for the parts ; consequently, the product of the parts is expressed by {m + x) {m — x) — m^ — ^^ 98 EXAMPLES. which is clearly a maximum when x = 0, which shows that the parts are equal Remark. — It is hence easy to perceive that the number nffi, when divided into n equal parts, gives m" for their maximum product 5. To find the maxima and minima of y = a ± (x — lif. Here, we have -^= ±^ («» — ^)~^ = ± ^ r ; which dx 3^ ^ ^{x-bf shows that x = h makes —■ unlimitedlj great, or reduces ~- = - {x—hy to naught, agreeably to 2 of the rule. By putting x^h — h, we evidently have y = a ± A* ; which by (2.), p. 94, makes y = aja. maximum when — is used for ± , and the reverse. 6. To find the maximum and minimum of y = a ± {x—Vf. get, by putting -^ = 0, aj = J, which makes ~ = infinity. Hence, by 3 of the rule, put x = h -{• h, and we get y =z a db (A)' ; which shows that by using — for ±, x =: b makes y = a, a maximum, and the reverse. 7. Given y = i^a^x — ax^ to find when y is a maximum or minimum. i/ Here we easily get - = a-x — x"^ for which we may evi- dently take u = (]^x — x" , and, agreeably to the remarks at the bottom of p. 95, find the maximum and minimum of u. From -^ = 0, and -7-5 = — 6a7, we get x = — -r and fix (M ' o ^3 EXAMPLES. 99 X = — -— ., ; and by putttnor _^ for x in -7-^, we have a negative result; wliicli sKows that x = —^ makes y a yo maximum : noticing, that x= — makes y imaginary. y o 8. To solve the equations x -\- y + 2 = a, x" -\- y^ = P, and xy- = a maximum or minimum ; or to find x, y, and 2, from the equations and the maximum or minimum con- dition. Putting, according to the second and third conditions, their differentials equal to naught, we evidently have xdx + ydy = 0, and 2xydy + y^dx = ; consequently, since the first of these gives ydy = — xdx^ the second, by substitution, becomes (y^ — 2,^) dx = 0, or 2/2 = 2z^ Hence, the second of the proposed equations, by putting 2x^ for y-, is reduced to Sx^ = ¥; whose solution gives a? = -— and x — ~\ noticino^, that x = — :, makes xy^ a maximum, and x = — — makes it a minimum. Having found x^ we easily get y from a^ -}- y'^= J-, and thence 3 will be found from x -\- y -\- z = a. Kemarks. — 1. It is hence easy to perceive that we may proceed in much the same way to solve all questions of an analogous nature. 2. The preceding solution may be modified as follows: From the second equation we have y'^ = h'^—a^, which reduces the maximum or minimum condition to the maximum or minimum of ¥x — x^ ; consequently, representing this by w, 100 EXAMPLES. we have to find x suclx that u = h^x — Tf shall be a maxi- mum or minimum. Hence, from y- = 0, or J^ — Sar' = 0, and -r-^ = — 6.7?, we get the same results as before. 9. Given x -\-y -\- z= a^ and a?"* y" s^, or m log X + n log y + p log s, a maximum, to find a?, y, and z. By taking the difierentials, we have dx -\- dy -{- dz = 0, -, 7 7 1 ^c?a? Tidi/ pdz or dz =■ — ax — dy. and H =^ 4- - — = ; ^' X ^ y ^ z consequently, substituting the value of dz^ we have mdx j)dx ^ ndy pdy ^ X z y z "* which, on account of the arbitrariness of dx and dy^ is clearly equivalent to the equations m p ^ m X ^ n y = 0, or — = - and - = -. X z p z pi z Hence, to the sum of these equations adding the identical equation - = -, we have m-\-n-{-v'x-\-y-\^z a ap ^ =L =: - or z = p z z 7n -\- n -\- p TltZ 71 Z and thence from x = — and y =■ — , we readily get am T an X = and y = m + n -^ p * rn -\- n -\- p To perceive that the preceding results satisfy the required conditions, the reader may consult Lacroix, " Calcul Dif ." vol. I, p. 380. <1 n ^ ^- - ^ ' f) EXAMPLES. 101 10. To find a?, sucti tliat -^ s shall be a maximum. ' X- -{■ G^ According to what is stated at p. 95, the question will be solved by making the reciprocal of the proposed ex- pression, or = a? H — , a minimum. Because a? x — =^ , or BD = B A, from which it clearly follows that making BD = BA = J, and drawing DPE, DBA will be the required triangle ; and P i)isects DE. 12. "Oiven the sum of the base and curve surfice of a right cylinder, to find when its solidity is a maximum." Let r and A stand for the radius of the base and height of the cylinder, and tt = 3.14159, &c. == the semicircumference of a circle whose radius = 1 ; then, if A stands for the sum of tlie base and curve surface, we shall, from the known prin- ciples of mensuration, get 2?'-A + /'^rr = A and t-tJi = s = the solidity of the cylinder = a maximum. From these con- ditions, we readily get 25 = Ar — rV — a maximum, which gives -7— = or A — dm =0 or r — y -- . EXAMPLES. 103 From tlie addition of 2/'7rA + rV — A and A — 3rV = 0, we get h — r, or the height of the cylinder equals the radius of its base. Remark. — In much tlie same way, if the whole surface is given, when the cylinder is a maximum we shall have r = y -^ , and h — 2/", by using A to represent the whole surface. 13. Find the longest straight pole that can be put up a chimney, when the height from the floor to the mantel = a, and the depth from front to back = h Let D represent the mantel, and AB the pole passing through it, meeting the floor in A, and the back of the chim- ney in B ; then BE = a and DG = EF = h. Representing AE by x, the right-angled triangle ADE gives AD = |/ (a^ + a?-), and then from the similar triangles ADE and ABF we have the proportion AE : AD :: AF : AB =. 4^. AF= ^^''' + '^'^ {h + x) = AE X ^ ^ ^ the length of the pole = a maximum; consequently, (a- -\- ar)l- + 1 1 must be a maximum. Putting the difier- ential of this equal to naught, we readily get the equation X -\-h ;, (a^ + «") = 0, X- ^ ^ which gives x = Vcvl) , as required. 104 EXAMPLES. Otherwise. Supposing AB to be tlie position of tlie rod, let it be slightly changed into the position A'B', by revolv- ing about D ; then (ultimately), its change A'C at the ex- tremity A must equal its change at the extremity B, and have a contrary sign ; consequently, the approximate position of the rod can be easily found by trial. It clearly follows from what has been done, that we shall have AD : DB : : tan ang A : tan ang B,or x : h : ~ : — , or — = — , which gives x = \r^ the same as before. Cb X 14. To find when the cylinder DIGrF inscribed in the cone ABC is a maximum. Eepresent the base and height of the cone by A and «, and the height of the cylinder by a?, then a —x represents the height AE of the cone whose base is DF the upper base of the cylinder. From well-known principles of geometry, we have AH^ : AE^ :: baseBC : baseDF = j-TfT, X base BC = -^ x (a — xf: consequently, multiplying this by x, the height of the cylin- der, we have — j- (« — a?)^ a? lor its contents. EXAMPLES. 105 Hence, because —r and a are invariable (a — xf x must be a maximum, whose differential, put equal to naugbt, gives — 2xdx (a — x) -\- (a — xy dx = or — 2x -^ a — x=^ 0. This solved, gives x= ^ ,ov the height of the cylinder is o one-third' of that of the cone. Remark. — It may be shown, in much the same way, that the height of the maximum rectangle in any triangle is half the height of the triangle. 15. " To cut the greatest parabola from a given cone." Let ABC be a triangular section of the cone by a plane passing through its axis at right angles to its base, and sup- pose that the sought parabola passes through F in BC, then, drawing the lines GE and FD through F, parallel to the tangent to the circumference of the base at and to the side of the cone AC, meeting the circumference of the base in the points Gr and E and the side AB of the cone in D, the curvilinear section GDE of the conical surface and the 106 EXAMPLES. plane of the lines GE and FD will, according to the com- mon definition, be a parabola ; having DF for its axis and D for its principal vertex, FE and FGr, which are evidently equal and perpendicular to BC, being called ordinates to the axis. If through D, DH is drawn parallel to FC, and DI drawn above DII so as to make the angle IIDI equal to the angle A or FDB ; then HI, the part of the side of the cone between the lines HD and ID, will be what is called the principal parameter or latus rectum, of the parabola, it being the parameter or latus rectum of its axis. Since the angles D and H of the triangle HDI are equal to the angles D and F of the triangle FDB, these triangles are clearly equiangular and give the proportion HI : DH or FC::FB : FD, or its equivalent, HIxDF=CF xFB4fE^ by a well-known property of the circle. Kepresenting HI by ^, DF by a?, and FE by ?/, the preceding equation be- comes px = y^, the well'known equation of the parabohi ; which, by knowing p and assuming a?, will enable us to find the corresponding values, + y and — ?/, of y, so that the curve may clearly be constructed by points, according to the common methods. Because the area of the parabola 2 4 GDE equals - G-E x DF = - xy^ it is clear, since the area is a maximum, that xy must also be a maximum. If we rep- resent the diameter of the base BC by a and BF by 2, we shall get CF = a — s ; which give 2/' — az — s-, from a well- known property of the circle. Because the angles of the triangle BDF do not change for different positions of the parabola, it is clear tliat DF will vary as BF or .s ; conse- quently, xy may be represented by z \/{az — z-) and a/^ — z^ must be a maximum. By putting the diiferential of this equal to naught, we have Baz^ ■— 42^ = 0, which gives z = —j EXAMPLES. 107 wbicli, of course, gives the position of ttie parabolic section, wlien it is a maximum. 16. "To find the position of a straight rod or beam, when it rests in equilibrio on a prop in a vertical plane, having one of its ends in contact with a vertical wall, which is at right angles to the vertical plane of the rod." Let BC be half the beam (supposed of uniform density and dimensions) on the prop PR, and having its end B in contact with the vertical wall EB\ whose plane cuts the ver- •tical plane of the rod perpendicularly ; then, through P draw DE perpendicular to the plane of the wall, and DO through G, the center of gravity of the beam, perpendicular to the direction of EP, meeting its production in D ; then, since the beam is in equilibrio, it results from well-known 'principles of mechanics that DO must be a maximum. Put BO = half the length of the beam = Z>, and PE the dis- tance of the prop from the wall = a, and represent the angle BPE = CPD by (^ ; and we shall have BO sin = 2> sin = BE + CD, also we have BE = PE x tan <^ — a tan (/>, and hence, by subtraction, we have Z* sin = DC = a maximum. Hence, putting the differential of this equal to naught, we 108 EXAMPLES. have h cos ^ required. 3 /(L a —- QO^ ^ = 0, whicli gives cos «/> = y ^, as Another solution. — ^Let the figure be constructed, and the same notation used as before : then let the beam be slightly changed in position, first, by giving it a slight angular motion, in its vertical plane, about P until it takes the position C'Gr ; second, by sliding it along P, without any angular motion, until G coincides with B' in the vertical wall. From P as a center, with radii PC and PB describe the arcs CC^ and BG, cutting BO and B'C in the points C and Q>\ B and G, and suppose the horizontal line through C meets a vertical line through C in I ; then IC clearly represents the vertical motion of the point C (or the center of gravity of the beam), resulting from the angular motion. Because the arcs QG' and BG, on account of their (sup- posed) minuteness, may be regarded as right lines, which cut BC and B'C perpendicularly on account of the minuteness of the angular motion cZ, and of course CI -{- BR = h cos (pdcp. If B'H = CI, it is easy to perceive tliat the center of gravity of the beam will be raised by sliding it along P (or keeping its end in contact with the vertical wall), through CI ; consequently, since the center of gravity neither ascends nor descends, the beam must clearly be in equilibrio, as required. Now from the triangle BPE, we have BP =a X sec (b = , . cos represents the versed sine of the angle aNh. Since ij} + xf — a^_{b + x + a) . (5 + x — a) _ 2bj llx 118 EXAMPLES. a maximum, by taking the hyperbolic logarithm of this, we must have log {b -\- x + a) + \og(h + x — a) — S log x — log 2h = a maximum ; consequently, putting the differential of this equal to naught, we shall have (see p. 54) h -^ X + a b + x^a x ' or its equivalent — Sh^ — 4:bx — a^ -{- Sa^ = ; consequently solving this quadratic, we have x= —2b -\- Vb' + Sa^ and x=: —2b— Vb^ -f 3a-, which, giving x negative, must be re- jected, and of course we shall have x= —2b ■{- Vb' + Sa\ By representing a and b by their proportional distances 1 and 0.72333 nearly, we get, from tlie preceding equation, X = 0.4304:6 for the distance of Venus from the earth. Hence, we easily get SEY = 39° 44' for the elongation of Yenus seen from the earth, and ESY = 22° 21', the elonga- tion of Yenus seen from the sun, which being less than 43° 40', Yenus's greatest elongation, shows that she is bright- est between her greatest elongation and her inferior conjunc- tion, being nearly half way between the inferior conjunction and greatest elongation. Because the preceding reasoning does not give the posi- tions of Yenus when she reflects the minimum light, we shall determine these positions after the following method. Thus, from p. 95, we have ¥{x- h) = F^ - '^^ h + fP (F.x) ^ -, &c., and F(. + /0 = F.4--A_JA+_A_^__+,&e.; where it will be noticed, that we have shown Yx can not be a EXAMPLES. 119 maximnm or minimum, unless it is determined on the sup- position that the term —^ — - h is made to disappear from the equations. Now it is easy to perceive, that we have thus far made the term disappear from the equations by assuming d (F,r) dx ; we now observe that we may, when necessary, make the term disappear from the equations by putting A = 0, • ^ c^(F.r;), .. ^. . d{Fx). or, since lor -~ — -n^ we may evidently w^rite — ^ — - dx, we ctx CCX may assume dx = 0; which, in this question, clearly indi- cates the inferior and superior conjunctions of the planet ; since x is a minimum and maximum at those points. Kemabks.— 1. By putting h =0.3871, we find x =1.00058, and thence get SEV = 22° 19' for the elongation of Mercury when brightest. Also, the angle ESY = 78° 56', while it is only 67° 13 '.5 at the time of the planet's greatest elonga- tion ; consequently Mercury is brightest between its greatest elongation and the superior conjunction. 2. Because the motion of Venus about the sun relatively to that of the earth is about 37' ; by dividing 22° 21'=1341' by 37', we get 36 days for the time when Venus is brightest before and after her inferior conjunction. 3. If we apply the formula x = — 2h -{- Vh'-i- 3a^, to find the j)Osition of a superior planet when brightest, it w^ill be found to be impossible ; for, since Scr will be less than Sir, it follows that VP + 3a^ will be less than 2h, and, of course, X = — 2h + Vh^ + 3(x^ will be negative, which is impos- sible, since x, the distance of the planet from the earth, is always positive. Hence, it is manifest that — ^— dx = 0, CiX when applied to the superior planets, can only be satisfied 120 EXAMPLES. by putting dx = 0; which clearly indicates that they reflect the most light in their oppositions, and the least in their conjunctions with the sun. 21. From the extremity of the minor axis of an ellipse to draw the maximum line to the opposite part of its pe- rimeter. Let AEB be an ellipse, having AB and DE for its major and minor axes ; and let EF be the maximum line required, having EG and FG for the rectangular co-ordinates of its extremity F. Then, a and h representing the major and 2 minor axes, we have ¥ :a^ : : hx — ay^ : y^ =-j^ (hx—x% from a well-known property of the curve. Hence, adding sc^ to y', we have E'F'^EG' + ¥0^ = 0^ -i- ^ = 0^ +-^,{hx-x^); consequently, gince EF^ is a maximum, ar^ + -to {^^^ ~ ^) must be a maximum ; and putting its differential equal to naught, we have 2a? -j- -y^ (J — 2x) = 0, which gives 1 a'h ^ = 9 "2 — Jfi 1 which is clearly a maximum, since the differential of 2aj + j^ {b — 2x) is (2 v^ 1 dx, which is negative, when dx is positive, as it ought to be. EXAMPLES. 121 Eemarks. — 1. It is evident from the nature of the ellipse, that the relation of its axes must be such that x shall not be greater than the minor axis J, in order that the" preceding value of x may be applicable to it ; since EG must clearly not be gi-eater than ED — K Hence, to find the greatest value that h can have when the preceding solution is possible, we put h ior x m x = ^ -^ — p, and thence get 2¥ =z a^ or h =^—^ ; consequently, when 1) has this or a less value, the maximum line wiU be found from x=z^ —^ 75. Z a^ — 0- The question and the substance of what has been said are substantially the same as given by T. Simpson, at pages 35 and 36 of his " Fluxions," for the purpose of showing whether the solution found in any case falls within the limits required by the nature of the question, a- 2. Eesuming, EF^ = x^ -\-~ (px — x% and putting its dif- ferential equal to naught, we have l2x +'jt{^ — 2x)\ dx=0; which, by putting dx = 0, clearly gives the minor axis ED for the maximum line when the minor axis is not less than — - . When the minor axis is not greater than — , by putting the factor 2x + jj{h — 2x) of the preceding differen- tial equal to naught, we get x= - — r^, to determine the maximum value of the line to be drawn ; and by putting dx = 0, we get the minor axis ED for the minimum value, as is manifest from the consideration that when h is less 6 122 EXAMPLEa than ^ tliere will be two maxima values, represented by EF and EF', and of course ED must be a minimum. It is lience clear that the (common) rule for finding the maxima and minima of a function of a single variable, given at p. 96, is not always sufficiently general 22. Given x -\- y -\- s = a and x^ifz^ or m log X + n log y -r p log {a — x — y) a maximum, to find a?, y, and z. It is manifest that (since x and y are independent varia- bles) we may put the differential of the preceding equation, with reference to y, equal to nothing, and thence get y in terms of a? ; by which means we shall reduce the question to that of making a function of a single variable equal to naught Thus, we shall have 2^__^^^=:0 or ^ ^— =0, y a — x—y y a— x—y which gives y = — ; consequently, putting this value for y in 771 log x -^ n log y +p log {a — x — y\ it becomes mlogx i-n log (a-x) + log ?i" — (p + n) log {n -hp) 4-i? log (a — a?) + log^^, Vv'hich must be a maximum ; consequently, putting its dif- ferential equal to naught, we have \x a — x I whose diilerential gives \x^ {a — xyj EXAMPLES. 123 From the first of these equations we get am and the second equation shows x"^y'^z^ to be a maximum, as required. Eemaeks. — 1. This example has been before solved at page 100, and it is easy to perceive that we obtain the same results as there found, by substituting the value of x in that of ?/, and then substituting the values of x and y for them, in 2 = a — x —y. 2. We have here given the preceding solution of it, for the purpose of showing the facility with which a function of any number of independent variables may be made a maxi- mum or minimum, by reducing it to the maximum or mini- mum of a function of a single variable ; since it is easy to perceive that we may proceed in like manner, whatever may be the number of independent variables. To make what is here said more clear, we will apply the procTess to the following example from page 33 of Simpson's "Fluxions." 23. To find such values of a,', y, and s, as shall make ij/ — x^) {x"2 — B^) (xy — y'^) a maximum. From making y alone variable, we have xdy — 2ydy = ; o X X" which gives ?/ = ^ , and thence xy —y'^ ^'-r . In like manner, making z alone variable in the proposed X equation, we have x'dz — Sx^dz — ; which gives z — —^^ 4/0 and thence x^z ~ z^ = - — -■ . By substituting the preceding values in the proposed expression, it becomes 124 EXAMPLES. (¥ — ar^)x TT-^ X 7- = maximum ; consequently, we must make &V — a?^ a maximum. Hence we "have 6h^x*dx—8ju^dx=0, and {20h^x^—oQxyix'; tlie first of these gives ^= ^ |/5, which put for x in the sec- ond expression makes it negative, and of course shows the proposed expression to be a maximum as required. SECTION y. TANGENTS AND SUBTANGENTS, NORMALS AND SUBNOR- MALS, ETC. M N (1.) Suppose AM = x and MS = y to stand for the abscissa and ordinate of the point S of the plane curve ASY, having A for their origin ; then, conceiving the curve to be described by the extremity S of the ordinate, while the other extremity M moves uniformly from the origin A of the co- ordinates along the line of the abscissae toward P, so as to keep the ordinate constantly parallel to itself during the motion, we may clearly suppose the ordinate to increase or decrease in such a way as to describe the curve. (2.) If now we suppose the right line TS^ to pass through the extremity S of the ordinate of the curve AS V in any one of its positions, in such a way that the first differential coeffi- cient of the equation of the curve equals the first differential coefficient of the right line ; then, the right line is said to touch the curve^ or the line and curve are said to touch each other ^ and the point S is called their points of contact. Thus, if y = {x) and y = Ax + B represent the equa- 126 DRAWING TANGENTS, ETC. tions of tlie curve and right line, bj taking their differential coefficients, we shall have -4- = — >— and -~ = A: conse- dx ax ax d-, and a-Xy — ¥Yx =. {a^ — Ir) y.», for the equations of the tangent and normal, which are well-known forms. Supposing a and h to be the half major and minor axes of the ellipse, bj putting Y and X successively equal to naught in the preceding equations, we have X = — and Y = — , and X — r — x. Y = -nr y ; which are well-known forms for drawing tangents and nor- mals to the ellipse. Kemarks. — 1. Because ay — h^x^ = — a/U^^ the equation of the hyperbola, is deduced from that of the ellipse by changing the signs of its terms that involve h- ; it is clear that, by cha,nging the signs of the terms that involve Ir in the above results, they will give X = — and Y = , X y and X = 2 — a?, Y = — ry — y, for the corresponding (X quantities in the hyperbola, 2. From X = — and Y = , by supposing x and y to X y be infinitely great, it is clear that X and Y will become un- limitedly small, or that tangents to the hyperbola at points infinitely remote from the center will pass through it very nearly, and have ay — J V = 0, or its equivalents y = - x Ob and y = a?, the equations of right lines passing through TANGENTS WITH HYPERBOLIC ASYMPTOTES. 131 tlie center of tlie hyperbola, for their limits ; in sncli a sense, that they will ultimately differ insensibly from coinciding with these equations: noticing, that these limits are called the asymptotes of the hyperbola ; and from i ctw y = r m) w, we readily get dx dr . , dy dr . -^■=—- cos G)—r sm w and ^ = -7- sm w + r cos w. aw aw aw aw Hence, from the substitution of these values of TAKGENTS AND POLAK CO-ORDINATES, ETC. 133 dx , (ill . ^ ^ -7- and — m tan b -- we get, after obvious reductions, ydx — xdy xdx -\- ydif r tan S = dr dco r-dd) = r: — , and thence rN dr -7-, the same as before; see p. 117 of Young's ''Differential Calculus." Since the angles SrP and SrT — 180°, if we represent Sz-T bj e, we shall get w = 180°— 0, which gives do) = — dd. Hence, the preceding equations become ''' (3) r tan S = /" X dr and r cot S — 7* tan rSN" = dr dd' (4): noticing, that 0, the angle T/'S, is the length of an arc of a circle whose radius is unity and center r, which is measured from tT toward 7'S, and may contain one or more circum- ferences, or any part of the circumference, according to the nature of the case. Remark. — The substance of what has been done may be expressed in the following simple manner. Thus, from r draw the right line rh to make the small angle dd with rS, and from /■ as a center, with radii rm = 1 and rS = r, describe the small arcs mn and Sa, cutting rS 134 TANGENTS AND POLAR CO-ORDINATES, ETC. and rh in tlie points m.n and Sa ; then, mn = dd, and from the similarity of the circular sectors, we have 1 : dd::r: Sa = rdd. Then, through S draw So perpendicular to rS and equal to rdO^ and ch' perpendicular to Sc, meeting the tangent T^ in h' ; then, the triangle Scb' is evidently equiangular to the triangles rSQ and rSN. From the equiangular triangles Sob' and SrQ, we have the proportion h'c : So : : Sr : rQ, or rdd rO — Vc: rdO :: r : rQ= -77-7 x r ; consequently, by com- paring this value of rQ with that of (3) at p. 133, we must have h'€ = dr ; and of course if dd represents the differential of 6, taken for the independent variable, J/g must represent dr^ the differential of 7* ; r being a function of 0, see (4.) at p. 2. We also have, from the triangles Sch^ and S/'N, the dr proportion So \Vc'.'. S/' : 7'N, or rdQ : dr :: r : ?'N = -^, which is the same as (4) at p. 133. It is hence evident, that if the angle dd is infinitely small, the triangle Sch' and the curve line triangle Sab will come infinitely near to coincidence; consequently, according to the method of limiting ratios, we shall ultimately have la : Sa : : Sr : rQ or dr : rdO : r : rQ = -j-^ dr for tbe subtangent, and rdO : dr :: r : 'r'N = -j^ for the sub- normal; results in conformity to the method of limiting ratios ; see p. 46. To illustrate what has been done, we will show how to draw tangents and normals to the common parabola, whose equation is 4aa3 = y^, when the pole is taken at the focus. Thus, let PAQ represent the parabola, having AM for its ILLUSTRATIONS, ETC. 135 axis and A for its vertex ; tlien S, being tlie focus, we liave 4a = 4AS, and representing tlie perpendiculars AM and PM by X and y, the equation 4AS x AM =: PM' of the parabola, becomes 4aa? = y^ ; which expresses the equation referred to rectangular co-ordinates. Supposing the right line T^ touches the parabola at P and intersects the axis in T, we have SP = r and the angle PST included by PS and TS = 0. By trigonometry, we have PS cos ang PSM — — r cos — SM — x — a, or x = a — 7' cos 0, and PS x sin PSM — rsin.d=zy ; consequently, from the substitution of these values in the equation 4«a? = y-, we have 4a (a — r cos 0) — /^ sin" 0, which is easily reduced to the form 4(2^ — 4:ar cos -f r- cos^ 6^ {2a — r cos 6)^ = T^ sin^ + 7^ cos^ Q ^=1 r- (since sin^ + cos- Q -z V)\ consequently, we readily get r =: - — 2a cos a cos-- for the polar equation. By taking the differentials of this equation, --- , and thence -y- = cot ^ , which, we have dr a sm - dQ 136 ILLUSTRATIONS, ETC. substituted for (3) in p. 133, gives /J r cot 2 = PS X tan ang SPT = the perpendicular from S to SP, limited by the tangent PT = the sought subtangent; and from (4), at p. 133, we . d , « sm 2 n have ^7: = — = the subnormal = r tan - = the cos«2 perpendicular to SP through S, produced to meet the perpendicular to l^t through P, which gives the limit of the required normal. Eemarks. — 1. By taking the differentials of the members of the equation 4:ax = ?/^, we have ^adx = 2ydy^ which gives — - = -^ and thence yx~=z-f-=z2x = the subtansrent, di/ 4a^ ^ dy 4:a ° agreeably to what is shown at page 127 ; consequently, by taking MT = 2AM = 2,r, and joining P and T by a right line, it will touch the parabola at P. 2. From 4,adx = 2ydy, we get 2a = ^~ = subnormal (see p. 128), is constant, and equal to -^ = half the parameter of the axis of the parabola ; since 4a is called the parameter or latits rectum of the axis of the parabola. For another example, we will show how to draw the tan- gent and normal to the logarithmic spiral, whose equation is r=:a^; by using polar co-ordinates. Let 1, 2, 3, Y, represent the spiral, having r for its pole, and r, 1, T for its angular axis, such that the positive values of are the arcs of a circle (rad. = 1), which increase arith- metically in the order 1, 2, 3, Y, while ?' = a^ increases geo- ILLUSTRATIONS, ETC. 137 metrically ; then, because = at 1, it is clear that since r — a^ =: aP = Ij that we must have r = 1 represented by rl, and in r^=^a\ 6 — 1 must be represented by the arc of the circle whose length is 1, which we may suppose to equal the length of its radius. By taking the hyperbolic logarithms of the members of r = a^, we have log r = 6 log a ; whose differentials give —p = tan ang ?^ST = log a; conse- quently, the angles at which the radius vector cuts the spiral having the constant, log a, for its tangent, must be constant or invariable ; noticing, if log a = 1, that the radius vector cuts the spiral at an angle of 45° or half a right angle. By (3) and (4), given at page 133, if we divide r by —J- = log a, and multiply r by --,- = log a, we shall have and r log a = r'N, for the subtangent and sub- dr -, ,^ — =z loo: a ad, or ra, log a normal, as required ; consequently, the tangent and normal can readily be drawn. (6.) When a right line touches a curve at an infinitely 138 ASYMPTOTES DllAWN, ETC. remote point from the origin of tlie co-ordinates, and at the same time passes at a finite distance from the origin of the co-ordinates, it is said to be a rectilinear asynijptote to the curve. Thus, by assuming the equation of the tangent from (1), given at page 129, we have Y — y = -j^ {X — x), in which X and ?/ belong to the point of contact of the tangent with the curve, while X and Y are the co-ordinates of any other point of the tangent ; then, by putting Y and X successively equal to naught, we have -,= J(X-.), or X^.-'^....(5), and Y = 2/-a,| (6); from which, it clearly results, if X and Y, or either of them, is finite when a? or y is infinite, that the curve must have a rectilineal asymptote, while if X and Y are both infinite or impossible, the curve has no rectilineal asymptote. Thus, from y = - Vx^— «^, the equation of the hyperbola, (Xi dx Vx"" — rr dy- = X- - X a x'-a? we have ay =^- xdx h- \/{x^— a^) or which witli the value of y reduce (5) to X which, by making x infinite, and rejectmg a^ on account of its minuteness, with reference to ar^, becomes X = a? = X — X — 0; X consequently, the hyperbola has an asymptote passing through the center. ASYMPTOTES DRAWN, ETC. 139 Again, reducing the equation to i» — j V(f^'^'^y% ^^ have ~ z=- — ^, and thence (2) is readily reduced to Y =^ y — ; consequently, making y infinite and re- jecting V^ on account of its comparative smallness, we have Y=^y — -^=y — y=^0, and of course, as before, the curve has an asymptote passing through the center. Eesuming the equation y == - Vx^ — a^, and supposing x unlimitedly great, by rejecting a^ on account of its compara- tive smallness, we have 7 = — a? ; or, since the radical oue-ht to have the ambiguous sign ±, we get 2/ = ± - a?, or its UlC old equivalents, 3/ = — and ?/ = ^ ; which clearly are the equations of two right lines that are asymptotes to the hyperbola, passing through the center of the curve, in ac- cordance with what has been before shown; noticing, that equivalent conclusions result immediately from %^^^_j^ ^^^ dy ^ h V¥Tf ^ dx a \/{x^ — a?) dx a y making x infinite in the first and y in the second, and reject- ing a? in comparison to a?^, and U^ in comparison to y"^ ; and we thus get dy = ± - dx, or its equivalents dy = - dx and c?y = d.x, which are clearly the differentials of the equations given above. 140 ASYMPTOTES DRAWN, ETC. Kemarks. — 1. If we convert into a series arranged according to the desc3nding powers of Xj we shall have Ix ( . a'x--\ a^xr^ a^xr^ „ which, when x is very great, clearly gives hx , hx /. a}x~'^\ for a succession of lines that are clearly asymptotes to each other, and to the hyperbola; noticing, that these are some- times called hyperholie asymptotes^ because the first of them are right lines. It is evident from that we may, express the asymptotes in terms of the descend- mg powers of 7J. 2. From ay = x^ -¥-- = xH1l- -¥x^\ we in like manner get y = 4('- "2 ¥x-^ 8 16 12 x , &c. which gives y = ± a' I/=± x}f ¥x-\ a V 2 r for asymptotes to the curve whose equation is ay = x* — h\ and to each other ; and because none of these are rectilineal, they are from their forms said to be parabolic asymptotes. ILLUSTRATIONS, ETC. 141 3. It is hence easy to perceive how we may proceed to find the asymptotes of curves that admit of them. Tlius, to find asymptotes of the curve whose equation is 2/'V — jpx^ — l)x -^ c ■=■ ^. Dividing its terms by a?^, the equation is reduced to 'if' - ■ jpx — l)x~^ + cx~^ = ; consequently, if — px = 0, and y^ — px — hx~^ — 0, are the successive parabohc asymptotes of each other and the proposed curve. If we take the curve whose equation is " mf — x^y = m^V' and develop y into a series of the descending powers of ,t, we shall in like manner get y = ~ 7n, y = — tti — in'^x^^^ y z:^ —in — tn^x~^ — ^mx-^^ &c., for the hyperbolic asymp- totes of the proposed curve ; noticing, that the first of these is a right line parallel to the axis of x on the side of y nega- tive, drawn at the distance m, from x. 4. When a curve is referred to polar co-ordinates, it is clear that there will always be an asymptote when 7', the radius vector, is infinite, and the corresponding value of Q is finite ; but if r and Q are both infinite, there is no asymptote. (7.) To illustrate what has been done more fully, take the following EXAMPLES. 1. To draw a tangent and normal to any point of the logarithmic curve, and to determine its asymptote. Let OACB be the logarithmic curve, having for the origin of its rectangular co-ordinates, and OA, BO, &c., for its ordinates on the side of x positive, and B'C, D'E', &c., on the side of x negative ; then, y = a^, representing the equation of the curve, by taking the hyperbolic logarithms of its members, we shall have log y — x log a, so that x, 142 ILLUSTRATIONS, ETC. D'^ W B +X being supposed to commence at 0, we shall have OA = 1 for the unit of length, and log BC = OB x the hjrperbolic logarithm of a, and, changing the sign of a?, we have log B'C = — OB' X log a, and so on. By taldng the differ- entials of the members of the equation log y = x log a, we have — = dx log a, which eives ^7— = , = 7/1 = the y dy loga ydy subtangent = const, and ^^-~ = y^ log a, the subnormal [see (1) and (2)], which is clearly correct, since y is a mean pro- portional between the subtangent and subnormal; hence, joining the point of contact of the tangent with the extremi- ties of the subtangent and subnormal, the tangent and normal required become known. To find the asymptote : by changing the sign of a?, the equation y = a^ becomes 2/ = a^ = ~ ; consequently, since a is supposed to be positive and sensibly greater than unity, it clearly foUows, from y=z—, that if x is unlimitedly great, y ct is unlimitedly small, and thence the axis of x is plainly an asymptote to the curve. Eemark. — It is evident, from what is here done, and from what has been done at p. 136, that the logarithmic curve and r k ILLUSTRATIONS, ETC. 143 the logaritlimic spiral, liave resulted from different ways of expressing tlie relation of a system of numbers and tlieir logarithms by linear description. 2. To draw a tangent to the curve whose equation is xy — A", and find its asymptotes. By taking the differential of the members of the equa- tion, since A- = const, we have ydx + xdy =. 0, which gives ^- — 1- a? = or ^-r- = — a? = the subtano^ent ; which, dy dy being negative, shows that it lies in a contrary direction from what has heretofore been supposed, or that it falls in the direction of the positive values of x. If the extremity of the subtangent is joined with the point of contact of the tangent and curve, the tangent will of course be drawn as required. Because the proposed equation is equivalent to A^ A^ . . either of the forms y = — or a? = — ; it is clear, from ^ X y the first form, that making x infinitely great, reduces y to an infinitesimal ; and, from the second form, it results that indefinitely great values of y give infinitesimal values of x ; consequently, the axes of x and y are asymptotes of the curve. Remark. — It is easy to perceive, that the equation xy=^A? is that of the hyperbola, when referred to its asymptotes as axes of co-ordinates. 3. To find the subtangent in the hyperbolic spiral, whose equation is rO = a. By taking the differentials, as in the preceding example, we have rdd + Odi' = ; which gives rdd tHB -T- ~ —0, and -T— = -~rO= —a= the subtangent 144 ILLUSTRATIONS, ETC. [see (3) at p. 133] ; consequently, the subtangent is negative and equal to a. It is hence easy to perceive how the curve may be constructed, and its asymptote drawn, since it mani- festly has an asymptote. Thus, describe the circle ACBD with the unit of distance as radius, and draw the perpendicular diameters AB and CD, taking the first of them for the angular axis and the center P of the circle for the pole of the spiral ; then, pro- ducing PC to E, such that PE = a^ and drawing EF perpen- dicular to it, EF will clearly coincide in direction with the asymptote, on the supposition that the values of in the equation rO z=:a are estimated from A, in the order of the letters ACBD, to include any number of revolutions that may be desired. To describe the spiral by points ; we put its equation in the form r=^-^ and thence, since the semicircumference of u the circle ACB == tt = 3.1416 very nearly, from knowing a, we readily find the corresponding values of r. Thus, by taking the arc Al equal to the radius of the circle = unity, by drawing a line from P through 1 to equal a, we have a point in the spiral ; and setting the arc Al from 1 to 2, and making a line from P through 2 = X, we have another point in the spiral; and in like ILLUSTRATIONS, ETC. 145 manner, by setting tlie arc 1, 2 from 2 to 3, and drawing a line from P througli 3 = ^ , we have another point in the o spiral ; and so on, indefinitely. Hence, drawing a curve with a steady hand through the points found, we shall have •an approximate representation of the spiral, which evidently has EF for its asymptote. K EM ARK. — It is manifest that this curve took its name from the striking analogy between its equation rO = a, and that of the hyperbola xy ■=^ a^\ see page 119 of Young's "Differential Calculus." 4. To find the subtangent in the spiral of Archimedes, whose equation is r — a0. By taking the differentials of the members of its equation, dv we have —=:«.== its subnormal = const, and of course T^' ■— -T-^^ r Y. -y— zzzT^ -■= rO ; consequently, the subtangent equals the length of a circular arc radius r, and angle that between r and the angular axis ; see Young, page 118. Eemark. — The equations of this and the hyperbolic spiral are included in the class of spirals represented by the equa- tion r = ad'^ ; noticing, that n may be positive or negative, according to the nature of the case. 5. To find the subnormal and subtangent in the spiral, whose equation is {r^ — or) 6^ = J^. Solving the equation with reference to r, we have A" 4) which gives dr ¥ l^ '^' Va^ + 4 ^Va^^-^ + J^ = the subnormal. 146 ILLUSTRATIONS, ETC. Dividing 'r by the subnormal, we "have —r- = — ^^ jr^ for tbe subtangent^ which reduces to —h when 6 = 0. Remarks.— 1. From r = yW + -^), it is evident that the last value of r is a ; which immediately follows from the proposed equation, when put under the form (P = -^ s . Hence, if from the pole of the spiral as center with a as a radius, a circle is described, it will clearly be an asymptote to the spiral ; since, when r = a^ must be unlimitedly great, or must include an unlimitedly great number of cir- cumferences. 2. In much the same way it may be shown that the spiral whose equation is r = y la^ — -^1, lies wholly within the circumference of the circle whose radius is a ; the circum- ference being an asymptote to the spiral. 6. To find the subnormal and subtangent of the spiral whose equation is (/^ — ar) ^ = 1. From the equation we readily get 7^ — ar = -^^, and thence /• = I + Y^j + ^), which gives -^ = — *Ko -^V T- +h= y— = "t^® subnormal ; consequently, dividing r* by this, we readily get the sub- tangent. Remarks. — From r = -^ + y\^ '^ ey' ^^ ^^^ proposed equation, it follows that the circumference of the circle whose radius = a, is an asymptot-e to the spiral, being an ILLUSTJIATIONS, ETC. 147 interior asymptote ; while tlie circle whose radius is a, is an exterior asymptote to the spiral whose equation is {ar-r')6' = l, or ,. =.. | + |/(^ - 1), the spiral falling wholly within the circle: see Young's " Differential Calculus," p. 123. 7. To find the subnormal and subtangent of the paraboUo spiral^ whose equation is r^ =: a-0 or 7' -- a0 . By taking the differentials, we have dr 1 a a .i i i consequently, 7*^ divided by the subnormal, gives ~d^ -j^ = 2a0^ = the subtangent, and since, from the proposed equation 0* = - , we have r'dO _ 2/'^ Eemaek. — It is manifest that the spiral is called parabolic, from the analogy of its equation r- z= o/d to that of tbe parabola y" = ax. 8. To find the subtangent and subnormal at any point of the cissoid of Diodes. ■i Let ABCD be a semicircle, baving AD for its diameter, to 148 ILLUSTRATIONS, ETC. whicli tlie perpendicular ordinates BF and OE are drawn, at equal distances OF and OE from the center ; then, drawing a right line from A to C, the extremity of the ordinate OE, it will intersect the other ordinate at a point Q- of the cissoid. Kepresenting AF, FGr, and AD, severally, by a?, y, and a ; the equiangular triangles AFGr and AEG will (from known principles of geometry) give the equal ratios expressed by y CE . . BF - = -:-^ = (by construction and the nature of the circle) zf^^ , a; AE ^ -^ ^ DF' y" BF^ AFxDF AF ^ ^ ^^ . or ^ = YSTTTJ = — TTf^5 — = TTPT = by the nature of x" DF^ DF^ DF a — x ^ the circle ; consequently, we shall have if- — , for the (.1 — a? equation of the cissoid. By taking the differentials of the members of this equation, we have , . , . ydy a? (Sa — 2,??) , , , "which gives ^-j^ = —^— — — ^ = the subnormal, dx £1 \Oj — x\ and dividing y^ by the subnormal, we have 2a7 {a — X) 3a — 2a; r3 the subtangent. If in y- = we put x — a, we shall have if — = ^ » consequently, if is infinitely great, and of course there must be two infinite values of y, one of which is expressed by + y and the other by — ?/ ; which must be asymptotes to the cissoid. Thus, from ?/- =: we have ?/ = + a? i/ ; con- 149 sequently, any positive value of y being expressed by y = xi/ , the corresponding negative value must be ^ a — X FG of course, tlie expressed by 3/ = lower part of tbe curve must be identical with tbe upper, being described in the lower semicircle, after the method that has been used in describing AGL. It is hence manifest that a perpendicular to the diameter AD through D, when produced infinitely both w^ays, will represent the asymptotes of the branches of the curve. Eemaek. — It is easy to perceive, that the upper and lower branches of the curve will touch each other at A, and will form with each other what is called a cusp of the first hlnd^ since their convexities touch each other. It may be added, that if two branches of a curve touch each other in such a way that the convexity of one is in contact with the concavity of the other; then they form, at their point of contact, what is called a cusp of the second kind. 9. To draw a tangent and normal at any point of the common cycloid. Let BFD represent the circumference of a circle having for its center, OB = r for its radius, DE and EF for the 150 rectangular co-ordinates of the extremity F of the arc DF, the point D of the extremity of the diameter DB being taken for the origin of the co-ordinates ; then, if the ordinate EF in the circle is produced to G so as to malce FGr = the circular arc DF, G will be a point in the cycloid. Hence, representing DE by a?, and EG by 2/, we shall have y = EF + the arc DE, or since DE =x = the versed sine of the arc DF, and EF = the sine of the same arc, when r, the radius of the circle, is taken for the radius; then, denoting the arc (according to usage) by ver sin-^a?, we shall have y =z ver sin~ ^ a? + sin ver sin~ ^ x for the equation of the cycloid, when the origin of the co-ordinates is taken at D, called if Ad? vertex of the curve. By taking the differentials of the members of the equation, we shall have , 7'dx rdx — xdx . /2r — x , dy = - H =:=zirrrrr=: = Y dx ] V2rx — x^ V2rx — x^ -^ since (see page 73) -—7:- — '■ ^r is the differential of the 4/(2/'a? — xr) arc whose versed sine is x and radius r, and that - — r=z V2rx - x^ is the differential of the sine of the same arc. Hence, dx / X X DE V^27^ dy y 2r-x V27-x-x^ EF ' consequently, since -j- equals the tangent of the angle which the tangent to the cycloid at G makes with EG, and that DE ^^r equals the tangent of the angle which the chord of the arc DF makes with EF, it results that the tangent to the ILLUSTRATIONS, ETC. lol cycloid at Gr is parallel to the chord DF of the corresponding arc DF of the circle. Hence a right line drawn through Gr parallel to the chord DF will be a tangent to the cycloid at G ; and because the chords of the arcs DF and BF cut each other perpendicularly, it follows that a right line drawn through Gr parallel to the chord BF, and extended to meet DB produced toward B, will be a normal to the curve at G. Eemarks. — 1. If in ^— = the subtangent, we put for y its value, we shall have the subtangent = / TT X (ver sin~^ a? -[- sin ver sin~^ a?) ; consequently, having computed the value of this, and set it off from E on BD produced toward D, by joining the extremity of the produced part with Gr, we shall clearly have the tangent, as derived from the subtangent and the point of contact. 2. Admitting the construction of the figure, and that KL, the diameter of the semicircle KIL perpendicular to AB, is equal to DB ; we shall have AB =: BC = the semicircum- ference DFB or KIL, arc KI == arc DF = FG = HE = LB (since EF = HI), and of course arc IL == AL. Hence, we shall have AM = AL — ML = arc IL — its sine IH ; conse- quently, if y represents AM, and IM = HL = ver sin arc IL, we shall have y = ver sin~^i» — sin ver sin~^a? for another form of the equation of the cycloid ; which may be regarded as being a transformation of the equation previously found, when the origin of the co-ordinates is changed without changing their directions. 3. The preceding equation clearly suggests the ordinary 152 method of describing the cycloid. Thas, conceiving the circle whose diameter is KL, to have the point I placed at A, and then rolled (without any sliding) from A toward C, the point I, in one revolution of the circle, will manifestly describe the cycloidal arc, ADC ; noticing, that AC and BD are called the base and axis of the curve, and that the circle described on the axis is called the generating circle. It may be added, since x and Virx — x^ are (from the princi- ples of trigonometry), not only the versed sine and sine of the arc IL, but of the arc IL increased or diminished by any number of times the circumference of the circle whose diam- eter is KL, we may suppose the circle to roll on infinitely along the right line AC produced to infinity toward C, and thereby to describe an unlimited number of successive cy- cloids, which will all be comprehended in the preceding equation. 10. To draw a tangent to the curve whose equation is y = Zx-\-Ux'- 2ar\ A B . D By taking the differentials of the members of the equa- tion, we have -j- = 3 -f 36a? — 6u?-; consequently, the equation of the tangent (see p. 126) Y — ?/ = -^ (X — a?), is easily found for any proposed value of x. Thus, if we put 1 for a?, we have ^ = 3 + 36 - 6 = 33, and ^^ = 3 + 18 - 2 == 19 ; ILLUSTRATIONS, ETC. 153 consequently, tlie equation of the tangent is Y - 19 = 83 (X - 1), or Y =: 33X - 14. It is easy to perceive that this tangent cuts the curve ; since, by putting x and y for X and Y, it is immediately reduced to x" — ^j? + 15a.' — 7 = 0, whose roots are a? = 1, a? = 1, and a? == 7. The first two of these roots belong to the point of contact, while a? = 7 is a point at which the tangent cuts the curve, having y = 257 for the correspond- ing ordinate of the curve. If we put X = 4: in -^ = 3 + 8 6a? — 6x^, and proceed in ClX the same way as before, we shall get -~- = 51, and thence the equation of the tangent to the curve at the point whose abscissa is 4, is Y — 172 = 51 (X — 4). Putting x and y for X and Y in this, we readily get x^— 9x' + 24aj — 16 = 0; whose roots are a; = 4, x = 4, and x = l, the first two of which are the same as the abscissa of the proposed point ; consequently, the tangent cuts the curve at the point whose abscissa = 1, and whose corresponding ordinate is y =: 19. Because the tangent to the curve at the point whose abscissa is 1 cuts the curve at the point whose abscissa is 7, v/hile the tangent, to the curve at the point whose abscissa is 4 cuts the curve at a point whose abscissa is 1, it is mani- fest that the first of these tangents must touch the convex part of the curve ; that is, that part which is convex toward the axis of x ; while the second tangent touches that part of the curve which is concave to the axis of x. It is hence evident that there must be a point in the curve whose abscissa is between 1 and 4, such that the tangent to the curve will not cut the curve at any other point. Thus, 1* 154 ILLUSTRATIONS, ETC. the tangent to the curve at the point whose abscissa is 3, by putting 3 for x, reduces ~ = 3 + S6x — Gxr to -p = 57, and 2/ = 3^ + X'^j? — 2jd^ becomes?/ = 117; consequently, the equation of the tangent becomes Y — 117 = 57 (X — 3), or Y = 57X — 54. Hence, putting x and y for X and Y, we have Sx + ISar' - 2ar»- 117 = 57 (a; - 3), or, 3^ -f ISr^ - 2x^ = blx - 54 , which is equivalent to a?5- 9^ + 27^' -27 = 0; whose roots are a? = 3, a? = 3, a? = 3, and, of course, the tangent to the curve at the point whose abscissa = 3, cuts the curve at the same point. Because the curve changes the direction of its curvature at C, or at the point whose abscissa is 3, it is said to have a point of inflection or contrary flexure at C. Bemarks. — 1. It clearly results from what has been done, that in curves which suffer an inflection, a line which touches the curve on one side of a point of inflection may cut it on the other side. 2. Because the tangents cut the curve at their point of contact, it is clear that the points of contact of the tangents may be regarded as the union or coalescence of the two points in which the curve is cut by a secant, by regarding the points of intersection of the secant as being unlimitedly near each othen Also, because the tangent at the point of inflection does not cut the curve at any other point, it is clear that the tangent at this point ought to be regarded as being both a tangent and secant; that is, as cutting the curve and as tangent to its convex and concave arcs at the point, when taken separately. RULE FOR THK POINTS OF IN"FLECTI0:N". 155 3. If the point of contact of a tangent and two iinlimitedly near points of intersection of a secant witli a curve are sup- posed to be equivalent, it is easy to perceive that it results from what has been done, that the curve may be cut by a right line in three points, or as many as there are units in the degree of its equation. It is also manifest that curves which admit of a point of inflection must be at least of the third degree. cixi 4. Supposing the equation of the tangent Y— y= -i^ (X— a?) Y — y cly to be reduced to the form ^ — ^ =: ; then, if the X — a? dx ' ' tangent cuts the curve, or can be made (as in the question) to cut it at the point whose co-ordinates are X and Y, such Y — y that Y — ^ay ^^ regarded as a consecutive value of -^^ it is clear that for the preceding equation we may write d-y j^=z or infinity, according to the nature of the case. See page 13. Hence the points of inflection of a plane curve may be found by the following RULE. 1. Let X and y represent the co-ordinates of a point of inflection, and suppose -^^ = F (x) = a function of x. Then, proceed, as in finding maxima and minima, to find the maxima and minima of F (x) ; that is to say, find those roots oi — j~-' = which do not reduce —rV^ to naught, and cix cLx dY ix) they will correspond to points of inflection ; if —}— = in- 156 POINTS OP INFLECTION. dx finity, we must find the roots of .^, ^ = 0, and then, as m *'' dF{x) ' ' maxima and minima, find those which correspond to maxima or minima which will give points of inflection, while, if there are no maxima or minima, there can not be any points of inflection. If the roots of — ~-^ = are also roots of — j~~ = 0, (PY (x) we must find the roots of — j— = 0, provided they do not reduce , ^ ^ to naught ; and so on, as in finding maxima and minima, 2. To determine for any value of a?, whether the curve is convex or concave toward the axis of a?, we substitute the value of a? in -T^ = — ^ ; then, if the result has the same sign as y, it is easy to perceive that the convexity of the curve is turned toward the axis of x, and vice versa. Thus, from ^ = S + S6x - 6.zr^, we have ^^^ = 36 - 12^ ; which is clearly positive when x is less than 3, and the con- vexity of the curve is turned toward the axis of x ; noticing, d¥ (x\ that — ^ = gives 36 — 12aj = 0, or a? = 3, and that — j-~ = — 12 shows F (x) to be a maximum, when the curve passes from being convex toward the axis, to being concave. To illustrate what has been done, take the following EXAMPLES. 1. To find the point of inflection in the curve whose equation iay = x^ + a^. EXAMPLES. 157 Here we liave -^ = Y {x) = -^ x~^ + 2x!, wliich. gives d¥{x) 1 -I o 1 ^'F(.^) 3 , __iJ ^ _ _^ t + 2 and -^ = p ; consequently, „ dF{x) . _ 1 -I , O A ^ A from — :r~ = 0, we nave — t ^ ^+2 = 0, or a? = t 5 ^'^^ dx ' 4 ' 4' c^F (;») . . . ' , ^. . . , ^ ^ Since — Tx ^^ positive, ± (a?) is a minimum. And because — -T-^— — — J a?~^ + 2 lias a sign contrary to tliat of y wlien X is less than j , it is clear that its concavity before a? = j is turned toward tlie axis of a?, but after a? = j? "^^^ ^^g^ ^^ "^ is the same as that of y ; consequently the curve has an inflection at the point whose abscissa = -r . 2. To find the point of inflection in the curve whose equation \^y- ^=^x + x". From this equation we have dx -"^ y^')- 2y ' , ^, dF (x) 6x dyl + Sx^ ^ and thence — x^- =^ ^ —77-^— = 0, dx 2y dx 2y^ III/ X -r ^<^j \ J. 1- t)\jo ^ or _ % 1 + 3a;^ _ (1 + 3a^y «;a? y "~2(r^+ar^)' which readily reduces to x^ + 2«^ = - , whose solution gives 3. To find the point of inflection in the curve whose equation is 2/ = a?^. 158 POINTS OF INFLECTION. Here -^ = F (x) = 3a^, gives — -,-^-^ = ox and — r^- = 6 ; ax ax^ clF (x) consequently, from , ^ = 0, we have 6.v = 0, or a? = 0, (PF(x) and from , ^ ^ = 6, it follows that a? = makes F {x) = Ss^ cCx^ SL minimum. Because -^ = Sx^ equals at the origin of the co-ordi- nates, it is clear that the curve has an inflection at the origin of the co-ordinates, where the curve touches the axis of a? on the side of x positive and negative, so that the convexity of the curve above and below the axis of x is turned toward it. Thus the curve must be of the general form expressed by the adjoined figure. Remark. — Any curve whose equation is cf the form y = a?", such that ?i is an odd positive integer greater than one, must clearly be of the same general form as before. 4. To find the point of inflection of the curve whose equation is y = a -\- (h — x)\ ^^^^ ^ "" ^^^ "" ~ 8 (^-^)% gives —^^ = — {h-x) * or ' = "in ' consequently, putting this equal to naught, we have x = l)^ and a point of inflection may, of course, exist at this point From -^ = — - {h—xy, for EXAMPLES. 159 x = h, we have -~ = 0^ and, of course, tlie tangent to the curve at tlie extremity of the ordinate y = a is parallel to the axis of x. From -t^^t— , = ^,^ — it is clear that when dF (x) 10 (Ix X is less than 5, -tt^tt-t will be positive, and when x is greater ctij \X) than h, -w^j-- will be negative; consequently, the curve crosses the tangent at the extremity of the co-ordinate a, where it has a point of inflection. "b Thus, as in the scheme, the curve passes through the ex- tremity of the ordinate a, the point of inflection, and touches the tangent at tlie point, above and below, so that from the nature of a tangent its convexities will be turned toward the tangent. 5. To find the point of inflection in the curve whose equation is y — rnx -\- (h — x)^. As in the preceding example, we take the differentials, and get ¥ (x) = —- =: m — )r (b — x)^ : ^ ^ ax 3 ^ ^ hence, as before, ^-^^= Jq (^ - ^)\ and thence the curve has a point of inflection. Hence clearly, if we change a in the preceding scheme into 7nh, and draw the tangent at its extremity to make an 160 POINTS OF INFLECTION. angle with the axis of a?, whose tangent := m ; then, the curve, when drawn with reference- to the tangent, as in the scheme, will express the curve and its point of inflection, as required. Remark. — For the curve whose equation is y z= mx + {x — 1))\ we obtain the same results as before; with the exception, that the part of the curve toward the origin of the co- ordinates lies below the tangent, while the remaining part .of it is above the tangent 6. If, as is generally the case with spirals, the equation is referred to polar co-ordinates, then we may proceed to find its points of inflection as follows. Thus, let WSY represent a spiral having r for its pole, T'T for its angular axis, and rS for its radius vector, which makes the angle with rT; then, supposing the curve to have a point of inflection at S, it is manifest, since the tangent to the curve at S cuts it, and touches its convex and concave arcs at the same point, we may suppose the perpen- dicular from the pole r to the tangent at S to be constant, when a small change is made in the position of the point of contact; see Vince's "Fluxions," pp. 123 and 124. Supposing with Yince, as we clearly rnay do, that the - I , by taking EXAMPLES. 161 the differentials of its members, we have dO = 7)i 1-) — , or rdd = 771 I -) dr. - \aJ a \a/ Hence, for simplicity, referring to the figure at p. 133, we have rdd = S = — , we have y _ (4:7)1^ H- fr _ o ('^^ + '^'^y _ (normal)* ~ 4cm^ ~ m^ ~~ 4w^ (See Young's " Differential Calculus," p. 131.) 4. To find the radius of curvature in the ellipse, whose equation is a^f + h'or^ = d/W ; a and J representing the half major and minor axes. Differentiating, we have (£'yp + J^a? = 0, or j9 = ^ , which gives , _ _ J^ ^2!^ _ _ h^^f_ _ _ l\aY--¥'b^3^) _ _ _^ ^ ~ a-y a-y'^ a^y ~ a^y^ ~ crif Hence, ^- + 1 = ^ , , — , and thence (p- + 1)^ = -^^—t-^—^ ] consequently, the radius of curvature r = ^ ^ iu ~~ ' USEFUL FORMULAS. 167 and since c^y^ — Iryp' = — a-I/^ is the equation of tlie hyper- bola, its radius of curvature is evidently of the same form. From what is shown on p. 128, ^j^ =jpy — the subnormal, and from a^j}^i/ -\-h'^x= we have py = ——j = the sub- normal ; and thence, from what is shown at the same place, we have consequently, from substitution, we have r = —p- ' noticing, since y^ =z —^ (a^ — a?"), that N may be written in the form or, according to custom, representing g— by e^, we shall have ^ ^^-{a'-e'x'f. Substitutinsf this value of iN" in /• = ,. , we shall have r z=i — - . in which (a^ — e^x^y = the semidiameter of ao ^ the ellipse, parallel to the tangent passing through the point of contact of the circle. See Young, p. 132. Because x^ = -jj {1/ — y% we readily get am a ' consequently, if N makes the angle L with the major axis of the ellipse, we shall clearly have y := N sin L, and thence 168 USEFUL FORMULAS. ^^ |/(//+aVN'^siQ^L) a ' wliich gives N = r- (i — 6^ sin^ L)"*. Substituting this value of N in r, we shall readily get ¥ __ g (1 - ^) a {1-e' sin^ L)' ~ ^-e' sin^ L)^ ' a formula that is very useful in determining the figure of the earth, L being the apparent latitude of the point of contact of the circle with the ellipse. See Young, pp. 132 and 133. Eemarks. — 1. The radius of curvature r = , may be put under several different forms, which are often used. Tbis, smce i? = -r^, we have^-+l = -~j:^ — ? which gives (f+lf = ^'^^'^'^'^ , and from p' = d'^-r ilx, this be- A A- 'A- -u ' {p' + ^f W^-dx^f comes, after dividmg by p\ — — 7—^ = ^-^ 7—- . ^ dx'd 5^ dx From what is shown at pp. 125 and 126 (see fig. at p. 125), it is manifest, since SQ = dx and GR = df/ are common to the tangent line and curve, that SR = Vdy-+ dx^ must be the differential of the right line TS and the curve AS. Hence, if s represents an arc of a curve and ds its diffcr- ^ .. ^ W + cU)^ d^^ . . ential, we shall have r = ^^ 7—^ = r- for its dx dx radius of curvature ; in which, without destroying its gener ality, W3 may for d -~ take its differential oa the supposi- USEFUL FORMULAS. 169 tion that eitlier dx or dy is constant, or a? or y taken for tlie independent variable. Thus since d"^ - ^V^^-^^^V ds^ we shall have r = d^ydx — d^xdy ' which, by taking x for the independent variable, reduces to ds^ . r = -, since dx = const, gives d^x =z ; and if y is cL ycLx ds^ taken for the independent variable, it becomes r = — ~p~f~ • If we take ds for the independent variable, we shall have d]^ + dx^ = const, and, of course, its differential gives dydy"^ + dxdrx = ; which gives d^x = 4 » ^ dxd^x -^ ~ dy' From the substitution of these values in d^ydx — d^xdy^ we have d^ydx — d^xdy = ^ X d^y = , ^ , and d?ydx — cT'xdy = -^ — ; consequently, from the substi- tution of these values in r = -^^ — = — — ,., , , we shall have dyax — dxdy dxds , d'lids r = -75—, and r = ^r— . d^y ' d^x 2. By referring to the figure given at p. 125, it is manifest from the nature of the right line, that if we pass along the tangent and assume SQ to be constant, EQ = dy will also be constant, while, if we pass along the curve concave to the axis of X and suppose dx to be constant, KQ — dy will 170 E VOLUTES, ETC. is clear that d^y and -y^ must be positive, when the con- vexity of the curve is turned toward the axis of x. Hence, because the radius of curvature must always be positive, it is clear that in applying r = ^^ — -, — ' and the preceding derived formulas to practice, ])' and cVy must be taken with the negative sign in them when the curve is concave toward the axis of », and with the positive sign when the convexity is turned toward the axis of x. (3.) Eesuming the equation y = F (a?), and the equations {y — y')'p-\-x-x' r=^, {y — y')p'= — (y + 1), from page 163, it is manifest that if we find x and y from any two of these in terms of x' and y' and known terms, that by sub- stituting them in the third equation we shall have an equa- tion between x' and y', which mil be the equation of the curve in which the centers of all the radii of curvature of the proposed curve must lie. Where it is to be noticed, that the equation y = F (a?) is called the iiivolute of the curve thus found ; which is called the evolute of y^=¥ (x), or of the involute. The reason for these denominations is plain, from the cir- cumstance that we may regard the involute as being generated by the unlapping of a thread placed in contact with the evolute, in such a way that the part unlapped at any point equals the corresponding radius of curvature, when its ex- tremity will be in a point of the involute. Where it is maniiest, that the radius of curvature is always a tangont to the evolute, and constantly perpendicular to a tangent to the involute at its extremity. For convenience in practice, we may give the last two of . the preceding equations the forms EVOLUTES, ETC. 171 y' = y+^^ and .' = .-^j+l), wHcli may be freed from jp ^-—^ and ^' =1 cl~ — dx^ by finding their values from 2/ = F (u?), the equation of the in- volute, or that of the proposed curve ; when we may proceed as directed above. To illustrate what has been done, take the following EXAMPLES. 1. To find the evolute of the parabola, whose equation is y- =z 4:mx. Here for 2/ = F (a?), we have y^ = 4:mx ; which gives di/ 2 m T , ^dy , 2 pm ^iv^ p—-f- = — and p =d-~-~ dx= "—■ = ^ . ■^ ax y -^ ax y y^ Hence we have ;/ -j- 1 = „ , and thence y , 1/ xy (7ny"^Y and y = — {4:m-y')'\ We also have , /if \ 2m « ^ x' — 2m ^ =""+ \i^^ + 2/j — = ^-^ + 2/^i OY x= — ; consequently, by equating these values of a', we readily get 4 y''^ = ^ {x'— 2my-~ m for the equation of the evolute, which is of the form of the well-known equation of the s^micuhiccd parahola If the origin of the co-ordinates is moved in the direction of x positive through the distance 2???., or that we put x' — 2m,=^x and use y for y\ the equation of the evo- 4.^-3 lute may be more simply expressed by the form y^ = -^ . Thus, let CAD represent a parabola having AB for its axis, A for its vertex, and E for its focus ; then, by setting 172 - E VOLUTES, ETC. off El in tlie direction of x positive equal to AE, it is clear that we shall have AI = 2/7i for the radius of curvature of the parabola at its vertex, which equals .4?7^ -^ 2, or half its principal parameter, or latus rectum. Then, since 'if- = ^=— gives 2/ ^ ± |/ -r^— , we construct the curve HIK, having 1 for its vertex by setting off y = y ~^~ at the distance x /~4^ above IB, and ?/ = — y -^— at the distance x below IB ; consequently, a curve passing through all the points thus found, wiil represent the evolute of tlie parabola, or the semi- cubical parabola. Supposing the evolute to be correctly constructed, then a thread stretched from A to I, and lapped on the branch IK so as to coincide with it, and made fast at its unlimitedly remote extremity, when unlapped, by moving the extremity A toward C and keeping it stretched, the point A will clearly describe that part of the parabola represented by AC. By E VOLUTES, ETC. 173 lapping tlie thread on IH instead of IK, we may in like manner describe the branch of the parabola represented bj AD. Remaek. — Mr. Young, at page 140 of his " Differential Calculus," says that the evolute does not extend on the side of X negative, or from I toward A, since x negative in y- ■= ^=— will make y imaginary, which is undoubtedly true ; yet from the first form v'^ = —^—p^ of the evolute, which, for a? = in the second form, gives x' = 2r/i, clearly shows that the point A is so connected with the evolute, that AI must be taken in conjunction with it, as has been done in the preceding construction ; and it is manifest that like observations will be applicable in all analogous cases. 2. To find the evolute of the common cycloid. From Ex. 2, at p. 165, we have dy /2r — x . ^ 2r ax -^ ^ X a? T and i>' = -^ sr ; hence, v' == V + - — 7~ and x' =^x — ^ ^ , , from p. 171, will, by substitution, become y' — y — 2 V2rx — x' and x' = x -{- 4:r — 2x = 4:r — x. Hence, from the substitution of the value of y, from p. 150, in that of y\ we shall have y = ver sin-^a? + sin ver sin~^'r — 2 |/(2ri» — a;^) = ver sin -^ a? — ^\2rx — a?^), or y'=z ver sin -^ a? — sin ver sin-^a?; which, from what is 174 EVOLUTES, ETC shown at p. 151, is the equation of a cycloid, the origin of the co-ordinates being at the extremity of its base. From x' = 4?' — a? we have a? = 4r — x\ from which it is manifest (see fig. at p. 149), since DB = 2r and that D is the origin of the co-ordinates, that if DB is produced below B to the distance DB or 2/', and then x' subtracted from 4/*, we shall readily get — a? = a?' — 4r ; which clearly shows that we may change the origin of the co-ordinates from D to the point distant 2/* below B, and reckon the positive values of x from the new origin upward instead of downward, when the origin is at D. Hence, supposing ABC to represent the proposed cycloid, by removing the origin of the co-ordinates from the vertex B to E, so that GE = (xB, we may reckon x positive from E toward G, and y' = ver sin~^ x — sin ver sin-^ a?, the equa- tion ol the semicycloid EC, whose semibase is EL and vertex C, will be that of the evolute of the proposed semicycloid BC ; and in like manner the semicycloid AE, whose semibase is EM and vertex A, is the evolute of the semicycloid BA : EVOLUTES, ETC. 175 the proposed semicycloids and their evolutes being clearly identical. Eemarks. — 1. The cycloid DEF being drawn (as in the figure) equal to the proposed cycloid ABC, it is evident that the semicycloid EF and ED will be evolutes of EC and E A ; and so on, indefinitely, for semicycloids that may be de- scribed below the cycloid DEF, like EC and EA below the cycloid ABC. And it is easy to perceive that a series of cycloids may in this way be continued indefinitely, both above and below the proposed cycloid ABC. 2. To describe the involutes by the evolutes, we take a thread equal to the semicycloid al arc EC, and fasten one of its extremities at E ; then, having lapped it on the arc EC, we carry the extremity C from C through B to A, when the cycloidal arc CBA will evidently have been described. To describe the arcs EC and EA, we use two threads tied to the points D and F, equal in length to the arcs DE and FE ; then, the extremities at E, being carried from E to A and C, will describe the semicycloidal arcs AE and EC. 3. To find the equation of the evolute of the ellipse. From p. 166, we have c^if' + V^x- = a'lP' for the equation of an ellipse, and V'x , , ¥ and P+^ = -^~^ — , and ^, = -^ . Hence, from p. 171, the equations y' = y + Pl+l and a,' = «-^(£jLl), become 176 EVOLUTES, ETC. From the equation of the ellipse we have ay = a^h' — 5V, which, substituted for ay in the value of y\ reduces it to y' — y- ^-^ -^^ ^-^ , which, putting a'- V^ (?^ is easily reduced to ,_ y {a> — (?d^) _ and, in a similar way, we have a?' = — j- . Hence, we readily get ! 6 ^V o /by'\^ , , (a'x''\^ y' = -jr or f=[~i-), and ^^ = [-r) ; consequently, from the substitution of these values of y^ and ar^ in ay + h'x^ = a^b\ we have a^ (^') + 5^ (^-) = a=5= or (%'0^ + (a'^^'')* = ^y')^ H- (aa^O^ = ^*, and of course (JyO 4- (ax'y = (a^5-)* is the equation of the evolute of the ellipse. By putting x' = 0, the equation reduces to a' or its equivalent hy^ = a? — ¥, which gives y' z=z—- — b] and in like manner, by putting y' = 0, the same equation gives ^' X = a . a Thus, let AB = 2a and CD = 2h be the major and minor axes of the ellipse, and let the points c and d be taken on the minor axis at the distances j- —b from the center, while a and b are taken on the major axis at distances equal to b'^ a from the center ; then, curves drawn, as in the figure, 177 witli their convexities toward the axes, so as to touch them at their extremities, will represent the evolutes of the ellipse. It is manifest that -y and — are the radii of curvature of the a ellipse at the extremities of the minor and major axes. The ellipse may be described by means of its evolute as follows : Take a thread, in length equal to the arc cb + BJ, and fasten one end of it at (?, and lap it on the arc cJ, and bring down the remaining part of it to B ; then, carry the thread around from B to A, and its extremity B will describe the half of the ellipse represented by BDA ; and it is manifest that having fastened the extremity of the thread at <:7, we may in like manner describe the remaining half of the curve, represented by BOA. Because the arc c5 + B6 = the arc cb + J- :'D = It follows that the arc cb^= i = -. — ; a ao consequently, since the four branches of the evolute are clearly equal to a^ — ¥ each other, we shall have 4 % — • for the entire length of the evolutes. Hence, if J is very small in comparison to a^ it is clear that the arc 8* will differ but little from ^ ; con- 173 EVOLUTES, ETC. sequently, the points c and d will fall ultimately without the ellipse, and the semi-ellipses BDA and BCA will have for their limits arcs of circles whose centers are at c and c/, and are drawn through the points B, D, A and B, C, A. Remarks. — 1. Resuming the equation of the evolute from p. 176, then, since the equation ay + Px'' = (jt-V' becomes a- if — V^x^ = — y')*' the equation of the evolute of the hyperbola. If we put y' z=z 0, the equation reduces to ax' =z a^ -\- ^', or x' = a -] , a which clearly shows that — equals the radius of curvature at the extremity of the major axis (2a) of the hyperbola. By assuming y\ we can, from the above equation, evidently find the corresponding value of x' ; and in this ^ay find any number of points, at pleasure, of the evolute. 2. It is easy to perceive that we can not, from the preced- ing equation, find the evolute of the conjugate hyperbolas ; which clearly shows that their evolute is different from that which has been found. To find the evolute of the conjugate hyperbolas, we must proceed in much the same way as before, by regarding h as RADII, ETC., IN POLAR CO-ORDINATES. 179 their principal semi-axis, and a as its semiconjugate ; conse- quently, we shall, as before, have ipyy = (a^ + Irf -f {ax' f for the equation of the evolute of either of the conjugate hyperbolas. By putting x' ^0 in this equation we readily 2 2 get hy' = a- + §-, which gives y' z=zh -\- y , and shows that -y- is the radius of curvature at the vertex of either of the con- jugate hyperbolas. By assuming x' we can, from the pre- ceding equation, calculate y\ and thence find, at will, any number of points in the evolute. (4.) We will now proceed to show how to find the radii of curvature of curves, whose equations are expressed in polar co-ordinates. Supposing, as at p. 131, that r cos w = — r cos = x and y sin 0) = r sin = y, by taking the differentials of a? = — r cos 9 and y = r sin 0, on the supposition that is the independent variable, we shall have dx=z — cos dr -\- r sin ddO^ dy — sin ddr + r cos Odd^ whose squares added give dx^ + dy'^ = dr^ -f r^dO\ . , „ ^dy , / sin Odr -f r cos ddd\ And from d ~ =d{ -, : ) dx \— cos ddr -\- r sm QdQl we readily get d^'ydx — ct'xdy = — r^dd' — 2drde + rd^rdO ; consequently, since (see p. 168) the radius of curvature, r\ {dy' + dx'-f _ {dr^ + r'^dd^f , 2 7 dy d~ydx — d^xdy ' dx we shall of course have for r' the expression (_ r'dd' _ ^dr" + r(^r) dd ' 180 EXAMPLES. noticing, that the expression can be put tinder the more simple form, which must be taken with the positive sign. If N represents the polar normal^ since — W we readily get i ' "-==(1)-' cPr' ~dd' to he taken with the positive sign for the radius of curva- ture in curves expressed in polar co-ordinates ; noticing, that r stands for the radius vector in the polar equation. EXAMPLES. 1. To find the radius of curvature of the spiral of Archim- edes, its equation being r = a6. dr d?7' Since -j^ = a, we shall have -7-, = 0, and thence »= m = a (1 + Gf^f + r' and ^+2(|y-^g=an2 + n give r' = -~^ — — - for the radius of curvature. 2i -\- tf « 2. To find the radius of curvature of the spiral whose equation is r = aO\ dr Since -^ = na&^~'^. we have do and W = a^e^^-' {n'' + e'f ] EXAMPLES. 181 also, since -j-^ =zn{n — 1) aO''-^ we shall have Hence we readily get r =z a0"-i (n^ + 0^)^ -h (0^ + tv" + n) for the required radius of curvature. 3. To find the radius of curvature of the logarithmic spiral, its equation being r = a^. Since -J- ■==La? log a and -^ = a^ (log df^ we easily get .'^ + 2(1)2-. J = a^-^[l + (log«)T; consequently, we shall get r' — <2* |/[1 + (log a)-], the same as the normal: noticing, that log a means the hyperbolic log- arithm of a. 4. To find the radius of curvature of the curve whose equation is r = a cos Q, Here we have -tH = — « sin 0, and — ^ =: — a cos 0, and, of course, ^ — y y~\ -f r^ =: «, and thence the radius of curvature r' = ^r— — :-> = ?:• a' + a^ 2 Eemaeks. — By referring to the figure at p. 131, it is mani- 7^ T 7^^^ dv I, fest that g^ = ^, and g^ = _- (N being the normal), ; represent the cosine and sine of the angle NSr, the angle I made by the radius of curvature r', and the radius vector r ; 182 EXAMBLES. -N' r* + 2 \ddl and N^ r _ d^ dr \ddl 7-= + 2 r» + 2 (if-^ dd' are t"he projections of tlie radius of curvature r on tlie radius vector r, and a line perpendicular to r ; noticing, that the first of these projections is sometimes called the co-radius of curvature. Kepresenting dr , ^ \de) difi \ldry .dr\ I , ^UlrV- d}A by y' and x\ we shall have fdr\^ ^ r + and "^ + ^ V/0/ "^ d(^ W0/ ' = y dr /^ + 2/~ dh \dd) ^ dO^ for the rectangular co-ordinates of a point in the evolute of the proposed curve, whose origin is the same as that of the proposed curve. Thus, in the case of Example 3, we have found in the EXAMPLES. 183 logarithmic spiral r' = a^ |/[1 + (log a)"'] = N, the normal, and, of course, we shall have r — '/ X ^= r — r = =y' ; so that the center of the evolute coincides with that of the proposed spiral. And , dr ^^ dr dr ^ . , or, representing x' by r'\ we shall have r" = a^ log a ; con- r" sequently, since r — a^ ^ we shall have — = log a, a con- stant ratio. . Hence, since r" is perpendicular to r, and has a constant ratio to it, it is manifest that the evolute must be a logarithmic spiral similar to the proposed spiral, their radii vectors making equal angles with their arcs. (5.) There is another method of finding the radius of curvature, that is often very useful in polar co-ordinates, that may be noticed in this place. t 1. Thus, let the curve ABC be supposed to be described by the extremity of PB = r during its angular motion around P in the same plane, in the order of the letters A, B, C ; then, if BKL is the circle of curvature at the point B, having O for its center, and its radius OB = R drawn to its point of con- 184: ANOTHER METHOD, tact with the curve ABC, or the tangent T^ of the curve at the same point B, by drawing PD and PF perpendicular to BO and the tangent, we shall have the right triangle POD, which gives PO- = PD= + OD-2 = PD= + (BO - BDf =P]» BO^- 2B0 X BD + BD-= PB^- 2B0 x BD + B0», which, since PB = ;• and OB = E, by representing BD.= PF by ^>, becomes PO^ =7^—2dB.-\- K" ; or, denoting PO by r', we shall have r'' = '^ — 2v'R + RK Since the points P and are fixed for the same circle (by regarding the curve and circle as having a very small com- mon arc), we may take the differential of this equation, on the supposition of the constancy of 7*' and R, and shall thence get rdr — Rdu = ; which gives R = -j— for the re- quired expression for the radius of curvature. Admitting the construction of the figure, the equiangular triangles PBD and LBK clearly give the proportion PB : BL : : BD : BK, or its equivalent r : 2R ::v: BK = = — : — = the chord ^ r do of curvature which passes through the pole, or origin of the co-ordinates ; which is a result that is very useful (as is the radius of curvature) in the doctrine of central forces. (See Vince's "Fluxions," pp. 149 and 242, together with ISTew- i ton's " Principia," vol. i., p. 68, &c.) There are one or two forms of v that are often useful, which it may be well to notice. 2. Thus, if the angle PBF made by r and the tangent T^, is represented by <^ ; the right triangle PBF gives PF = ^ = 7" sin <^. Also, if PGr is assumed for the angular axis, and the WITH EXAMPLES. 185 angle GPB = d ; then we shall, from tlie principles given at p. 134, get v=.rx ^^^^^^ = -77-— /^r From this expression, we readily get dv =^ — rdr I consequently, E = '-^ will be reduced to the form which agi'ees with the form of r\ the radius of curvature, found at p. 179 ; noticing, that the radius of curvature must be taken with the positive sign. Hence, it follows that the equation of the evolute of the proposed curve may be found from the expressions for y' and x\ given in the remarks at p. 182. It may be added, that having found E, we can easily find PO or v\ from the triangle FOB, and also the angle BPO ; consequently, the evolute can be constructed by points. EXAMPLES. 1. To find the radius of curvature in the ellipse when re- ferred to polar co-ordinates, the origin being at the focus. Taking a and h for the half major and minor axes, we have, from a well-known property of the curve, the equation la — r h^ , ,.„ .. . adr h'dv , , = -^ ; whose differential gives —^ — — T' ^^^ thence 186 EXAMPLES. rdr _, 7'^ ¥ -T- = R = -3 X - rdr 7'^ y^ h^ = R = -3 X - . If we put J? = - = the semiparameter of the major axis, this becomes R = -3 x ^ ; which is easily shown to agree with the value of the radius of curvature found at p. 167. This expression will enable us to find the radius of curva- ture either of the ellipse, hyperbola, or parabola, by putting p for the parameter of the major axis, and observing that r equals the distance of tJie point of the curve whose radius of cwi^ature is to be found from, the focus or origin of the co-ordinates, and that v equals the perpendicular from the focus to the tangent to the curve at the same point. 2. To find the radius of curvature of the parabolic spiral whose equation is ?• = a0 . By taking the differentials, we have -j- = - ad ^, and consequently, we shall get r" 2ad^ A(tH (402 ^ l)i' or shall have v = . Hence we readily get a (40^ + 3) e^dd dv= -^ ^-3 — ; . , , a'dO ^ ^ a {46^ + if consequently, from rdr = -^r— we get K == , as ^ -^ 2 ^ (4^ + 3)0^ required EXAMPLES IN INTERSECTING LINES. 187 3. To find the radius of curvature of the equiangular or logarithmic spiral. From r sm (f) = v, since is constant or invariable, we Tclv get dv = sin (jxlr ; and thence from -y— , we easily get R = -: — -, as required. It is hence manifest that R and r sm (f> are the hypotenuse and leg of a right triangle, having the angle opposite to r equal to ; consequently it follows, as at p. 182, that the evolute must be an equiangular spiral simi- lar to the proposed spiral, and having the same center. (6). Sometimes, as in finding the radius of curvature in (2), at p. 164, by regarding the evolute as being formed by the intersection of successive normals to the involute, we obtain a convenient method of finding the locus of the in- tersections of lines or surfaces drawn according to some law, which are sometimes called consecutive lines and curves. EXAMPLES. 1. Suppose we have the equation x^ — yz ■\-a^= 0, such that z is arbitrary; then it is required to find the curve resulting from the elimination of z from the equation, on the supposition of the constancy of y and x when z varies. By putting the differential coefficient with reference to z equal to naught, we have ^xz — y = 0, which gives s = -^— . Hence, putting this value of z in the proposed equation, we y^ 2?/^ have J -^ + a = 0, or y* = 4:aXj the equation of a parabola whose parameter equals 4a. Remark. — ^If the proposed equation had been xz'^ — yz^ -i- a = Oj 188 EXAMPLES IN INTERSECTING LINES. 2?/ by a similar process we should have found 2 = -^ , and 27 thence have obtained y^ ^ -^ aar^ the equation of the semi- cuhical pa7*ahola^ for the result of the elimination of 2 from the proposed equation. 2. Given x'- + y" = r'^ and {x — xj + (// — yj = r» for the equations of two circles, to eliminate x' and y' from them. By taking the differentials of the equations by regarding x' and y' alone as variable, we have x'dx' + y'dy' = 0, or dy' x' -^, = -, , and from the other equation we in like man- \Anll y ner get ^, = 7-7; consequently, from equating these J if T X — X values we get — = 7 , or x'y = y'x^ which gives a?' = — . From the substitution of this value of x' in y t'v jjj/2 _j_ y'i _ ^n^ ^Q gg^ y' _ __^_^ ^ which reduces 1/ X 7* X x' = ^— U) x' = — -j^, ^ . Ilence, from the substitution y Vip' + y') of these values of y' and x' in {x — x'y + {y — y'f = r^, we readily get x^ -i- y^ — 2/ \/ {x^ + //") + r'^ = 7'^, or by ex- tracting the square root of both members of the 'equation, we have ^(a^ -{- y^)— r' = ± r, or its equivalent, x" + y^ = {/ ± ry, equivalent to two circles, represented by ay^ + y'^= (r' -j- ry and a^ -j- y^ = {r' — 7')l Hence, the series of circles repre- sented by (a? — x'y + (y — y'y = r^, are touched on their outside and inside, or said to be enveloped by the circles EXAMPLES IN INTERSECTING LINES. 189 ^ -\- y^ z=z {^r' + ry- and a?^ + 2/' = i^r' — rf^ which clearly have the same center as the given circle x'^' + y'~ — r'l Eemark. — The preceding solution is merely a modifica- tion of that given by Young, at pages 146 and 147 of his " Differential Calculus." 3. Supposing ACB to be a triangle, such that the position of AB being changed, the area of the triangle shall be inva- riable, then it is required to find the curve to wliich AB shall always be a tangent. Eepresenting AC and CB by x and ?/, and assuming 2/ 1= aa? -f 2> for the equation of AB when referred to x and y as axes of co-ordinates, by putting a? = 0, we shall have y =z h or CB = 7> ; also, by putting j/ = 0, we have ax + 5 = 0, which ejives x =1 or AC = . Because, from the principles of mensuration, the area of ,, . , . ^-p. AC X CBsinano^C l/'miG .^ the triangle ACB == ■'^— ■= — —^ — ; if we represent this by ^, we shall have 5 = — — ^ or a=— —^ . Hence, from tlie substitution of a, the equation y = ax -{- h, becomes y =: -; — x -\- h; whose ZiS differential coefficient taken by regarding h alone as variable, gives -1-1=0, which gives h = — .— ^ . Sub- s X sm \j 190 EXAMPLES IN INTERSECTING LINES. stituting this value of J in y = x + h, it becomes lent xy = —^—^^ the equation of an hyperbola between its asymptotes, as required, since the curve must clearly always touch the side AB in all its positions. SECTION VII. MULTIPLE POINTS, CUSPS OR POINTS OF BEGRESSION, ETC. (1.) Multiple Points. — If two or more brandies of a curve cross eacli other at a point, the curve is said to have a multiple point of the first kind ; the point being called double, triple, &c., when two, three, &c., branches cross at the point : also, when any number of branches of a curve touch each other at a point, it is said to be a multiple point of the second kind. If/* (cc, 2/) = represents the equation of a curve, it is manifest that we may find its multiple points of the first kind, by determining those points of the curve where we have y-rr) = J9": such, that n is a positive integer equal to the number of branches that cross each other at the point, fly and -~- =z p represents the tangent to any one of the branches at the same point It is hence manifest, that to find p^ we may differentiate y (a?, 2/) == n times successively, by regarding x and y as being independent variables, or by considering dx and dy each as being c(5nstant or invariable, when the successive differentials are taken. EXAMPLES. 1. To find the multiple point of the curve whose equation is ay"^ -f cxy — hx^ = 0, at the point whose co-ordinates are 192 MULTIPLE POINTS, ETC., a? = and y = 0, or at the origin of tlie co-ordinates. By taking the successive differentials, we have 2ayd?/ + cxdy + cydx — Shx^dx = 0, and 2a + - = 0, or /> = ; consequently, the curve has a double point at the origin of the co-ordinates, one of whose branches touches the axis of a?, since one value of p equals naught, and the other branch makes an angle G with the axis of x^ whose tangent = . Another Solution. — Solving the equation by quadratics, V , ex ex (^ , 2abx . \ ' wehave y=--±^-(l-f ^^ ,&c.); whose roots hx^ . , ex hx^ are y = , &c., and v = h, &c. By taking the differentials of these values of ?/, regarded as a function of a?, we get do 21)x . T du G 2hx - — ■ = , &c., and -y- = h, &c. ; dx G dx a G consequently, putting a; = 0, we get -^ = ^ = 0, and ~ = , the same as before. dx a 2. To find the multiple points of y'^ = (a? — a)V. By taking the differentials, we have WITH EXAMPLES. 193 ^ydy = 2 {x — a) xdx -f- {x — a) ^cLc, whicli is satisfied so as to leave c?y and dx undetermined, by putting y = and x — a = or x = a; consequently, if there is a multiple point, it must evidently be at the point represented by y = and x=: a. To find wbetber y = and x = a correspond to a mul- tiple point, we diflferentiate 2ydy = 2 {x — a) xdx -\- (x — dfdx^ by regarding dx and dy as constant, and get 2dy'' = 2xdx' + 4 (a? — a) di(^, which, by putting x — a^ reduces io l-j^\ z= a^ or dy = \/a and -~- •= — \fa\ consequently, a double point exists at the point whose co-ordinates are y = and a? = «. 8. To find the multiple points of the curve whose equa- tion is 2/^ = h^x + 2W -\- arl Here we have 2ydy = Irdx + 4:hxdx + Sx'^dx, which is satisfied so as to leave dy and dx undetermined by putting y = and J' -f- 4:bx + Sx^ = 0, or x=: — h. Hence, as in the preceding example, we have 2dy^ = 4:hdxF + 6xdx% or (-— ) = 45 + Ga?; or, putting x = —5, we have ( -^J = —2h; ^y ^ A/— in. .^A ^y consequently, -^ — V —2h and — ^ = — V— 25, which is dx dx a double point when h is negative. If, however, 5 is posi- tive, the point represented by y = and x = 'b^ must clearly be detached from all the other points of the curve, though connected with them by the same equation; and such a point is called an isolated or conjugate point. (See " Cal. Dif.," p. 101, of J. L. Boucharlat; and Young, p. 150.) 194 MULTIPLE POINTS, ETC., 4. To find tlie multiple points of the curve, whose equa- tion is y' = (a; — a)V. By taking the differentials, we have 3y Wy = S(x — afardx + 2{x — dfxdx, which, bj putting 2/ = and a; = a, leaves dy and dx unde- termined ; consequently, if there is a multiple point, it must clearly correspond to y = and x = a. Hence, taking the successive differentials, regarding dx and dy as constant, we readily get [— ) = x^^ or, putting a for a?, we have Since this has but one real root, it clearly results that the point corresponding to y = and a; = a is not a multiple point. Kemarks. — ^It may be shown, in much the same way, that the equation y" z=z {x — a^x"^, when n is an odd integer, can not have a multiple point; and that when n is an even integer, it has a double point 5. To find the multiple point of y^= (x — dfx. It is easy to perceive, on account of the inequality of the exponents of y and a? — a, that the curve represented by the proposed equation, can not have a multiple point of the first kind; consequently, we will proceed to determine whether it has a multiple point of the second kind. Since by putting x = a, the equation is satisfied, and gives y = 0, by taking its differentials we have 2ydy = 4 (a? — dfxdx -\- {x — dfdx^ which is also satisfied by putting a? == a and y — 0, and by taking the differentials of this by supposing y to be a func- tion of x or dx constant, we have WITH EXAMPLES. 195 2yc^V 4- 2iy= = 12 (;» - afxdj(r + 8 (aJ - afdxP, wliicli is satisfied by putting x = a, y = 0, and dy = 0, which leave drt/ undetermined. By taking the differentials as be- fore, we have 2(/d^y + Myd^y = 24 (a? — a) xdx^ + 36 (a? — afdx^^ which is also satisfied by putting x = a, y =z 0, dy =^0, and leaves d^y undetermined. Taking the differentials of this, we have 2ijd'y + Sdyd'y + 6 (dyf = 24:xdx' + 96 (a? - a) dx^ ; which, by putting x = a, y = 0, dy = 0, is reduced to 6 (d'^yf =z 2^adx\ and is equivalent to (y'^j =4a, or, extracting the square roots of the members of this, we have 3 = V« and g=-2ya. It is hence evident that the curve has two branches that touch the axis of x on opposite sides, and each other at the point whose co-ordinates are x = a and y = 0, since dy^=0 or -^ = ; and that the order of contact of the branches dx with the axis of a?, and with each other, may be expressed Otherwise. By taking the square root of the members of the proposed equation, we shall have y=:{x — ay \/x. Which, by taking the differentials of its members, gives -£ = 2{x-a)^x + ^{x- afx~\ which, by putting x = a^ gives -^ = 0. 196 MULTIPLE POINTS, ETC., Hence, taking the differentials again, gives which, by putting x = a, reduces to -^ = ±:2\/a', since the square root ought to be taken with the ambiguous sign ±. Hence, as before, two branches of the curve touch the axis of x on opposite sides, and each other at the point (y = 0, x = a) with contact of the order -~ , and of course the curve has a double point of the second kind at the point (y = 0, x = a). Remarks. — It is manifest that this process is more simple than the preceding. And it is manifest that in either method we may take the differentials of the right members of the equations (since it will not affect the results), without taking that of 37, the factor of {x—a)\ (x—af, &c. 6. To find the multiple point corresponding to 3/ = and a; = a in the curve whose equation is y = {x — af \/x. Since this curve evidently can not have a multiple point of the first kind, we proceed to determine the multiple point of the second kind (by regarding \^x as constant), as in the otherwise of the preceding example. Hence, we have dy = S {x — af\/xdx^ which for x ^= a gives -.-- = 0, and shows that the curve touches the axis of x at the point (y = 0, a? = a). Taking the differentials again, we have ■3^ = 6 (a? — a)\/x^ which x = a reduces to -y^ = 0, and, of course, the curve has contact of the order — ^ = at the dx^ point (j/ = and x = a). By taking the differentials again, WITH EXAMPLES. 197 we liave -7^ = 6|/a?, which, by putting a for a?, and taking ± before the square root, gives -~-^ = ± 6\/a ; consequently, the proposed curve has a double point of the second kind, at the point (y — 0, x = a), whose order of contact is expressed Remarks.— 1. If y = (x — a)" a?"*-}- h, in which m and n are positive integers, then, it is clear that we may pro- ceed in the same manner as heretofore to find the multiple * points. Thus, if 71 =1, by taking the differentials we havo df/ 1 /dyY , ■ n -. P -^ = a;"*, or l-~j = a?; and puttmg a for a?, as heretofore, (dy\"^ ~j =0, which, when m is an even number, gives (-—I — ± t^a; which clearly gives a double point of the first kind, at the point (a? = a, and y = !>)', noticing, if m is an odd integer, that ^ = |/a is not a multiple, but a single point, since "^/a can not have but one real odd root, the remaining roots being repetitions, imaginary or impossible. If n is greater than 1, and 771 odd, the curve will have a single point, the order of contact being expressed by -^r'i i but if m is even, the curve will have a double point of the second kind, expressed by ± |/rt, at the point (x = a and y =zhy^ see Young, pp. 151, 152 ; observing that Mr. Young is clearly incorrect, when he says that a radical of the third degree gives a triple point, and a radical of the 7;7tli dcgreo 198 CUSPS, OR POINTS OF REGRESSION. will indicate that m branches of the curve meet at the point (a? = a and y = J). 2. If y (a?, y) = represents an explicit function of x and y, then, by finding y in terms of a?, after the manner of solving equations, and then proceeding as before, we may find the multiple points, as above. (2.) Cusps^ or Points of Regression. — -A cusp, or point of regression, is generally considered as a species of double point, at which two touching branches of a curve stop or terminate. If the convexities of the branches touch each other, the point is called a cusp of the first kind; while, if the concavity of one branch is touched by the convexity of the other, the point is said to be a cusj} of the second kind. It is evident, that the particular co-ordinates of points where cusps exist must be found by particular considera- tions, and not by the application of Taylor's Theorem ; for otherwise the branches would be continued through the cusp, and make it a multiple point, instead of a cusp; against the hypothesis. Remark. — When more than two branches of a curve touch each other and stop, it is plain that we may regard them as being cusps, and proceed to treat them in the same way. EXAMPLES. 1. "To find the cusps of the curve whose equation is y^ = x'{l- xj = x^ - Sx' -f 3aj« — x^V' It is manifest that the equation is satisfied either by put- ting x = and y = 0, or by putting x= ±1 and y = 0. Hence, to determine which of these gives cusps, we may take the differentials of the members of the equation on the CUSPS, OR POINTS OF REGRESSION. 199 supposition of tlie constancy of dy and dx in the successive dilierentiations. Hence we shall have ydy = (2.»^ - 9a^ + 12^^ - 5x') dx, which is satisfied hj x = 0, or cc = it 1 and y = 0, while dy and dx remain undetermined. By taking the differen- tials again, we have dy"" = {6x^ - 4:5x^ + 84:x' - 'iBx^) dx\ which, by putting iz; = or a? = ± 1, gives consequently, by extracting the square root, we have ^ = 0, and ^=-0, dx ax both when a? = and when a? = ± 1. It is hence clear that the axis of x is touched on opposite sides at the origin of the co-ordinates, and at the extremities of the axis of a?, represented by a? = ± 1? or by a? = 1 and « =: — 1, on the positive and negative sides of the axis. Where it is manifest that the extremities of the axis must be cusps of the first kind, since the convex branches of the curves touch the axis of x and each other at the extremities of the axis, and stop at those points. It is also plain that a? — and y ^=^^ correspond to a double point of the second kind, since the curve is evidently continued through the origin of the co-ordinates. Otherwise. — Eesuming the equation ?/^ z= x^ (1 — a?-y, to find its cusps we may differentiate successively y^ and (1 — a?Y without a?*, or by regarding x as constant, except so far as it is contained in I — a?l Hence we have ^ydy rrr - 6^ (1 - xjdx ] 200 CUSPS, OR POINTS OF REGRESSION. and differentiating again with reference to (1 — a?^/, as before, we have 2 {y(Py+df) = 2^^ (1 - ar*) c^ ; and differentiating again with reference to 1 — ar*, we have 2 {ycPy + ^dyd'y) = - ^Sx^da^. By putting x = ±1 and y = in this equation, we have dy ^y _ ^ . consequently, the curvatures at the points a' = ± 1 are of higher orders than as found by the preceding method, and they are cusps of the first kind. To find the multiple point corresponding to a?*, we may reduce j^ = a;* (1 — aj^) to y = x^ (JL— x^)% and take the differentials with reference to ar' by regarding (1 — ar*)^ as being constant Ilence we shall have dy = 2x{l— Qpf dx ; and in like manner, d-y = 2 (1 — x^f ds?' ; ^X =2'{i-xJ. Putting a? = in this, we have consequently, the curve has a multiple point of the second kind, at the origin of the co-ordinates ; the curvature being clearly of the sec'ond degree. 2. To find the cusps of the curve represented by y =: x^ -^ x^. The equation is satisfied by putting x = and y = 0, or at the origin of the co-ordinates ; and by taking the differentials, we have -j-= 2^ + ^ a?^, which, by putting 201 du cc = 0, gives -^ =: 0. It is hence clear that the curve has ° dx a cusp of the second kind at the origin of the co-ordinates, its two branches touching the axis of x and each other at the origin. 3. To find the cusps of the curves represented by y- =r ± a^. The equations are satisfied by aj = and 2/ = at the origin ; and by taking the differentials, we have which, by taking the differentials again, regarding dy and dx as constant, gives 2 (^J = ± 6,»; which, by putting a? = 0, gives -y- — ± 0. u,x It is hence clear that the carves have cusps of the first kind at the origin, the convexities of the curves touching each other. The cusps may be represented by 0< and >0, in which is the origin, the positive values of x being reckoned toward the right ; and the first figure corresponds to the sign +, while the second corresponds to the sign — , in the proposed equations. 4 "To find the cusps of the curve expressed by {y-lf = {x-af." The equation is clearly satisfied by y = 5 and x = = the angle xO:c'. Since OP' cos = a;' cos <^ = the projection of x' on the axis of .r, and that OP'' cos I ^ — ) cos = y' sin (^ cos Q equals that of y' on a?, we shall have x=^x' cos (p -\- y' sin cos 0, and in like manner y = x' cos (o ~ *A) + y' ^^s (""" ~~ ^) ^^s ^ = x' sin (j)— y' cos «^ cos 0, and ^ = y' sin 0. Hence, if we substitute these values of x, y, and z, in the equation of any surface, the resulting equation in x' and y' will give the equation of its section by the plane of x' and y\ through the origin of the co-ordinates, and thereby give us a clearer view of the nature of the surface. Thus, if we take a?^ -f- y" + 2^ = r^, the equation of the sur- face of a sphere, and make the preceding substitutions, we shall have x'^ (cos^ (p + sin^ 4>) -f y'^ cos^ d (cos^ + sin^ «/>) + y'^ siu^ 6 consequently, the section of a sphere by a plane through its center is a circle whose radius equals the radius of the sphere. If x' cos -{- y' sin (p cos ^ + a, a?' sin

sin — c cos oy^ for the equation of the section of the surface by the plane of x' and y\ which is clearly the equation of a circle whose radius is the square root of the right member of the equa- tion. If a — 0, J == 0, and = 0, the equation reduces to X'' + y'2 ^ ^2 _ ^.2^ the cutting plane being at the distance c from the center of the sphere. If ^^ '^^^ equiva- lent ap+hq=z\^ which is called the general differential equation^ or, more properly, that of partial differential co- efficients, of cylindrical surfaces. The same equation results immediately from Z — s = j9 (X — a?) + l{x- aj -\- {y - IJ + ^"] for the general equation of surfaces of revolution. li a' = and h' = 0, the equation is reduced to 3-\-ax + hy = ({){x'' + y'' + b% which, when a = and h = 0, or when the axis of revolu- tion coincides with that of z, becomes z = (f) (x^ + ^^ + z^)^ or, more simply, we shall have z =' t/> (x^ + y^). If we regard the right member of the general equation of surfaces of revolution as being a function of z^ we shall, from the elimination of the arbitrary function, as heretofore, . , , , . p + a X — a' -{- pz ffet the equation ^ = r-. — ^— , ^ ^ q-{-h y-h'+qz' or its equivalent, (y'~ h'—lz) p— {x—a'—az) q-\-a {y~J/) - h {x—a') = 0; 216 BEPRESENTING CYLINDRIC SURFACES, ETC. which, when z is the axis of revolution, reduces to yp — xq = 0. The same equations among the preceding partial differen- tial coefficients may readily be obtained from the equations, x — x'-\-p{z — z') = 0, and y — y' + q {z — z) = 0, of the normal to any point {x\ y\ z') of the surface of revo- lution, as is evident from the consideration that the normal must pass through the axis of revolution, whose equations must clearly be coexistent with those of the normal. Hence, eliminating x and y from the equations of the nor- mal by means of the equations of the axis, they will be reduced to az + a' — x' +jpz — pz' = (a +p) z -\- a' — x' — pz' = 0, and {h + q)z -}- y—y' — qz' = 0; consequently, eliminating z from these equations, we shall , a + p x' — a' ■\- pz' have 1 — = —, 77 ^—:, h + q y' - b' -^ qz" which agrees with the equation at p. 215, when we use a?, y, and z for x\ y\ and s', as at the place which has been cited ; hence, all the preceding results will be obtained, as alcove. To illustrate what has been done, let a; = as + a', and y = Js -f- h'^ represent the equations of a right line revolving around an axis parallel to the axis of s, to find the nature of the surface of revolution described by it. From s = t/> (ar -f ?/*), we clearly get a?^ + y^ = t/»'s = a function of z ; which clearly becomes x' + f = {az^- aj + {hz + bj, from the substitution of the values of x and y from the equa- tions of the revolving line. To determine the nature of the surface more fully, we KEPHESENTING CYLINDRIC SURFACES, ETC. 217 shall find the nature of its section by a plane through its axis. Thus, substituting the values of a?, y, and ^, from 7T p. 210, since = - = 90°, we have sin ==: 1 and cos == ; and thence, get x ^=^ x' cos 0, y == a?' sin 0, and z = y'. Hence, from the substitution of these values in the preceding equation, since sin^ + cos^ = 1, we shall have x'^ = {ay + ay + {hy' + hj for the equation of the section of the surface by a plane through the axis of s, which is perpendicular to the plane of X and y. Developing the right member of the equation, we have x'' = {a' -}- ¥) y'^ + 2 {aa' + W) y' + a"" + l'\ or its equivalent, ,. = (a» + V) (y' + ^t|^) - ^^J^' + «" + J'^ which, by representing x' and 2/' H 2~rT2~ ^7 ^ ^^ and q to be constant, or their total differentials dz to equal naught, while a?, y, and z are changed, since -r- =z p dz and — z= q^ we easily get in differential coefficients, the 220 DEVELOPABLE SURFACES. general equation of developable surfaces. Thus, representing dp _(r-2 dq _d^z , ^M. — ^ ^'^g _ <^-g dx ~ d?' d^ ~ dif' dy ~~ dx' dxdy ~ d/ydx^ severally, by r, t^ and 5, by putting the differentials of p and q equal to naught, we have '^^dx + ^dy=0 and "^ dx ^ ^ dy = 0, ' dx dy "^ dx dy ^ or rdx 4- sdy = and sdx + tdy = 0, - . , . di/ r , dy s which give ^=-J ^^d d-x=--i' Equating these values of ;t^ ? we have - = -, or rt — ^=0. s t which is equivalent to d^ d^ _ / d^2 Y_ dx^ dy^ \dxdy) ~ ' for the equation of partial differential coefficients of the sec- ond order of developable surfaces. Eesuming, rdx + sdy = and sdx + tdy = 0, and multi- plying the first by dx and the second by dy, by adding the products we have rdx^ + 2sdxdy + tdy'^ = 0, which is called by Monge (at p. 82 of his work), the charao- teristic of developable surfaces. Because dz ^z^jpdx + qdy, we shall clearly have ^-z = rclji?- + 2sdxdy + tdy^, consequently, since the right member of the equation equals naught, we shall have d-z == in case of a plane ; that is, cZ^^ = is the characteristic of developable surfaces. DEVELOPABLE SURFACES. 221 Since d^z == belongs to a plane, and that tlie contact of the tangent plane with the surface is clearly a line, it is evi- dent that all the points of the line may be regarded (see Monge, p. 82) as constituting a plane line. Because 'rt ^= r, we shall have ,9 = Vrl, which, being put for s in the characteristic, reduces it to rdx^ + 2dxdf/ Vrt-{- tdif = {djc^/r -f di/^{f — 0, whose square root gives dxVr + dy^t = 0, ov ^=-/'-, or J=-/^; these, when the surface is developable, are dy _ r ^ dy _ s dx s dx~ t as before shown. dy V s Because -p=: ,or — -= the tangent of the angle which the projection of the line of contact on the plane a?, y, makes with the axis of x, it is clear, from what has been done, that the line of contact must be a right line ; which may be regarded as being the generatrvx of the developable surface. Hence the developable surface (by Monge called the envelope of the infiniteshnal tangent plane) may be consid- • ered as composed of plane elements of unlimited lengths and of infinitesimal, breadths, which successively cut each other in right lines. Hence the first of these elementary planes may be turned about its line of common section with the second, until its pLane is brought into the same plane with it ; and in like manner the plane thus formed, may be turned about the line of common section of the second and third elements, 222 developable: surfaces. until it is brouglit into the plane of the third element ; and so on to any extent that may be required. It is hence evident that a developable surface may be spread out on a plane without being torn or rumpled. Because (see pp. 212 and 213) the equations of cylindrical and conical surfaces are represented by op -\- hq = 1, and z' ^=px' -\- qy' -\- z —px — qy, it clearly follows, from what has been done, that they are developable surfaces ; since they evidently come under the form z =pX + ^Y -\- z—px — qy, in which the differentials of p and q are put equal to naught, X, Y, Z are constant, whUe a?, 3/, z are variable. Remarks. — If we assume z = x^ (a) + yip (a) + a to rep- resent the equation of a plane, in which (a) and i> (a) repre- sent any arbitrary functions of a ; then, by putting the first and second differential coefficients taken with reference to a equal to naught, we shall have the forms (see Monge, p. 85) a?0' (a) + yxj)' (a) + 1 = and a?" (a) + y^'' (a) = 0. Young (at p. 208 of his " Differential Calculus") says, that for a, in the equation of the plane, the function /" (a) ought to be used, since Mongers form excludes those forms com- prehended in the form z = X(f> (a) + yip (a) + c ; but this objection is clearly invalid, since, if we please, we may for z put s — («) H- a and xcp' (a) + yxp' (a) + 1 = 0, represents a developable surface. For the first equation in virtue of the second, gives DEVELOPABLE SURFACES. 223 ^ = ^ rrr (a), and -^ = q = ^{a)- consequently, since p and q are functions of a, we shall evi- dently liave p = 6{q) = SL function of q. Hence, we shall have ^ = ^ = ^ X '^^ = se'is), ay ax dq ^^' consequently, eliminating Q' {q) from these equations, we get rt — s^ = 0, the equation of developable surfaces, and, of course, the assumed equations jointly represent a developable surface. Kegarding a as being an arbitrary constant, that ought to be retained in the equations, it is clear that the equations may both be regarded as being functions of the characteristic, since the position of the characteristic clearly depends on a. Hence, the first equation being that of a plane, and the second that of a right line on the plane a?, ?/, it is manifest that the characteristic must be a right line, which is the same as the generatrix of the surface. (See Monge, p. 85.) If we eliminate a from the equations z = x(}> (a) + yi> (a) + a, x(}>' (a) -{- yrp' (a) + 1 = 0, and x(p'' (a) -f yf' (a) = 0, we shall clearly get two equations in terms of a?, y, and s, which will clearly be the equations of the line in which the intersections of the successive characteristics must lie, which must evidently be on the developable surface ; this line being called by Monge, the edge of regression of the envelope^ or developable surface. (See Monge, p. 85, and Young, p. 212.) 224 ILLUSTRATIONS. We will illustrate what has been done by one or two simple examples. 1st Let z = xi^ -{- ya + h represent the variable plane, to find the developable surface and its edge of regression. Here, by taking the differential coefficients relatively to a, we have the remaining two equations expressed by 2xa -{- y = 0, and 2x = 0. Hence, eliminating a from the first of these and the proposed equation, we get 4.x (0 - 5) + f = 0, for the equation of the envelope ; which will be found to be a developable surfaca If we eliminate a from the three equations, since 2a? = 0, we shall get s = 5, a point in the axis of Sj for the edge of regression. 2d. Let the variable plane he 2= xa? + ya^ + a ; then, the other equations are Zxd? + 2ya + 1 = 0, and ^xa -j- y = 0. Solving the second of these equations by quadratics, we get a = — — ± ^—~: ; which, substituted for a in the proposed equation, will give the envelope or developable surface. Also, eliminating a from the three equations, since the V 1 third gives a = — ^, we have Sx = y^ and y2= — -^ the first being that of a parabola on the plane a?, ?/, and the second that of an hyperbola on the plane y, 2, for the equa- tions of the edge of regression. 6. By a twisted surface we mean one described by a right line which is continually changing the plane of its motion. To represent such a surface, we shall suppose x = az -\- a' and 2/ = Js + 2»' to be the equations of the generatrix, which we shall suppose to be continually moving along three given TWISTED SURFACES. 225 curves of double curvature (or which do not lie wholly in the same plane) as directrices ; then, each of these curves will be expressed by two equations, when projected on the rectangular planes of x, ^, and <•/, z. Hence, if we eliminate a?, y, and s, from the equations of the generatrix by the equations of any one of the directrices, we shall have an equation involving a, J, a' ^ and V^ as un- known quantities ; consequently, if we eliminate a?, y, and ^, in like manner, from the equations of the generatrix by the equations of the remaining directrices, we shall have two more equations, each involving a, J, a', and 5', as unknown quantities. Hence, from the solution of the three equations thus found, any one of the quantities a, 5, a\ and h'^ as a, may be supposed to be taken for the independent variable, and each of the others to be a function of it. Thus, for a', J, and h\ we may put ^ {a\ (p (a), and 6 {a) ; which reduce the equations of the generatrix to the forms X = as + 'ip (a), and y = £(}){a) + (a), in which V', 0, and 0, that precede a, are used to denote any arbitrary functions of it ; so that if ip (a) is assumed to equal naught, {a) — a^, and 6 {a) = a^, our equations will become X — nz and y = a% + a^, which, by eliminating a from the second by the first, give ys^ = qi?^" + x^^ for the equation of a twisted surface. Generally, if a is found from one of the equations, X = az + i/^ (a) and y — z(f){a) + 6 (a), and its value substi- tuted in the other, z will become a function of x and y, or we shall have the equation of a surface, since ^ is a function of X and y ; consequently, as heretofore, we shall have dz = pds0 -h qdy^ in which x and y are the independent variables. 10* 226 TWISTED SURFACEa By taking the differentials of x = as -{- i> (a) and y = e:){a) -f 0{a), supposing a to be constant, we shall have dx = adz and dt/ = (f) (a) dz ; which show that, in taking tlie differential oi dz =pdx-[-qdi/^ when a is regarded as con- stant, on the supposition of the constancy of dx and dy^ or that X and y are the independent variables, we must regard dz as also being constant; consequently, we shall have, in this way, dpdx-\-dqdy = 0, in which djj and dq stand for the total differentials of ^ and q. Since (see p. 220) dp = ~- dx -\- ~ dy =z rdx + sdy and ^9.^=^ -f ^^ -^ -^ ^^'J = ^dx + tdy, we shall have the equation dpdx + dqdy = rdxP + 2sdxdy + tdy^^ = 0. Because dx and dy are constant, by taking the total dif- ferentials of this equation, we shall have d^pdx + (f-qdy — drdx" + Msdxdy + dtdy^ = 0. If we ut then, since and dr _ dx = u, dr _ dy ds ~ dx~ % ds dy~- dt ~d^ = w, and dt dy dr^ dr ~ dx dx + ds = ds "" dx dx + ds , dy'y^ dt = dt "" dx dx + dt ^ TWISTED SURFACES. 227 tlie preceding equation is reducible to dydar + drqdif- — drdx^ + Idsdxdy + dtdy^ « = udx^ + Zvdx'^dy + Zwdxdy^ + u'd]f' = 0, or, we sliall have an equation of the third order of partial differential co- efficients of twisted surfaces. From dx = adz and dy ^=^(1) (a) adz, by division, we get -—- = — -^ , which must clearly be the same as found from dx a -^ or Its equivalent y + _ _ = _ -, whose solution gives dy _ - s± i/{&^-rt) _ ,^ dx- t - "^ ' consequently, we shall have —^ = a' ^ or {a) = aa', and the remaining eqaation becomes, by substitution (see Monge, p. 198), u'a'^ + Zim"" + ^va' -\- u ^ ^. If the three directrices are so given as to enable us to find the forms of V («), («)' ^^^ ^ W? *-^^^ ^7 finding the value of a in one of the equations a? = as + V' {d) and y=-z4* {(') + ^ («)? and substituting it for «, in the other, as at p. 225, we shall have an equation in x, y, and 2, for the equation of the twisted surface, and what is called the integral of the equa- 228 TWISTED SURFACES. tion of partial differential coefficients of the third degree, given on p. 227 ; if, however, a can not be eliminated from the equations, the equations, in their undetermined form, must be taken for the integrals. KEMARK& — It is manifest that whatever may be the natures of the directing lines, we may proceed in much the same way, as has been done, to find the equation of the twisted surface described by the motion of the generatrix. SECTION IX. CURVATUEE OF SURFACES, AND CURVES OF DOUBLE CURVATURE. (1.) Curvature of Surfaces. — ^Let z' — z =:p ix' — x)-{-q{f — y) represent the equation of a tangent plane at a point of a curve surface whose co-ordinates are a?, y, and z] then, from what is shown at p. 207, »-X+^(2-Z) = and y -Y + q{z -Z) = 0, are the equations of the normal to the curve surface, at the same point. By taking the differential of the tangent plane, supposing a?, y, 2, alone to varj, we have dz ~jpdx + qdy^ and from ^^=^i^^ + |^2/ and dq = ^£dx^r^f^dy, or (see pp. 220 and 226), dp =: rdx + sdy and dq = sdx + idy. Taking the differentials of the normals, supposing X, Y, Z, not to vary, we have dx + fdz -]- {z — Z) dp — dx 4- p^dx + pqdy -\- (z — Z) {rdx + sdy) = 0, and dy + pqdx + q^dy -h {z — Z) (sdx + tdy) = ; which are equivalent to 280 CURVATURE OF SURFACES. l+y+(3-Z)r+to + (3-Z).]g = 0, and jpq + {z-Z)8 + [1 + f + {2- Z)f]'^^=0, dij Eliminating -j- fi'om these, we have [1 +y + (2 - Z) r] [1 + r + (2 - Z) i!] or (2 - ZY{rt - s") + (2 - Z) X [(1 + q')r-2j>qs + (1 +y) j5] +y + ^2 ^ 1 = 0; and the elimination of s — Z from the same equation, gives ©'[(i.^V-i>.O.J [(1 + q')r- (1 +i>^) ^] - (jy^ + 1) s +pqr = 0. These formulas may be much simplified by supposing the tangent plane at the point (a?, y, z) to be parallel or coinci- dent with the plane a?, y, imagined, to assist the imagination, to be horizontal, the concavity (or hollow) of the surface being turned upward, and the axis of z vertical, its positive dz dz value being reckoned upward ; then, ^ = — and q = -,— will evidently be reduced to naught, and the formulas will be reducible to and (fy + !:^^^_l=0. \ax/ s ax Solving these equations by quadratics, we shall have ^ 2 irt - s') CURVATURE OF SURFACES. 231 and dy _— jr-t) ± V{r — tf +"4.y' dx 2s which, on account of the ambiguous signs, clearly show that s — Z and -^ , each admit of two values. Because dy ^-^.-t) + ^/(r-tf + if dx 2s and ^' = -(r-t)-V{r-tf + i^ _ dx 2s evidently represent the natural tangents of the angles which two vertical sections of the surface, by planes through the axis of 2, make with the plane of the axes of x and s; by taking the product we shall have ~ x -^ = — 1, conse- quently, if A stands for the angle whose tangent is -^, A + 90° must stand for the angle whose tangent is -j- , ctx A J. / A . AAox sin A cos A ^ smce tan A x tan (A + 90 ) = r x -. — r = — 1, ^ ^ cos A — sm A and, of course, the two planes passing through the axis of z cut each other perpendicularly. If we represent z — Z by E, we shall have for the equation z-Z = -^ 2{rt-s^) the transformed equation 2 (rt - s') which clearly represent the radii of curvature of the pre- ceding perpendicular sections at their point of contact with 232 CURVATURE OF SURFACES. the plane of the axes of x and y. Representing these radii separately by R and R', and taking the sum of their re- ciprocals, we have E R' r + t-V{r-(f + 4:^ r + t-\- V{r — tf + 4v _, dp d^z 1 ^ do d}z Because t* = -f- = -7-^ and ^ = -r^ = — ^ , dx ddcr ay ay dx" dij and that -^3- and ~- are clearly the radii of curvature of the vertical sections of the surface which pass through the axes of X and y^ it ^clearly follows that 7' -\- 1 expresses the sum of the reciprocals of these radii. Consequently, since the position of the axes of x and y in the plane of x, y, is arbitrary, it clearly follows that the sum of the reciprocals of the radii of curvature of any two vertical planes through the axis of s, which cut each other perpendicularly, is always equal to -p + ^7 ; and of course the sum of the reciprocals of the radii of any two such sections, is always equal to the sum of the reciprocals of the radii of any other two. If, according to custom, we represent the curvature of the circumference of a circle by the reciprocal of its radius, we shall have the sum of the curvatures in any tvjo vertical sections that pass through the axis of 2, and cut each other perpendicularly^ equal to the sum of the curvatures iii any other two vertical sections that pass through the axis of s, and cut each other perpendiculariy. It is hence clear, that if the curvature in one of two perpendicular planes is a maximum, that it must be a minimum in the other plane. CURVATURE OF SURFACES. 233 and vice versa. It is also clear tliat E and E' — called the principal radii — are such, that the first is less than any other radius of curvature, while the other is greater than any other radius. If we take the principal sections for the axes of co-ordi- nates, then drj -{r-t)+ V{r- tf + ^s^ _ 25 dx 2s r-t-\- V{r-tY + 4.s' and dy' _ -{r-t)- V{r~tY + 4:S^ 2s_ dx 2s r—t— V{r — if + 4/- given at p. 230, may be taken for the tangents of the angles which a pair of perpendicular planes through the axis of z makes with the first of the principal planes through the axis of X. If the perpendicular planes are brought to coincidence with the axes of co-ordinates, we shall have ~- := ; and of course, from the first of the preceding formulas, we must have 5 = 0. Hence, putting s = in ^^rJ^-Vp^P^ and r, ^ r+^+ i/(.-.y+4,^ _ 2 {rt — s^) 2 {rt — 6'-) they will become E = - and E^ = - for the radii of cur- -^ r t vature of the principal perpendicular sections. Supposing E^' to stand for the radius of a perpendicular section through the axis of ^, which makes an angle whose tangent = ~ with the axis of a?, then we shall clearly have _,„ dm? + dip' dx^ + dip- d?z rdx^ H- tdy^ ' 284 CURVATURE OF SURFACES. since 5 = 0, or K ' = - r + t \dx) wliich, bj supposing -^ = tan 0, becomes ^„^ l+tan»(A _ 1 r -\-t tan- 7" cos^ + ^ sm- «/> whicli, by putting for r and ^ their values -^ and :p7 , is easily reduced to T>// 5^5 K' cos^ + K sin^ ~ W cos"-^ — R sin^ 1 _ cos* sin* W ~ ~R ~W ' noticing, that what is here done corresponds to a circular wheel with a groove in its circumference, R' representing the radius of the wheel whose convexity is turned upward, and R the radius of the groove whose convexity is turned downward, and its concavity upward. If R sin* <^ = R' cos^

we shall have 'R" = -jr- — infinity ; CURVATURE OF SURFACES. 235 consequently, tan = y =5- and tan =r — y — - indicate two riglit lines on tlie surface drawn to make angles with the axis of x, or width of the groove, passing through its center and making angles with it, whose tangents are |/ -5- and — y tp- on the positive and negative sides of x positive, and on the positive and negative sides of a? negative; the surface between these tangents being clearly concave, while the remaining part of it is evidently convex, so that the tangents separate the concavity and convexity of the surface from each other. Supposing the right line x' is drawn from the origin of the co-ordinates in the plane of x, y, to the surface, so as to make the angle \ ,._j then, by assummg z = [^^- ± -2^) ^ , since cos^ (px'^^ = x^ and sin^ (jix"^ = y^, ' we shall have ^ ^ Ie ^ ^ ' which is the equation of the surface of a paraboloid of the second order. T^ /cos"(/) . sin^ ± R sm^ ' and R'' is the radius of curvature of a vertical section of the paraboloid, at the point whose co-ordinates are x and y^ as it clearly ought to be. Hence we perceive how to measure the curvatures at any proposed point of a surface by those of the paraboloid, and 236 THEOREM OF MEUSNIER. that whether the principal curvatures have the same or con- trary directions. Eemarks. — 1. Resuming R'^ = ' ,^ ^ , from page 233, di^ and representing VdJ^ + dy"' by ds^ it will become R'' — -,^- ; which is the radius of curvature in a normal section to the ^ curve surface at its point of contact with the plane of the axes X and y. Suppose now a j^lane making the angle with the normal section, also touches dn at the origin of the co-ordinates ; then, it is clear that -z^, will be the radius of Cu Z curvature in the oblique section with the curve surface, in which d^z' corresponds to d^z taken in the normal plane. Because d?z' and d^z are clearly the hypotenuse and side of a right triangle having for their included angle, we shall d'z have d?z' cos Qz^d^z or d/z' — cos 6/' which reduces -^-7 to -rp x cos = R'^ x cos 6 ; drz d-z ' consequently, the radius of curvature in the oblique section equals the (orthographic) projection of R^' on the plane of the oblique section, which is called the Theorem of Mcusnier. 2. Resuming the equations of the normal from p. 229, and putting, with Monge, g=rt-^, h=={l + q')r-2pq8 + {l+f)t, and ¥=^f+q'^-l in the fourth equation at p. 230, we shall have aj-X+^(s-Z) = 0, 7/-Y + ^(5-Z) = 0, THEOREM OF MEUSNIER. 237 and (^-Z)^ + -(3-Z) + - =0, whose solution ogives s — Z = -, — ■ — -—7^ ir^nk • ^ ± \\'(''' — ^gKT) Supposing E to be the radius of a sphere that touches the curve surface at the point (a?, y, 2), having X, Y, Z, for the co-ordinates of its center, we shall have for the equation of the spheric surface, which, by substitu- tion from the equation of the normal, becomes E;.:. {f + ^2 + 1) (3 - ZJ ^¥{z- Z/; consequently, from the substitution of the preceding value of s — Z in this, we readily get p _ 2^^ A ± |/(^' - ^g^") ■ for the two radii of curvature at any proposed point of the curve surface. To illustrate what has been done, we will apply it to find the radii of curvature of the surface, whose equation is > = f- Here we have -f^ dx =^ = |- and dz _ dy~ ■.q = X ^ A' which give B r^ + f + A^ ~ A^ and from d^z __dp _ _ d^ ~~ dx~ ~ 0, since p is not a function of x^ and, in the same way, dq_ dy 2^ = 0, but 5 = ^: dy dx 1 ~A ? 238 THEOREM OF MEUSNIER. wbicli give h= — 2pqs = ^ , and they also reduce g = rt — ^ to — 5^ = — -— . Hence the equation (2 — Z) s + - (2 — Z) H — = 0, is easily reduced to (2_Z/ + ^^(2-Z) = ^ + y' + A', ■whose solution gives „ -mi± 4/[A* + A' ( a;' + y') + !^ f\ z-Z= J , or Zi — s = ^^ which gives or R = ■ (A^ + a^ 4- 2/^)^ for the expression of the radii of curvature, at any proposed point of the given surface. If in the preceding value of R we put a; = and y = 0, we get R = ± A or R = A and R = — A for the radii of curvature at the origin of the co-ordinates, which is clearly that of the vertex of the given surface ; since these radii have contrary signs, it is manifest that the principal curva- tures of the surface at its vertex, are turned in opposite directions, and it is manifest that like conclusions are ap- plicable to any other point of the proposed surface, but their magnitudes are not equal, as at the vertex. CURVES OF DOUBLE CURVATURE. 239 It is easy to perceive, that by making analogous substitu- tions to those made in the equations for the radii vectores in the quadratic equation in -~ given at p. 230, we may, after the manner of Monge, at pp. 121, &c., of his "Application de r Analyse a la Geometric," proceed to find the integral of the equation, and thence to trace out the lines of curva- ture on any proposed surface, together with the correspond- ing radii of curvature. We shall not, however, attempt to do this, but shall satisfy ourselves with the following obser-, vations. Thus, from what is shown at p. 230, it results that there are, at any point of a curve surface, tw© lines of curvature at right angles to each other, such, that the successive nor- mals to the surface in each intersect each other and form a developable surface; the line in which the successive normals intersect being called the edge of regression of the develop- able surface^ while the lines in which the developable sur- faces cut the proposed surface are called lines of curvature, (2.) Curves of Double Curvature. — In treating of curves of double curvature, it will be sufficient to regard them as consisting of indefinitely small arcs, regarded as straight lines ; such that (in general) no more than two successive arcs can lie in the same plane. Suppose then x — x'-^K (y — y^) + B {z — z') = 0, to represent the plane of any two successive sides of the curve, having x, y, z, for the rectangular co-ordinates of the first extremity of the first of the two successive sides, and x\ y\ z\ for the co-ordinates of any other point of the plane ; then dx^ dy^ and dz^ being the differentials of x^ y, and ^, we shall have dx + Ady + ^dz = 0, when we pass from the co-ordi- nates of the first extremity of the first (short) side to those 210 CURVES QF DOUBLE CUBVATUBE. of its second extremity, or those of the first extremity of tlie second side ; and in passing from the first to the second extremity of the second (short) side, we in like manner get cT-x + AcPy -f- BcT'z = 0. From the last two of these equations we get . _ dzcPx — dxd-2 , TK — ^^^^'y ~ ^y^^ . ~ dyd^z — dzd^y ' " dijd-z — dz^-y ' and from the substitution of these values of A and B for them in the equation of the plane, it is readily reduced to the form {x — a?') {dy^z — dzdry) + {y—y') {dzd/x — dxdrz) + {z — z') {dxd-y — dyd^x) = 0, which is sometimes called the osculating plane of the point (iP, y, 4 Supposing R to he the radius of a circle passing through the extremities of the same two successive short sides, and that the point (a?', y\ z') is taken at the center, we shall, from the nature of the circle, have the equation W = {x - xj 4- Cy - yj + (^ - ^y for the first extremity of the first short side; which, in passing to the second extremity of the first side, gives {x — x')dx + {y — y') dy + {z — z') dz = 0, and this, when we pass to the second extremity of the second side, gives {x - x') drx + {y- y') dhj-^{z - z') d'z + dx" + dif + dz'. If ds represents the length of the first side, since dx^ dy^ dz^ arc clearly the projections of fZ.9 on the axes of a?, y, and ^, it is easy to show that we must have dsr + dy- + dz^ — ds^ ; CURVES OF DOUBLE CURVATURE. 241 consequently, the last of the preceding equations becomes (x - x') d'x + (y - 2/0 ^V + (^ - ^0 ^'2 + ^«' = 0. Bj successively eliminating z — z'^y — y\ and a; — a?', from the preceding equation, by means of the equation {x — x') dx + {y — y') dy ^{z — z') dz = 0, we have the equations {x — x') {dx(^z — dzd'^x) -{- {y — y') {dydh — dzd^y) = dzds% {x — a?') (dxd}y — dyd}x) + (z — z') {dzd^y — dyd^z) = dyds% and {y — y') {dyd^x — dxd^y) -\- {z — z') {dzd^x — dxd^z) = dxds\ Hence, supposing x\ j/, z', in the osculating plane, to corre- spond to the center of the circle, by adding its square to the squares of the three preceding equations, because the double products destroy each other, we shall have, since {x-xy + (2/ - yj + {b-z'Y = R' and dz^ds^ -f dy^ds'' + dx'ds^ = ds\ ■D2 _. {dxd^y — dyd^xf + {dxdC-z — dzd'xf + {dyd^z — dzd?yY * for what is sometimes called the square of the radius of the absolute curvature, corresponding to the point (a?, y, z). From the development of the squares in the denominator, and omitting the factor dx^ + dy^ + dz^ = ds^^ that is common to the numerator and denominator of the resulting fraction, we have R° = ,.-> >, , ^02.^0 -irz. d~x- + d-y^ -f a-z- — d's^ If ds equals its successive side,^ ds is constant, and d^s = 0] consequently, we shall have ~ d'i^ + dy -{- (fz"^' 11 242 CURVES OF DOUBLE CURVATURE. whicli can readily be deduced from the simplest principles of geometry, and that, whether the proposed curve is of single or double curvature. Thus, lei ah ^zhc — ds represent two successive sides of the polygonal curve, having the circular arc whose center is (x) dx ~ dx dx~ dx '' equal to P and Q, the equation will be more simply expressed by x-x'-\-{y-y')V + {2-z')Q, = 0. Representing -^ and ~j- by P' and Q', and taking the dif- ferential of this equation, we have (y - 2/0 P' + (^ - ^') Q^ + F + Q^ + 1 = 0, and from this, by putting -y- and -y- equal to V and Q'', we in like manner get (y - yO V" +{z- z') Q'' + 3 (PP' + QQO = 0. The first two of these equations, since they represent planes perpendicular to the first and second short sides, and that they intersect in a right line, clearly show the character- istics to be placed on a developable surface, or to have a de- velopable surface for their envelope. And it is evident that the three equations together represent the edge of regres- sion, or the line in which the successive straight lines (that characterize the developable surface) intersect, which is evi- dently the line in which the centers of spherical curvature lie. It may be noticed that if the origin of the co-ordinates is taken at a point in the curve having the axis of x for a tan- gent at the origin, the preceding formulas will be much simplified. For, at the origin we shall have a?, y, 2, each equal to naught, and P and Q are also reduced to naught, or be- come infinitesimals. Hence, we shall have CURVES OF DOUBLE CURVATURE. 245 x' = 0, 2/T' + b'Q' = 1, and yT^' + zW = ; consequently, we sliall have P'^2 _|_ Q'/2 and thence W = y"' + 2'' reduces to R- = /p rj^^ _ p//n/\ 2 • Again, if in the first two of the formulas we put -0 {x) and 9 (a?) for y and s, and eliminate x from the results, we shall obtain an equation in x\ y\ and z\ for the equation of the developable surface noticed above. li 3 — z' is eliminated from the same three equations, and ^ {x) is put for y in the two resulting equations, then the elimination of x from these equations gives an equation in terms of y' and x' for the equation of the projection of the edge of regression on the plane of the axes of x and y. Similarly, by first eliminating y — y' from the equations, putting (f> {x) for 2 in the results, and eliminating a?, we shall get an equation in x' and 2' for the equation of the projec- tion of the edge of regression on the plane of the axes of X and 2. If from any point in the common section of the first two perpendicular planes a straight line is drawn in the second plane to the second short side of the curve of double curvature, and produced in the opposite direction to meet the line of common section of the second and third planes, and a line drawn from the point thus found to the third short side of the curve, and produced in the opposite direction to meet, as before, the line of common section of the third and fourth planes, and so on ; then, a straight line drawn from the first (assumed) point to the first short side, and continued as a curve in the opposite direction 246 CURVES OF DOUBLE CURVATURE. through the points found, will clearly represent the evolute of the proposed curve, regarded as its involute. (See Sec. VI. p. 170.) It is hence easy to perceive that the proposed curve of double curvature has an unlimited number of evo- lutes. (j[y^ y y^ It is manifest that -j-y = —, represents a tangent of an evolute of the proposed curve, when projected on the plane of ar, y, and that by eliminating z — z' from the equations x-x' + (y -2/') P -f (5 _ s^) Q=: and {y-y') P' + (2 -^0 Q + P + Q^ + 1 - 0, ' and putting i/> (a?) for y in the resulting equation, and in «¥ ^ y-y ' dx' X — x' ^ then, eliminating x from these equations, we shall obtain a differential equation in x' and ?/', whose integral, according to the principles of the Integral Calculus, will involve one arbitrary constant The constant will enable us to make the evolute pass through any proposed point ; for if the co- ordinates of any point a?', y\ and z\ are represented by a^ h, &c., by putting a and 7j for a?' and y' in the integral, we easily get the value of the constant, and thence the integral is determined, so that the projection of the evolute on the plane x, y, passes through that of the proposed point ; and in a similar way the projection of the same evolute on the plane a?, s, may be determined, and the evolute will be found as required. For an example we will put y = x and z = x^ for y = V' (^) and s = (a?), which give ^ = P = 1, ^=Q=.2x, ^ = = F\ ^ dx dx ^ dx ' CURVES OF DOUBLE CURVATURE. 247 ^ = 2=:.Q', F^ = 0, and Q'^ = 0. Hence tlie formulas given at p. 244 become x — x'-\-y — y'-{-^x{z — z') = 0, 2 — s' + 2i»2 + 1 = 0, and Q = 2a?=:0, of whicli tlie first two give tlie developable surface, and all tbree give the edge of regression. If tlie values of y and z are put for them, in the first two of these equations, they will become 2x — x' — y' + 2x^ — 2xz' = and 1 — z' -\- Sx"" = ; consequently, putting a? = in these equations, we have a?' -j- y^ — and z^ = 1 for the equations of the edge of regression. Eliminating x from the first two of the preceding equa- "^^—^ — I = I — x^^ / ^^^ ^^^ equation of the developable surface, which clearly shows that z^ must be positive, and not less than 1. we put X for y, we shall have dy' {x — x'') = dx' {x — y'\ for the equation of the projection of a tangent to an evolute on the plane x^ y. And eliminating z' from the equations "Ix — x'-y'^ 2a?^ — 2xz' = and 1 — z' -{- Zar =^ 0, we get = -f^^f; consequently, putting this for x in the preceding differential equation, it becomes 248 POIITTS OF INFLECTION. for the differential equation of the projection of an evolute on the plane a?, y. Hence we must find the integral of this equation and determine its constant, so as to suit the nature of the case. It may be added to what has been done, that if a curve of double curvature has single curvature at one or more of its points, it is said to have lost one of its curvatures, and to have a single inflection at such points ; while, if it loses both of its curvatures (or becomes rectilineal) at one or more of its points, it is said to have a douhle inflection at such points. It is manifest from the definition, that the points of single inflection in curves of double curvature, may be found by putting the expression for the radius of spherical curvature equal to oo , or by putting its reciprocal equal to 0. p//2 I Q//2 Thus, by putting t^ojty' T>"C\'\^ *^® expression for E^, given at p. 245, equal to infinity, or putting its reciprocal equal to naught, we have P'Q'' — P"Q' = 0, which, from what is done at p. 244, is the same as to put drydPz — il^y(Pz = 0, or to find the points of the curve which satisfy the equation d^2 _ d^y d}z ~ d-y ' If this equation can not be satisfied at any point of a curve of double curvature, it clearly can not have a point of sin- gle inflection ; while, if it can be satisfied at one or more points of the curve, it is manifest that the curve may have single inflections at such points. To find the points of double inflection, we put the radius POINTS OF INFLECTION. 249 of absolute curvature of tlie curve of double curvature, cither equal to or oo ; consequently, from tbe expression for E"^ given at p. 243, we bave c^V + ^y + ^V == or 00. Since tbis expression bas been obtained on tbe supposition of tbe constancy of ds^ = dx^ + dy^ + dz'^, we put tbe differ- ential of tbis equal to naugbt, wbicb gives dxd^x-^dyd'y-^dzd'B^O or drx = - ^^^^-±^^ ] wbicb, put for cPx in tbe preceding equation, gives d^f + cT-^^ + (^^y + '^''P'J = or 00 . Because tbis expression consists of tbe sum of tbree squares, it is evident tbat we must satisfy it by putting eacb of its terms separately, equal to or oo ; consequently, tbese conditions will be satisfied by dy = or d^y = co ^ and d^z =0 or d^z= a:). Tbese conditions clearly follow from tbe projections of tbe curve on tbe planes a?, y, and a?, 2, wbicb will manifestly be plane curves baving (eacb) tbe same number of points of inflection ; consequently, from tbe rule given at p. 156, we must bave d^y „ d'^y dx . , d'^z ^ dx ^ --^=Oor-y^=oo or-^ = and — = or -^ —^, ax dx ay dx d-z wbicb are clearly equivalent to tbe preceding conditions. 11* INTEGRAL CALCULUS. INTEGRAL CALCULUS, SECTION L (1.) The Integral Calculus is the reverse of the Dif- ferential Calculus; the object being to find the function called the integral^ from which any proposed differential may be supposed to have been derived. Thus, since 2xdx and nx''~'^dx are the differentials of ix? and a?" or (more generally) of x'^-\-o and «" + J x^dx or J {aa^dx + hx^dx) —J ^dx, &c. (7.) It may be added, that the arbitrary constants in in- tegrals, are (generally) to be determined so as to satisfy certain conditions which the integrals must answer. Thus, if the integral / {Sx-dx + 6x*dx) must equal naught when X = a, we proceed as follows. By integration we have fiSx'dx + 5x'dx) = x' ■i-x' + C] IJ^EGRAL CALCULUS. 265 consequently, putting a for x in this, we must, from the con- ditions of the question, have ^-^ + a^ + C = 0, which gives G = — {a^ -\- aF) for the value of the constant Hence the integral, duly corrected, becomes /( {^x'dx + bx'dx) = u? -\-x'— {w" + a') ; and it is manifest that we must proceed in like manner in all analogous cases. To signify that an integral, as / {axdx + hfdx — ca^dx) is to be taken from x = A to a? = B, we write /{axdx -f- hx^dx — co^dx) = -^ -\- — 7- + ^» A Z o 4: which, by putting A for a?, gives ^ (aA' hA' cA'\ ^ = ~v^~^"3 — r)^ and thence the integral becomes {axdx + bx'dx — ca^dx) ~2'^3 4 V2"^ 3 4"r which, by putting B for aj in its right member, becomes / {axdx + ha^dx — ca^dx) oB^ hW_cB^_ /oA^ hA^ _ cA*\ 2 "^ 3 4 I 2 "^" 3~ ~T) = |(B^-A^)-f|(B3-A>^)-|(B^-A0, which is called a definite integral^ because a?, in its right member, is determined; consequently, when an integral is 12 . _ * /: 266 INTEGRAL CALCULUS. taken from one value of its variable to another value of its variable, the integral is d^'fiiiite or detecmined^ otherwise the integral is indefinite or not fixed, (8.) When the integral of a proposed differential is fcund, it is said to he integrated ; and when the integral is taken from one value of the variable (a?) to any other proposed value, it is said to be integrated from the first to the second value of the variable. (9.) To aid in what is to follow, and to show the natures of differentials and integrals more fully, we will now pro- ceed to give the solution of the following important PROBLEM. If a; and y —fix) = a function of a?, represent the abscissa imd corresponding rectangular ordinate of a plane curve, it is proposed to show how to find the area bounded by the ordinate drawn through the origin of the co-ordinates, by any other ordinate, and the intercepted parts of the axis of x and the curve : supposing the ordinate to be constantly posi- tive between the preceding limits. It is clear that we may suppose f{x) to be expressed by A -h Baj« + C^* + Dxf" +, &c., in which A, B, C, &c., a, J, c, &c., are independent of a?, which, for simplicity, we shall suppose to be positive, and that a?", a?*, «*', &c., are arranged according to the ascending powers of x. Let then, in the figure, be the origin of the co-ordinates, and suppose 04 represents any abscissa, and 4^ the corre- sponding ordinate ; we propose to find the area or quadrature of the curve, bounded by the ordinates Oa and 4e = y^ the abscissa 04 = a?, and the portion of the curve ae. Suppose a? to be divided into any number {n) of equal PEOBLEAf. 2G7 parts at the points 1, 2, 3, &c., and let oc' represent any one of these parts; then, the ordinates corresponding to the points 0, 1, 2, 3, &c., may evidently be expressed by A = y\ A -1- Ba?'« + Qx"> + &c. = y", A + B {^xj + C (2^7 + &c. = y'", and so on to A + B {nxj + C (?i£t'7 + &c. == 2/"'''- Allowing the rectangles to be drawn as in the figure, it is easy to perceive that the sum of all the inscribed rectangles will be expressed by (y' + 2/^' 4- . . . . + y"") x' = Anx' + B [1 + 2" + 3« + + {n — 1)^] x"'^'^ + C [1 + 2* + 3* + . . . . + (n - 1/] x"^^ +, &c., as is manifest from the principles of mensuration, while the sum of all the corresponding circumscribed rectangles will be expressed by 268 INTEGRAL CALCULUS. (/' + /" + .... +2/""0«'' = Knx' + B [I + 2-^ + 3« + .... + ^T a;'°+^ + C [1 + 2* + 3* + . . . . + n*] x"'^^ +, &c. It is easy to perceive that the difiference between the pre- ceding sums of the circumscribed and inscribed rectangles is expressed by (y"+^-y>'=B;iV +H C/iV*+^ 4- &c. = (B.r« + C.r'' + &c.) aj', since nx' = x. If x' is unlimitedly small, the difference is evidently also unlimitedly small ; consequently, since the difference is clearly greater than the difference between the sought area of the curve and the sum of all the inscribed or circumscribed rectangles, it is manifest that, by taking x' sufficiently small, the sum of all the inscribed or circum- scribed rectangles may be made to differ from the sought area of the curve by a difference which shall be unlimitedly small. (See Lemma IL, Book L, of Newton's " Principia.") It is clear that what has been done holds good, whether ae is a curve or straight line, or even if it is a curv^e whose con- vexity is turned toward the line of the abscissae or the axis of x. We now propose to put the above expressions for the sums of the inscribed and circumscribed rectangles under more useful forms. By putting n — 1 = ti', it is clear that we may assume 1 + 2« + 3« + . . . . 4- n'- = P^i"^-^^ + Q/i'" + 'Rn'--' +, kc, and suppose P, Q, &c., to be independent of n' and 1 + 2^4- 3*+, &c., clearly admit of like representations. By changing n' into 7i' + 1, and subtracting the assumed equations from the results, we get the identical equations PROBLEM. 269 P [(a + 1) W<^ + (^l)f ^/a-i + &c.] + a. [an^«-i + ^(^"Zi) ^,'a-2 4_ &c.] +, &c., and so on; of course, by equating the coefficients of like powers of n' ^ in the members of the equations, we readily get ~« + l' ^~2' 8.4' ' 1.2.8.4.5.6 ' &c. ; and so on, for the other representations. From the substitution of the preceding values of P, Q, &c., by putting for n' its value n — 1, and expanding the powers of 72. — 1 according to the descending powers of 7i (as hereto- fore) by the binomial theorem, we get 1 + 2« + 3« + ... + {n-ir - ^^ + ^.^ + &c. ^ a-\-l 1.2 ^a+l ^ ^a ^^^a-1 ^ (^ _ 1) (^ _ 2) n«-l a + 1 1.2 ' 8.4 1.2.8.4.5.6 and by changing a into J, 6^, &c., we get the corresponding representations of 1 + 2* + 3* + . . . + (ti — 1 )'', and so on. It may be proper to notice here, that the numbers Q = — r-^ , R = — - , and so on, called the numbers of l.Z {5.4: (James) Bernouilli, may easily be calculated to any extent, by solving the equations "^ 1.2 "^ 1.2.3 ^ 1.2.3.4 ~ ' and so on. (See p. 98, Vol. Ill, of Lacroix's "Traite du Calcul Differentiel," etc.) 270 INTEGRAL CALCULUS. Hence, from the substitution of the preceding values in the expression for the inscribed rectangles, at p. 267, we shall have ' Ar' 1 1 tI " O f ^ + ^ ^''^')" + ^ ^^"^'^^ + ^'-^ ''' -^%A [aB {r^y-^ + IQ {nxj-^ + &c.] x- - y:^^^ \a{a-l) {a-2)B(nx'y-^ ^Jj{h-1) (b - 2) C {nxy-' + kc] x'^ + &;c. = (since nx' = x) Ax -\ ^ + -^ — - + &c. ^ ^ a + 1 h-\-l [aBx<^-' + hCx'-' + &c.] x'^ - y;^jjq [a{a-l){a-2yBc(f-' + h{b-l){h-2)Cx'-' + kc.']x'' + ,kG- ' If, see p. 268, (B:»« + C^* + &c.) x' is added to the right member of this equation, the sum will express the circum- scribed rectangles, and we shall have (y" + y'" + .... + r*') x'=.Ax + ^^ + "f-^ + &o. Ay 1 .' + i^ [aB.«- + hCx^-^ + &c.] .'^ - j^L__ [a(a-l)(a-2)B^^-3-f2>(?y-l)(7>-2)0.iJ*-3+&c.] .»'' + , &c. It is easy to perceive that the part 6 + 1 Baj'^ + 1 Cx' of the inscribed an 1 circumscribed rectangles, which is inde- PROBLEM. 271 pendent of x\ or does not depend on tlie number of equal parts into whicli 04 == a? is supposed to be divided, must express the area of the curve bounded by the ordinates Oa and 4^^, and the intercepted parts 04 and ae of the line of the abscissae and curve, as required ; it is also evident that the terras in the rectangles, which involve x' and its powers as factors, must depend on the number of equal parts into which 04 = a? is supposed to be divided. T^ , , Bx^nx' Oxhix' J^rom Ana? + — \- -^ q- +,&c., (X + 1 /> + 1 which is the first form of . , Ba;«+i , Qx'^\ , - it is clear that (A + Bx" -f Ce»^ + &c.) x' =:f{x) x' == yx' is equivalent to the differential of the curvilinear area 046a, and may be expressed by writing dx for x' ; noticing that the x' here used need not be the same as the x' in the other terms of the rectangles described above. Also, multiplying by n, and putting nx' == ndx ■= a?, which gives is clearly the same as the integral of the preceding differen- tial, since the results are found by measuring the index of x in each term of the diiferential by unity or 1, and dividing by the measured index of a?, which is in conformity to the com- mon rule for finding the integral of the differential of a power. It is hence clear that the Differential and Integral Calculus are deducible from what has been done, without using infin- itesimals or limiting ratios. [See (17) at p. 44.] It is hence easy to perceive in what sense the Integral 272 INTEGRAL CALCULUS. Calculus may be regarded as being tbe reverse of the Differ- eatial Calculus, and vice versa. Representing A + Ba^ -h Ca?* +, &a, by y, tbe expression for tbe sum of the inscribed rectangle, becomes (y' + S^'' + .... + y")aj' = Jy<^ + -^ - ^2 + 3.4 ^ ^ 1.2.3.4.6.6 c^a^ ^ +' ^''- ' since aB.'-' + 50,.- + &c. = ^(A + B.° + & cO ^ I _ and [a (a-1) (a-2) Bir«-H J (J-1) (5-2) C»*-'+ &c.] = ^^^ [A + Baj« + Caj* + &c.] -f- 6/aj« = ^^ , and so on ; a form which is substantially the same as given by Lacroix, at p. 107 of his work, from very different principles. By adding {y—y') x' to the right member of the preceding equation, we shall have {y" -\- y'" + + y) «?', the sum of the circumscribed rectangles, expressed by It may be added, that in the inscribed rectangles y' is the first ordinate and y^ the last, while in the circumscribed rect- angles y" and ]p + ^ are the first and last ordinates. If the preceding equations are divided by x\ and ly is used to express the sum of the ordinates, taken according to the preceding directions, we shall have Jydx y' 2/1 dy ^> 1 ^y ^y~ x' ■^1.2 1.2 "^3.4^2^ ^~1.2.3.4.5.6c^^^""^'^''-' and r ly -ill ^ VL j^ y^ j^ 1.^^' ^L___^'3, ^c . •^ ~ x' 1.2 + 1.2 + 3.4 dx"" 1.2.3.4.6.6 ^' ^^' ' PROBLEM. 273 if we add y to tlie members of the first of tliese equations and y' to tliose of the second, and write S before y to signify the sum of all the ordinates, then the two equations concur in giving ^y^Iy^y - x' ^ "l ^ZAdx"^ 1.2.3.4.5.6 ^^-^ +'^'^- This formula enables us to find the exact or approximate values of series, whose terms follow a given law of forma- tion, and are equidistant from each other, or have equal in- tervals between them. Thus, to find the sum of the series 1^ + 2'^ + 3' + ....+ a?^ we have y ■= x^, called the general term of the series. Hence, Jyd,e = ^-\-C, J = 2a?, and ^| = 0, and since the difference of the successive terms of the series 0, 1, 2, 3, &c., equals 1, we put 1 for x', and ^y becomes — S" + ^^ + C, the arbitrary constant C being = since the value of y\ which corresponds to in the series 0, 1, 2, 3, &c., is equal to ; consequently, % is reduced to X 3 2 ' 2^3 + .0, to which, adding y = x^, we have _ _ _ _ , for the sum of x terms of the proposed series. In like man- ner, to find the sum of the series 1, 2^, 3^ ar', we have 12* 274 INTEGRAL CALCULUS. y' — and y = xr^^ and thence ^« i + 0, ..a |. W, g= 1.2.3, Jj = 0, te / Hence, the formula -J X' ^ 2 +3.4^a."^ 1.2.3.4.5.6 c/^'^ ^ +'*''•' since x' = 1, gives % = ^ + 2 "^ ^ ~ 120 "^ ^' in which C is the arbitrary constant To determine C, since Sy=0 when x= 0, by patting a? = we get = — j^ + C, which gives C = zr^-z ; consequently, we shall have for the sum of x terms of the proposed series. (10.) It clearly follows, from what has been done, that the diiferential of a function of a single variable as x, being of the form f (x) dx, by putting fx = y becomes f (a?) dx = ydx ; which may, if we please, represent the differential of the area of a plane curve, whose ordinate corresponding to the ab- scissa .T, is represented hy y =f [x). Thus (see the fig at p. 267), if Sd = y —f{x) and 3, 4 = dx^ the product ydx =zf{x) dx = the area of the rectangle 3(^D4, which may represent the differential of the curvilinear area to the right of Sd; consequently, the area to the right of the ordinate Sd, is the integral of the differential, supposing it to commence at the point where the curve cuts the axis of x. If ydx =f {x) dx is the differential of some known func- tion of a?, the integral / / (x) dx can be immediately found ; PROBLEM. 275 but if/*(^) dx, can not readily be reduced to tbe differential of a known function of x^ then j f{x) dx^ I ydx^ being reduced to the integral of the differential of the area of a curve, will, from the sum of the inscribed rectangles, given at p. 272, become (after a slight reduction) / ydx=.{^' + y" + ...,+y^)x' _ g' + g _ J^ dx"^ ^ 1.2.8A5.6 (^^ '^''•' a formula that will enable us to find an approximate value of the proposed integral, when x' is sufficiently small, par- ticularly when taken in connection with the integral fydx =. (/'+ y " + . • • + 2/ or y"^') ^'^ + rl - H - ^ ^.»^ + 1.2.8.4.5.6 (^^-^ '^'^•' deduced from the formula, given at p. 272, for the sum of the circumscribed rectangles. To illustrate what is here said, we will show how to find dx T when taken from the limit x = to the limit x=l, or be- z — ^ — ^ . [See (7.) at p. 2 64. ] Here y = z ^ , which, by putting x = 0, gives X ~f" X and putting a? = 0.1 or cc- = 0.01, gives ?/' = ^-i^ =0 990099+; /dx ' ^ by the first of the preceding formulas, J. -p X 276 INTEGRAL CALCULUS. X = 0.2, gives y'" = rSi "^ 0.961638+ ; and so on, to a? = 0.9, which gives y^" =0.552486+. By adding the ordinates, we have 2/' + 2/'^ + .... +2/^^=8.0998+; and since the ordinates are drawn at intervals of 0.1, wo have a?' = 0.1 , and hence get (y' + 2/'' + .. . . 4- 2^10) x' = 8.0998 x 0.1 = 0.80998 +. Also, _,|l^-=_0.05 and ||' = 0.025, since for y we must put ^ ^ = ^ , the last value of y. And we have dy _ J 1 , _ 2x Tx-"^' rr^ - ^^^ - - (iqr^y' which, by putting 1 (the last value of x) for a?, gives ^ _ _ 1. dx~ 2' consequently, - -— ^ x'^ = 0.000416 +. 0.4 cLx Hence, rejecting the remaining terms, on account of their com- parative minuteness, and adding the terms found, we have -^ = 0.80998-0.05 + 0.025 + 0.000416=0.78539+. ol + ar From the third form of the table given at p. 257, by put- J equals the length of an arc of 45° of the circumference of a circle PROBLEM. 277 whose radius =1, wliicli is well known to be 0.78539+ ; consequently, tlie arc has been correctly found to five decimal places, by a calculation of remarkable simplicity. By draw- ing the ordinates sufficiently near each other, it is clear that we may in this way find the circumference correctly to any finite number of decimal places. For a curve such, that the differential of its area is that of an integral of a known form, we will show how to find the area of a parabola. Thus, let ax = y- represent the equation of the parabola ; then, by taking the differentials, we have dx = ^ , which gives ydx — — — , whose integral is from the equation of the curve. To find the constant C, we shall suppose the area to com- mence at the vertex of the curve; then, x = gives / ydx =: 0, and, of course, we shall have = 0, and the area becomes / ydx = - xy = two-thirds of the semi-par ah- olaJs circmnscrihing rectangle ^ agreeably to a vjeU-Jcnow7i j^roperty of the parahola. (11.) Eesuming the figure at p. 267, and supposing it to revolve about the axis of x or 04, it is manifest that the curvilinear area will describe a portion of a solid of revolu- tion ; and that the inscribed rectangles will describe cylinders inscribed within the solid, while the circumscribed rectangles will describe cylinders circumscribing the solid, such that the solid will be greater than the sum of all the inscribed 278 INTEGRAL CALCULtJS. cylinders, and less than the sum of all the circamscribed cylindei*s. If TT = 3.14159, &c., the cylinder generated by the revolu- tion of the rectangle 01 Aa will, by mensuration, be expressed by "ny'-x'^ and in like manner all the remaining inscribed cylinders may be expressed. Hence, if y, y\ y'\ &c., are changed into ttz/^, ttz/'-, rry"^, &c., the formula for the sum of the inscribed rectangles, at p. 275, will become Jiry^-cU = -nffdx = tt \{:y" + y"'+ . . . . + 2/"') x' - Q^ -f 1.2 3.4 dx ^1.2.3.4.5.6 ^^ ^ ^^' the formula for the sum of all the cylinders inscribed in the portion of the solid of revolution ; and in much XhQ same way, the sum of all the cylinders which circumscribe the solid may also be found. Noticing, that this process will be unnecessary when the integral expressed by / y^dx can readily be found. Thus, in finding the contents of the paraboloid described by the revolution of the parabola ax = y^ about the axis of X. Since dx = ^ , we have iMx = -^-^ , and thence we get which equals half of the cylinder which circumscribes the paraboloid ; noticing, that no constant is necessary, since the paraboloid equals naught when x = 0. For another example, we will show how to find the con- tents {o?' cuhature) of a sphere whose radius equals R LENGTHS OF CURVES. 279 From wliat is shown at pp. 210 and 211, it is manifest that if the sphere is cut by a plane whose perpendicular distance from the center is x^ the section will be a circle, such that E- — »^ will equal the square of the radius of the section ; consequently, 7r (R2 _ rji^^ dx^Tx (:Wdx - xHx) is clearly the differential of the portion of the sphere, between the cutting plane and a parallel plane passing through the center of the sphere. Hence, by taking the integral from a? = to a? = E, we have for half the sphere ; consequently, the contents of the whole sphere is — - E^, which clearly equals two-thirds of the cir- o cumscribing cylinder. (12.) We now propose to show how to find the lengths of plane curves. Thus, let AB and BC represent the abscissa and ordinate of any plane curve AC, having A for its vertex, which we shall take for the origin of the co-ordinates, supposed to be rectangular. Then, representing the arc of the curve AC by 2, the abscissa AB and ordinate BC by x and y, wc may clearly take the very short line Q>^ parallel to AB, to stand 280 LENGTHS OF CURVES. for dx^ the differential of a?, and st parallel to BC or y, meet- ing the tangent to the curve at C in t^ to stand for dy^ then it is clear that C/, the hypotenuse of the right triangle C*^, must equal dz^ the differential of 2, or we shall have for the differential of the arc AC = z. What is here affirmed, is clear from the definition of a tangent given at p. 125, which irs that the differential co- efficient -^ at the point C in the curve must be the same as in the tangent; consequently, using Gs to represent dx^ st must represent dy^ and thence C^ must clearly represent dz^ as above. Because the approximate method of finding the integral of the differential is sufficiently evident from what has here- tofore been done, we shall not stop to give it. Thus, to find the length of the curve whose equation is y' = ax^^ called the equation of the semicubical parabola, by taking the differentials, we readily get dy= - a^x ^dx, and thence dz = (x^-\-^a^\x~^dx) whose integral is . = (.*^^f+c, C being the arbitrary constant If x and z equal naught at the origin of the co-ordinates, we shall have (|a^)-fC = 0, or C = -^a, LENGTHS OF CUKVES. 281 Hence the correct integral becomes consequently, tlie proposed curve is said to be exactly recti- fi'Cible^ because tbe integral of its differential can be exactly found. For another example, we will find the length of the com- mon parabola, its equation being ax = y\ By taking the differentials of its members, we have adx = 2ydy, or dx = ^— ^, 2 2^ 2 which, by putting -^ = h, gives dx^ — ^-j~ ; consequently, ¥ + v^ dx'^-h dy''= ./ dy% or ^idx^ + dy^) = dz = ^:^tt^ _ My yHy Hence, since and that y'^dy _ hdy we have reduced -^-^ — , ^ ' to + h:d.{lY + 'yf, 1^ 2i 282 LENGTHS OF CURVES. Hence (see the last example at p. 256), we stall have = I^^G^ + *') + ! log [y + 4/(y' + *^)] +c, . by representing hyperbolic logarithms by log, and using C to represent the arbitrary constant. By putting y = 0, we shall clearly have 2 -— 0, and thence C = — ^ log h ; which reduces the integral to S^^'-^-l ^^ + *=) + ! log yA^^±Ii. Hence the common parabola is rectifiable in algebraic and transcendental terms, but not in algebraic quantities, like the preceding example. For another example, it may be proposed to find an arc of a cycloid, reckoned from its vertex. By referring to page 150, we have dy — y dx, r being the radius of the generating circle, and x and y the abscissa and ordinate ; consequently, ^ X ' VWr dx or. which needs no correction, supposing the integral to com- mence with X. Hence, see the fig. at p. 149, it is clear that the cycloidal arc DG = 2DF = twice the chord of the corresponding arc of the generating circle ; consequently, DG^ or z^ — 8rx, SURFACES OF SOLIDS OF REVOLUTIOIT. 283 (13.) We will now proceed to show how to find the sur- faces of solids of revolution. Thus, supposing the fig. in (12), at p. 279, revolves about its axis AB, it will generate what is called a solid of revolu- tion, whose arc AC will describe its curve surfiice, which we propose to show how to find. Because Ct — (1.3 ^= the diiferential of AC = z, it is mani- fest that dz, multiplied by the circumference of the circle whose radius equals BC — y, will represent the differential of the surface described by the arc AC in one revolution about its axis AB. Hence, putting n ==3.14159 + , and representing the surface described by AC by S, we shall clearly have dS = ^-nydz for the differential of the described surface. Thus, to find the surface of a sphere whose radius is r, we shall evidently have r \ y\',dz\ die (or CS), (from similarity of triangles, since the radius drawn to C cuts C/^.perpendicularly, and that when the angle TCB is acute, tlie center is at the right of B in AB), or ydz =^ 7'dx ; consequently, c/S = ^rcydz reduces, by substitution, to c/S = lirrdx^ whose integral is I dS — I 2Trrdx or S = 27t/'^, which needs no correction, supposing the surface S to commence with x. If for x we put 2/'. the integral becomes S' = 4rr/'-, where S' stands for the whole surface ; consequently, since nr = the surface of a great circle of the sphere, it follows that S', the whole surface, equals four times the area of a great circle of the sjjhere / and from S = ^^rx^ it is manifest that the variations of S are proportional to those of a?. If we take dz^ in the parabola given at p. 282, we shall have ,S^?^f-+^y<^y, 284 SURFACES OF SOLIDS OF REVOLUTION. for the differential of the surface of the comjnon paraboloid, whose integral fdS>=^f^{V' + f)ydy gives S = g (J^ + 2^» + C, C being the arbitrary constant. To determine C, we suppose S and y to commence together, and thence get = -~h\ which gives S = 1^ [(5^ + 7/)^ - J^ for the correct integral. We will now show how to find the area of the surface generated by the revolution of the catenarian curve about its axis, supposing the equation between the length of the curve and the corresponding abscissa to be expressed by the equation z^=2ax-\-x^^ or by its equivalent, |/ (a- + 2^) = a+x, By taking the differentials, we have _ zdz consequently, since ds^ = ds^ + dy^, we have dj/ = dz^ — daf= dr 2— — ■„ = -j— — s ^ a^ -h s^ a^ + ^ adz Because o?S = ^'nydz = 27r {^dz + zdy — zdy) = 2 n- {dyz — zdy), we have, by taking the integral, C being the arbitrary constant, which equals 2na^ or SURFACES OF SOLIDS OF REVOLUTION. 28o when S = at the vertex of the curve. Because our equation is equivalent to S = 2Tr {yz — ax), as required. For the last example, we will show how to find the surface generated by the revolution of a cycloid around its base. Thus, by referring to the fig. at p. 149, since BD = 2r and DE = 37, we have BE =2r — x=^ the perpendicular from Gr to the base AC of the cycloid, and which revolves about the base ; and, fi-om the example at p. 282, we have (h = i/ — dx = V 2r x~^dx. ^ X Hence, putting 2r — x for y, and V 2rx~^dx for dz in dS = 2nydz, we shall have c?S = 2774/2? (2r — x) x~^dx = 27TV2r(2rx~^dx — x^dx) for the differential of the surface generated by the revolution of the cyclodial arc DG about BO, since this increases pos- itively, while that described by GC decreases. By taking the integrals, we have S:=27r V2^{f2rx-^dx -fx'^dx) =2 tt V2? Urx^ ~\^% which needs no correction, supposing the integral to com- mence with X. By putting 2r for a?, we have S= 277 V2^{4.r \^-\r V2?) = 2r:V¥rx \rV2^^= ^^ for the surface described by the semicycloidal arc DC about BC, and of course — ^ is the whole surface described by the revolution of the cycloid around its base, as required. (14.) We will now show how to use polar co-ordinates in finding the areas and lengths of curves. 286 USING POLAR CO-ORDINATES. Thus, let AC be for the polar co-ordinates of any point C of the curve ; then, shall —^ equal the differential of the curvilinear area APC. For, taking PB and the perpendicular BC for the rectan- gular co-ordinates of C, and denoting them by x and y, their origin being at P ; then, from what has been shown, ydbc is the differential of the area ABC. (See p. 266, &c.) Also, since the area of the triangle PBC equals -~^ and that the curvilinear area APC = the area ABC — triangle PBC = the area ABC — ^, by taking the differentials of those equals, we shall have the differential of the curvi- linear area APC = yd:c - W^A^ = ydx_-xdy^ 2 2 Because tan angle BPC = — tan

; cos- r^dr d<^=-, which gives -^- =^; whose integral, taken from ?' = 0, is /I'^dcp _ 7^ In like manner, from the equation r = a*, the equation of the logarithmic spiral, by taking the hyperbolic logarithms, we have log r = <^ log a, whose differentials give ^^ ^^ 1 ^^ ^^ — = a^ loff a or d(b = —^ , r ° r log a^ J ^, 7'-o?0 rdr aad thence -tt- = ^r-, ; 2 2 log a whose integral, taken from r = 0, gives J 2 ~ 4 log a ' By taking r = - , or = - , the equation of the hyper- bolical spiral, we get ,^ adr , ^, r'c?^ adr d(p= 5" , and thence -^- = p^- ; whose integral, taken from r = r\ is /7^d<\> _ a (/•' — r) ~2"~~ 2 ' which, taken to r = 0, or an infinitesimal, is /r'^d(^ _ ar' which equals the area of a right-angled triangle, whose per- pendicular sides are a and r' . USING POLAR CO-OKDINATES. 289 Remarks. — In treating of spirals, it will sometimes be convenient to consider the pole as moving, according to some given law, instead of being fixed, as is usually done. Thus, let a thread be wound from A around the circle ADB in the direction of the letters A, D, and B; then, when the thread is unwound from A, so as to be constantly a tangent to the circle, the extremity A of the thread will describe a curve AC, called the Involute of the Circle^ to which the thread is clearly constantly perpendicular, while it unwinds ; so that BO denoting any unwound part of the thread, it is manifest that BC cuts the curve AC perpendic- ularly at C, and is at the same time a tangent to the circle at B, and equal in length to the circular arc ADB. (See Sec. YL, p. 163.) We now propose to show how to find the area of the invo- lute bounded by the arc AC, the unwound part CB of the thread, and the circular arc ADB. Representing OC by r^ the radius OB of the circle by R, the right triangle BOO gives BO =: |/ (/"^ — R") = the circu- lar arc ADB. Hence, .^ '- equals the arc to radius = 1, which represents the angle AOB, which we shall take for 0, and for r we shall take y {f- — R-). 13 290 USING POLAR CO-ORDINATES. Hence, from (f> = -^ , we get # = gTT/^JZTRS) ' and this multiplied by t^ — K^ tlie square of the correspond- ing radius vector, gives (/^ - H') d _ i/{7^-'R')rdr 2 ~ 2K for the differential of the sought area: since the angular motion of BC is clearly the same as that of the perpen- dicular radius OB, it is clear that tlie angle <{> has been cor- rectly represented, while the pole moves from A, on the circular arc from A through D to B. By taking the integral of the differential equation, we have yii^__^_ = k___L, for the correct area ; supposing it to commence wben r = E, or when ^{?^—W) = 0. Hence, since the circular sector (fi'om the principles of geometry) ADB = OBC, it follows tbat we shall have the area AOBC-OBC = the area AOC= the area ACBD= ^-g— » which agrees with the area usually found, (See p. 76 of Vince's "Fluxions.") (15.) To find the lengths of curves by using polar co- ordinates, we proceed as follows : Thus, by using the figure and notation in (14), at p. 285, we have x = — 7* cos ^ and y = r sin (f>, which give dx= — cos (fydr + r sin and dy= sin. /^) = c?s, the differential of the arc AC, it results that in polar co-ordinates we shall have dB = V{rW + dr') = |/(/'^ + ^') # for the differential of the arc AC = 2, as required. Remarks. — 1. By referring to the figure at p. 131, and to what has there been done, taken in connection with what has been done above, it follows that the normal to the curve at C, limited by the perpendicular through P to the radius vector PC = r, equals /(,^ + '^, since (see page 132) -— = the square of the subnormal. Hence, by putting the normal y 1?'^ + -^-^l = N, we have Nc? = -^ ; whose integral gives supposing that the integral commences with r = B>] noticing, that in this solution the pole is supposed to move from A, around the circumference of the circle, in the order of the letters A, D, B, as at p. 288. For the last example, we will take the reciprocal spiral, whose equation is r = - or 6 =z ~. (p r By taking the differentials we have 294 VOLUME OF A SOLID. d(l> — —, wliich gives d(p' = —^-y or 7'^'d^ = —^\ and thence we have dz =^ ^{fd^ + d,^) = |/(~ + l^dr=^ ^/{i^'' + 1) ^ rdr dr ('^+1)? = dV{7^ + l) + df (log r) ^ (^ Slog [|/(P + r^) + l]i ; consequently, by taking the integrals, we shall have C being the arbitrary constant ; this integral is clearly the same as s= ^ {t^ + 1) + log -_-l^_- + C, which will clearly enable us to find the value of z that cor- responds to the interval between any finite values of r. (16.) We will now show how to find the contents or vol- ume of a solid, the equation of whose surface can be ex- pressed by an equation between the rectangular co-ordinates, a*, 2/, and 2, without regarding the body as being a solid of revolution. It is manifest that we may regard the very small parallel- epiped expressed by dxdijdz^ as being the differential of the solid, and represent its integral by j j fdxdydz^ by using the / successively to represent the separate integrations with reference to 2, ?/, and x. Thus, by perfbrmiDg the first integration with reference to z taken between the plane x^ y^ and the surface of the body, the integral is reduced to the form /Y zdxdy ; which may be ILLUSTKATED BY EXAMPLES. 295 integrated again by regarding z as being a function of x and y. If we at first integrate with reference to y by regarding X as constant, we sball 'h.Q.YQ J J zdxdy =^ I dx j zdy ^ in which / zdy denotes the area of a section of the solid by a plane that is parallel to the plane of the axes of z and y ; then, having found the integral / zdy^ we can find the in- tegral / d;c I zdy^ which being taken between proper limits of 0!, will give the required volume of the solid. It is manifest that we may perform the integrations in the forms J J zdxdy = I dy I zdx^ instead of using the preceding forms ; noticing, that those forms which are the simplest in the integrations are always to be chosen. It ought to be added, that to find the integrals in the simplest manner, the planes of the co-ordinates should be drawn, if possible, so that they may divide the body into equal parts. Thus, to find the volume of the ellipsoid whose equation f) O c> is -^+-4^+-^=l, it is manifest that the planes of its a- 0' G^ ^ axes are those of its co-ordinates. Then, to simplify the equation still further, we put X = ax\ y = hy', and z = cz\ which, by substitution, re- duce the equation to x'"^ + y"^ -f s'^ = 1 ; the equation of the surface of a sphere, liaving 1 for its radius. Since / zdx ■— ac I z'dx\ and / dy' I z'dx' = ahcjj z'dy'dx\ it manifestly follows, that if we multiply the volume of the 296 VOLUME OF A SOLID. sphere whose radius equals 1, by the product abc, the result wJU express the volume of the proposed ellipsoid Now, from x"' -f y'^ + z'' = 1, we have s' = y (1 - 1/' - x'') = ^ {r'' - x'% by putting r'^ = 1 — y'^ ; consequently, we shall have ffz'dy'dx'=ff^/(^r'^ - x'')dy'dx'= f dy' f x/{r''-x'')dx'. It is manifest that to find the integral / \/ {r'^ — x'-) dx\ r' must be regarded as constant, and that it will be sufiicient to take the integral from a?' =^0 to a?' = t\ since the whole in- tegral can thence be readily found. It is manifest that / ^ v'('-"-^'^)&'=^ = |(i-y"), which equals the fourth part of the area of a circle, whose radius is t' or y' (1 — ^r). Hence, Jdy' J{^r'^.^x")dx' becomes '^f{l-y'')dy', whose integral it will be sufficient to take from y' ^=0 to _ /»i _ y' = 1, which gives | J /I - v"") %' = g , for the eighth part of the sphere, whose radius = 1. Hence, —^ — is evidently equal to the contents or volume of the proposed ellipsoid, as required. It may be added, that, to simplify the integrals // zdxdy^ we sometimes put y = a?w, and thence, since x and y are independent variables, get dy = xdu, which reduces // zdxdy to J J zduxdx z= I du I zxdx. ILLUSTRATED BY EXAMPLES. 297 TTiTis, in finding the volume of the sphere whose equation is a?^ + 2/- + 2^ := E^, we have 2 ^ |/(E2_ x" - f) rr: ^/(R^ - x" - x'u") = ^/[R^ - x' {1 + u')], by putting y = xu. Hence, the integrals^^ zduxdx become fdu f\W -^ (1 + %^')'fxdx =^ by regarding u as constant in the integration, and using C for the arbitrary constant. Supposing the integral to commence when X — 0, we have C — -^-p. ^ ; then, if the integral is ■p extended to a? = —j- -^ , we shall have |/ (1 + 'i^^) =: — tan-^^ _ _^- 3 ^-" .« ' in which the arc tan"^ - must clearly be taken from - = X ■ ^ X to - — infinity, and of course the arc equals ^ ; conse- ■p3_ quently, ~-^ is an eighth part of the sphere, and its volume equals — ^ — , as required. o Remarks. — 1. If we change the rectangular co-ordinates X and y into the polar co-ordinates x — 7' cos <}> and y — /' sin , 13* 298 POLAR CO-ORDINATES. r being the radius vector (in the plane a;, y^ drawn from the origin), and the angle it makes with the axis of x ; then, by assuming dx = — r sin d(j> from x^ -\- if r=z r^^ we have ydj = rdt\ on acsoant of the independence of x and y, or 1x1 1' dr dy = — = ; consequently, dxdy = — rdrd(f> ; noti- cing, that if we had assumed dy =: r cos <^i. Hence, we have fj-z\lzd(^ = - Irrfz-dz, since the integml with regard to ^ ought clearly to be taken throughout the whole circumference. By taking the integral ~2n I z-dz from z = 'R to s = 0, we have ^ r ,, 2R«rr — 2^ I zHz — — r — J -& 3 for the volume of half the sphere, and of course that of the whole sphere is — — — ; the same as found by the preceding o methods. 2. It is easy to perceive that we may transform the infin- POLAR CO-ORDINATES. 299 itesimal solid dzdyilx by polar co-ordinates after the follow- in sr manner. o ' Tlius, let r denote its distance from the origin of the co- ordinates, and (9 the angle it makes with the plane of ir, y ; then, T cos being represented by r' ^ it will be the projection of T on the plane a?, y, and we also have r sin = z. Hence, if r' makes the angle with the axis of x^ we shall, from what has been previously shown, get dxdy = 7''d/d({). Since r^ = r'' + s", if we assume dz = r cos Odd, it results, from the independence of r' and z, that we must assume r'd/ =z rdr ; consequently, dzdydx is transformed to 7^ cos 6drddd(f). Hence, Jjj dzdydx = / dydxdz^ called a triple integral, is transformed to the triple integral J r- cos Odrdddcp — J iHr J cos Odd Jd(f) ; noticing, that two successive integrations are called a double integral, and so on, according to the number of successive integrations. It may be added, that the preceding trans- formation is essentially the same as that of Laplace, at p. 6, vol. II., of the "Mecanique Celeste," and that of Lacroix, at p. 209, vol. II., of his "Traite du Calcal Integral." By applying the preceding formula to find the contents of a sphere whose radius is R, it is manifest, as before, that the integral with regard to dd must be taken through the whole circumference, which reduces it to J r-dr J cos OdO fd(p = 2n J r^^dr cos Odd; 300 SUKFACES OF SOLIDS. whose integral witli regard to 6 must be taken frora sin = — 1 to sin = 1, which reduces it to 27r A-W/' /*cos Odd = infr^dr] whose integral, with reference to r, must be taken from r = to /• = R, which gives 47tW [Trjr-dr = 3 ' for the volume of the sphere. (17.) We now propose to show how to find the surface of a body or solid, on suppositions like to those in (16), and shall premise the following important proposition : Thus, let (%c, oy^ and oz^ be three rectangular axes having o for their origin ; then, the square of the numerical value of the face xyz of the triangular ^pyr amid oxyz^ equals the sum of the squares of the numerical values of tJie three 're- maining faces of the jpyramid. For representing ox^ oy^ and oz^ severally by a, 5, and c, the right triangles oxy^ oyz^ and oxz^ severally give V{a'-^l% V{^' + c% >/(«= + ^), for the representatives of the sides xy^ yz^ and xz^ of the tri- angular face xyz of the pyramid. Hence, since the triangular faces oxy^ oyz, and oxz^ are SURFACES OF SOLIDS. 801 severally represented by -^, -^, and -^, we propose to show- that the square of the face xyz equals -4- + -4- "^ X~ 4 If, for brevity, we represent the sides a?y, yz^ and xz^ by A, B, C ; by a well-known rule for finding the area of a triangle from its three sides, we shall have the area of the triangle xyz expressed by /A + B + C B + C-A A + C-B A + B-C\* ( 2— ^ -1 ■ ^ ^— ^ 2-—)' whose square equals (B H- Cy - A^ A^ - (B - C)^ 4 " ^ " 4 _ BN-C^-AM-2B C A^ - (B^ + C^) + 2BQ -" 4 "^ 4 _ 2BC + (B^+C^-A^) 2BC - (B^- + C- - A^) _. ____ X -^ ^ 4B^C^^ - (B^- + C^ - A^)^ 16 From the substitutions of the values y («^ + W)^ |/(J^ + (?\ and ^'{cC- + c-), of A, B, and C, in the preceding equation, we have the square of the face xyz equal to 4 (6^ + (T) (g^ + g') - 4:0' _ om + a^& -f l\^ 16 ~ 4 ' as required. It is clear that the triangles xyo^ yzo^ and xzo^ are severally equal to the projections of the triangle xyz^ by perpendiculars upon them. And since, from principles of geometry, the perpendicular from to the face xyz^ multi- plied by it, equals the perpendicular oz multiplied by the 302 SURFACES OF SOLIDS. triangle yxo^ to which it is perpendicular, each product being three times the pyramid, it follows that the triangle xyo equals the triangle xyz multiplied by the quotient resulting from the division of the perpendicular from o by 02^ which is clearly the cosine of the inclination of the face xyz to the face xyo. Hence, the cosine of the inclination of xyz to either of the other faces multiplied by xyz equals the other face; consequently, from what has been shown, it follows that the sum of the squares of the cosines of the inclinations of the face xyz to each of the other faces equals unity or 1. Hence, also, any plane in the plane xyz is such, that its square equals the sum of the squares of its projections on the three planes xijo^ yzo^ and xzo. We will now suppose the curve surface to be touched by a plane at any one of its points, and that an unlimitedly small portion of it at the point of contact, having two of its opposite sides pamllel to the plane of ar, 0, and the other two opposite sides parallel to the plane of y, z^ is taken for the differential of the curve surface. Then, the projections of the parallelogram thus formed on the planes a?, y, a*, 2, and ?/, 2, will evidently be parallelograms whose areas may be ex- pressed by the products dxdy^ dydz^ and dxdz ; consequently, from what has been shown, we shall have dx-df + dy^dz^ + dxHz" = dx'df fl + (^)' + (J)'} for the square of the differential of the curve surface, and of course if c?-S represents the differential, we shall have for the required differential of the curve BurfBuce. SURFACES OF SOLIDS. 803 It may be noticed that ~ and ~, wliicli suppose ^ to be a (Xi-C (Xi 'I function of x and y, have heretofore been represented, as in Sections 8 and 9, by^ and ^; agreeably to which, if we please, we may write the preceding equation, according to custom, in the form (i-S = dxd(/\/{i +jy"+ q^). It may also be noticed, that according to what has been shown, ^(l+y + ,=) = /{l + (J)V(|y equals the reciprocal of the cosine of the angle made by the tangent plane with the plane a?, y. To illustrate what has been done, we will apply the for- mula to find the surface of a sphere whose equation is ^= -f r + ^' = R' By taking the partial differential coefficients, we get dz ^ X J (^^ _ y dx~~ z dy ~^ ~~ z^ which give consequentlj, we shall have by putting R'= = R- — y\ By taking the integral relatively to X, or by regarding E' as being constant, we have dS r dx -D • 1 a' -D . 1 aj Ty = V 7(1^^ = ^ ^'^-' F = ^ ^'° TW^sO ' which, taken from » = to a? = V(^^— f)^ gives ^ = ^5 j 804 ARBITRARY CONSTANTS ILLUSTRATED. consequently, dS = -^- dy^ whose integral is S = -y y, which, ■pi>_ taken from y = to ?/ = R, is -^- , the eighth part of the surface of the whole sphere, which, of course, equals 4:R-7r, four times the area of a great circle of the sphere. Otherwise. — By putting the equation of the spheric surface in the form R2 = ar* + /- -f s' = r' + 2', we shall, by the notation at p. 298, get dxdy = rdrd(p = — zdzdtp^ and thence <^S = — zd2d(() x - = — d^Rdcj) ; whose integral relatively to (f> must clearly be taken through- out the entire circumference, and gives ds = — 2B,7Td2 ; and the integral of this must evidently be taken from 3 = — R to s = R, which gives ^ttR^ for the whole surface of the sphere, the same result as by the preceding method. (18.) We will now proceed to show the use of arbitrary constants in the development of functions, and in the integra- tion of differential equations, move than has yet been done. 1. To show the use of constants in the development of functions, we will give the following investigation of Taylor's Theorem. Thus, suppose the differential of any function of a? + A may be represented by the form dF{x-\-h) = 'F'{x + h)dh, when the differential is taken on the supposition that A alone is variable. By taking the integrals of the members of the equation, we have ARBITRARY CON"STAN"TS ILLUSTRATED. 305 F {x + h) = C +fY {x + h) dh', in wliicli F (a? + h) is tlie integral of the exact differential f/F (;» -f A), and C tlie arbitrary constant, while / F' (a? + A) dh indicates that the integral of F^ {x + h) dh is to be found, on the supposition that h alone is regarded as variable. If we determine C on the hypothesis that the integral / F' {x + h) dh vanishes when h equals naught, since A = reduces F {x+h) to F (a?), and / F' (a? + A) dh to naught, we shall have F {x) = C. Hence, by substituting this value of C, the equation F (aj + 7^) = C + I ¥' {x + h) dh is reduced to F (a? + h) — F {x) + J Y' {x + h) dh ; noticing, that F (a?) is not supposed to be unlimitedly great Because F' (a? + h) is a function of x + A, it follows, from what has been done, that for F' (a? + h) we may put Y{x)+f'F"{x + h)dh, which reduces the preceding equation to F (a? + A) == F (x) + Jy (^) dh + fdhfw {x + h) dh :=¥{x)-{-W{x)h+f'¥''{x + h)dh'', since / F^ (x) dh becomes F' {x) h on account of the con- stancy of F^(a?), and by using / (according to custom) iQxJJ . Similarly, because F^^ {x -f- A) may be represented by F" {x) + Jy" {x + A) dh, we have 806 ARBITRARY CONSTANTS ILLUSTRATED. pT' {X + h) dh' =fdhfw' {x) dh + f Y" {x + h) dh\ and hence F(^ + A) = F(a.) + r{x)j + Y'\x) ^ -\-fY%c + h)dIi\ If n represents any positive integer, it is manifest that we shall in this way get To find the values of F'(i»), V {x\ Y" {x), kQ.,wQ resume the proposed equation d¥{x + h) = l^'{x + h)d/i, I,- -u • d¥{x-\-h) T^,, , ,. which gives — -~. = ¥' [x + A), for which we may evidently put ^^--tA) = F (aj + A) = F' {x) +fY' {x + h) dh; for, since x and A enter the function F (a? -}- A) in the same manner, it is clear that the differential coefficient taken by regarding h alone as variable, must be equal to its differen- tial coefficient, taken by regarding x alone as variable. Because F' {x) enters the preceding equation, like the arbitrary constant C in the equation F {x Jrh)=Q+ Jy {x + h) dh, it is manifest that we may determine F' (.^) from the equation &J^±]!l^^'^,)+fr'i. + k)dh, ABBITRARY CON'STANTS ILLUSTRATED. 307 on the supposition tliat when h = 0, we must also have fF'Xx + h)dk = 0; consequently, by putting A = 0, we get Because the equation F'(» + A) = F'(^) +f¥"{x -f h) dh may be supposed to have been obtained from dY{x + h) = Y'{,iG-^h)dh, in the same way that F (a? + A) =: F ix) i-fr {x + A) dh has been derived from ^F (^ + A) = F' (,^ -f A) dh, it is clear that we shall (as before) get flY (x) , . Because F^ (x) = — ---^ , if dx is constant, it is ciear that . -,,,,. d.W{x) . d'¥{x) . ,. , d'F{x) ior J^ {x) = — -—^ we may write — •,-y-^ ; m wnich ■■ j ■ (XX ci x" (Xxr is called the second differential coefficient of F {x). It is evi- dent that we shall in like manner get and so on, for the third, fourth, &3., diiferontial coefficients. Hence, we shall have I (^ + A) _ F (..) + ^^. ^ + --j^ ^^ + ^-^- j~2j +, i&c, as in Taylor's Theorem, as required. 808 ARBITRARY CONSTANTS ILLUSTRATED. It will be perceived that, in the precediug investigation, we have virtually introduced an unlimitedly great number of constants ; since there must (essentially) be as many as there are equations like F {x-{-h) = C -hf¥'{x+h) d/i = ¥ {x) +fF' {x-\-h) dh, Y{x^-K) = F' {x) ■\- Jf" (^ + h) dh, and so oa But since these constants all result from C = F (a?), or are dependent on C, it is clear that the integral of d'F{x + h) = Y{x-\-h)dh contains only one arbitrary constant. Indeed, it is manifest that in enter as constants ; whose values result from (x), or depend on X and the form of the function represented by 0. It is hence evident, that in integrating any differential equation there will be as many constants introduced as there are integrations, which will be arbitrary when they are inde- pendent of each other. 2. Supposing an equation between variables and constants to be freed from fractions and radicals, and that its terms are all brought into the first member of the equation, and ordered accordmg to the ascending or descending powers of one or more of the unknown letters, then, if the equation has a term called the absolitte term^ which does not contain any variable, by taking the differential of the equation, the absolute term will disappear from the differcDtial equatic^ii; and the pro- ABBITRAUY CONSTANTS ILLUSTRATED. 809 posed equation, sometimes called the ^?rz7/247?'t'^, is said to have lost a constant in the ditferential equation, sometimes called the first derivative of the proposed equation, bj a direct differentiation of the primitive ; but if the form of the primitive is changed, so as to make the constant coefficient of any other term of the equation the absolute term of the changed equation, its absolute term will, as before, disappear from its differential equation, which may be called an indi- rect derivative of the proposed equation, which may be said to have resulted from an indirect differentiation of the pro- posed equation. It is hence easy to perceive that there may be as many direct and indirect differential equations obtained from the given primitive, to free it from each of its constants separately, as it contains constants. Thus, if y -f- «a3 + ^ = represents the given equation, hav- ing h for its absolute term, then, by a direct differentiation of the equation, we get dy + adx = or -~- -\- a = for the direct derivative of the proposed primitive, which does not contain the absolute term b. By putting the proposed equa» tion under the form a = 0, we have a for its abso- X ' lute term ; then, taking the differentials of the members of this, we have y_+h _ d{y + h) x x — dx (y + h) _ X m? . ~~ ' or xdy — ydx — hdx = 0, or its equivalent y -~- +5 = 0, which is the indirect derivative of the proposed equation, which is clearly the same result that the elimination of a from 810 ARBITRARY CONSTANTS ILLUSTRATED. y + ax -\-h = by -^ -\-a = will give ; it is also clear that the elimination of -j- from the differential equations will reproduce the proposed primitive. It is also manifest that the derivative equations -^ + a=0 and y— -j^ -f ^ = dx ^ dx are entirely distinct from each other ; the equivalent of the first dy + adx = being immediately integrable, while the integral of the equivalent of the second ydx — xyd -\-hdx = (or —^--j^ 2 = 0) 8/ X becomes integrable after it is multiplied by — -^ , the factor which is said to he requisite to the integrahility of the in- direct derivative, ydx — xdy + hdx — 0, of the proposed primitive. If we take the equation y -^1)X ■\- car ^= 0, it is evident that a constant can not be eliminated from it by a single direct differentiation, while the constants h and c can be eliminated by indirect differentiations. For, by putting the equation under the forms -, H h C = and ^ -r cx-^h = 0, mr X X ■ and taking the differentials, we have p REDUCTION OF INTEGRALS TO SIMPLER FOEM& 811 d%+d-=0 or ^*' _ (2y + fo) = 0, X' X ax and cZ - + cdx — or x ^ — (y — car) = 0. X dx ^ ' It is evident that by eliminating -~ from these equations, we dx shall get the primitive equation y -\- hx -\- cx' ^^ 0, which can not be found from the immediate integration of either of the derived equations. If, for another example, we take the equation . y — aa? + a^ == ; dxi then, by differentiation, we have dy — adx = or -~ =a. Substituting -~ for a, in the proposed equation, it becomes xdy df y dx ^ dx' ^' which is of the second degree in ~ , and of the iSrst order of differentials. Thus we perceive how differential coefficients of the higher orders may sometimes be introduced into differ- ential equations, by eliminating the different powers of a constant from it, by means of the powers of a differential coefficient; but it is manifest from the methods of finding multiple points in Section YII., that they may sometimes be introduced by differentiating as in finding multiple points. (See the examples at p. 191, &c.) 3. We now propose to show how to reduce such integrals as are of the forms / X.dx"\ j Xdx"", &c., vi and ?i being positive integers, to simple integrals^ expressed by the sign / . Thus, 312 REDUCTION OF INTEGRALS TO SIMPLER FORMa f\da? =zfdxfxdx = fdxf{Xdx H- x-yidx - X^rda?) = X I X^dx — I 'Kxdx ; which clearly results from integrating by parts (see p. 260). Similarly, f'xda^ =fdxf'xda^ =f{xdxfxdx - dxfxxdx) = ^ {a^fxdx - 2xfxxdx + fxx'dv), fxdx' =Jdxfxd^ = ^ fix'fxdx - 2xdxfxxdx 4- dxfx3?dx) = j^ [x'fxdx - Sx'f Xxdx + Zxfx^dx-fX;^dr), and so on, to A<*^" = i.2.3..'(,>- r) i^-'f^^- - ^ ^-i^-^ whose law of continuation is manifest (See Lacroix, voL II., p. 152.) If for / Xdx^ I Xxdxy I Xx^dx^ &c., in the preceding formula, we put fxdx + cfxxdx + C\fx/dx + C", &c., in which C, C, C", &c., are the arbitrary constants, they will represent the complete integrals indicated by / Xdx" ; be- cause there will be as many arbitrary constants as there are LIMITS TO INTEGRALS. 813 integrations, and tliey clearly enter the formula, as they ought to do. If the constants equal naught, it is clear that the pre- ceding formula is equivalent to provided y is regarded as independent of x in the integration, and that the integral is taken from the value of x at the commencement of the integral, to the value of x at the end of it ; for which last value (of x) we ought to put y, or y must represent it Eemarks. — 1. The preceding formula enables us to find limits to the integrals indicated by given in the investigation of Taylor's Theorem, at p. 306. For X may represent —j^ — -^ and h may be used for X in the preceding formula ; consequently, we shall have If we put y — h = yz^ or h = y (I — z)^ we shall have dh ~ — ydz^ since y is independent of h ; consequently, we shall get r^'^' = - 1.2.8 ■■'■(«-i) /%"^"-'^-. supposing the integral to be taken from 3 = 1 or A = to z — or y = h. If the limits of the integral are interchanged, it is evident that we shall have rXd/i^^=-— -. ~ fxy"z--'d2. . v 1.2.3 {n — 1) J -^ If M and m are the greatest and least values of X (re- 14 814 REPRESENTATIONS OF INTEGRALS. garded as having the same sign and as finite), in the interval trom a? to ic -f A, then we shall have A'^^"- i.2....;i-i) A^"-"-''^-' such that T-K-7Z and ^ ^ ^ — are its greater and 1.2.3. ... 71 1.2.3 n less limits ; noticing, that these limits are clearly the limits of the errors committed by rejecting 1'^''"= 1.2.3. ■■'(.-d A^"^"-'^- (See Lacroix, vol. Ill, p. 398.) 2. It is easy to find the integrals indicated by / Xc?a?", in such a way that they shall be freed from / , the sign of integi-ation. Thus, since r. (^ dX a^ (^X ar" , \ (see Bernouilli's series at p. 261), and by disregarding the arbitrary constants (for the present), we shaU, by integrating by pai-ts, get / Xdx^ = " 1.2 dx 1.2.3 "^ da^ 1.2.3.4 dx' 1.2.3.4.5 '^' From this result, we, in like manner, get which, integrated by parts, as before, gives J Ar/x_X— -- — ---- + — j^^—-,&c. KEPRESENTATIONS OF INTEGRALS. 815 Proceeding in this way, and supplying the arbitrary con- stants, it is easy to perceive that we shall have / Xdx- = X 1.2 n dx 1.2 {fi + 1) nOi_+l)^t! n (71 + 1) 01 + 2) 3 d'X 1.2 d'X~ 1.2.3 "^ dx" 1.2..., {71+ 2) dx' 1.2.... {n + S) "^ C, C\ &c., being the arbitrary constanta (See Lacroix, vol. IL, pp. 154 and 155.) Being now prepared, we will give a short section on the Calculus of Yariations. SECTION II. FIRST PRINCIPLES OF THE CALCULUS OF VARIATIONS. (1.) If y is an arbitrary variable, whicb depends on a con- stant ; then, if in consequence of a change in the constant it becomes Y', the difference Y' — V, represented by (JY, is called the variation of Y, which is expressed by writing (J, called the characteristic of variations^ before or to the left of Y. If (Y) represents any function of Y, and the alge- braic sum of all the changes in the value of (Y) that result from the separate variation Y'— Y, represented by dY, of each Y in (Y) is taken, it will represent what is called the variation of

(Y) _ dct> (Y) 6Y ^ dY ' which shows that the variational and differential coefficients CALCULUS OF VARIATIONS. 817 of a function^ loiih reference to the same variable^ are equal to each other. (3.) Since from (1.) Y' = Y -\- dY, we have, from Taylor's Theorem, • ,^(V') = 0(V + <5V) = 0(V) + ^-^ <5V +, &o. ; which, bj retaining only the term that contains the simple power of cJYj becomes which clearly shows that (Y') must be of a different form from (/) (Y), since 6Y results from the change of a constant contained in Y. Hence, if we represent the proper form of the first mem- ber of the equation by V (Y^), we shall get V(VO = 0(Y) + ^cyY; which gives i/,(YO-0(Y) = ^^(5Y. Since J^ 6Y is, according to what has been shown, equal to (5 (Y), we shall hence get ^(Y')-cp{Y)^6(J) = d6(f> (Y) ; or since d'lp {Y') is a change of the fonn d(p (Y), we shall have dijj (YO - di> (Y) = ddcp (Y), and thence d6(f> (Y) = ddcp (Y) ; and with equal facility we get C^M0(Y) = <5ri'^0(Y), 818 CALCULUS OF VAIIIATIOKS, n being a positive integer. Hence, m or in any expression to which d"" and 6 are prefixed, we may clearly interchange d and (5, the characteristics of differ- entials and variations, without affecting the value of the result ; noticing, that this is usually considered as being the fundamental principle of the Calculus of Variations. On account of the importance of what has been done, in what is to follow, we propose to illustrate it geometrically. Thus, if the line OC is taken for the line of the abscissas, on which the positive values of V are estimated from the origin 0, toward the right; then ab, being drawn as an ordinate to the curve he, representing the value of 0(V), which corresponds to Oa = V, by changing Oa or V into OA or V, and drawing AB parallel to ab to represent the changed value of ab=z(f)(Y) as an ordinate V^(V"), in the changed curve BD, we shall have i/^ (V) = ^ (V), the varia- tion of ab represented by AB — ab in the figure. Similarly, Oc and OC representing other values of V and Y^, we shall have CD — cd for the representative of the cor- responding value of V^ {Y') — (Y), which may be regarded as consecutive to the preceding value. CALCULUS OF VARIATIONS. 819 Hence, we sliall have (CD - cd) - (AB - al) = (CD - AB) - {cd - ab) ; the first member of this equation, from the definitions at page 2, being the differential of ( AB - ah) = W -cpY^Scp (V), since (AB — ah) is on the same curves with its consecutive value (CD — cd) ; while {cd — ah) in the second member of the equation has (CD — AB) for its consecutive value, which is taken in the curve BD and not in the curve he ; and of course, since (cd — ah) = d^ (Y), we shall have (CD - AB) - (cd - ab) expressed by ^d(l)(Y). Hence, from what has been done, we shall have ddcf) (Y) = ^d(f> (V) ; which agrees with what has been shown, from other considerations. Again, since Oa = Y, and ac = dV, and cC = cJ (Y + +c^|(a+2/)^} = 0. The integral of the first of these gives do) (« + 2/) ^ = ^ = const., which gives 836 TIME IN UNLIMITEDLY (a ■\-yfcb?=z h' {dx" + df), or [(a -f yf - J^ dx" = IHif, hdi/ or dx = V{a-\-yy-T'' and by putting the second equation in the form ^ (« + y) ;^ = d^i and integrating, we have («+2')| = * + C, C being the arbitrary constant. If -— = when 5 = 0, we have dif («+.)!= since C = ; and thence 2ad(/ + 2(/di/ = 28dSj whose integral gives 2ay -{- 1/ = ^, which needs no correction, supposing s and y to commence together and to be reckoned upward ; noticing, that the origin of the co-ordinates is clearly at the vertex, since ~ == at the origin. The preceding equation is the common catenary, the well-known curve, into which a uniform chain of unlim- itedly small, short links, when suspended from its extreme points, will form itself; and it is also well known that the equation dx = , — =£: r- , Sfi^a^yf^l?' previously found, is another form of the equation of the same curve. Eemarks. — Because the length of the curve in this ques- tion is given in addition to the maximum condition, the question is said to fall under the class of what are called uojpeinmetrical questions. / SMALL CIRCULAR ARCS. 837 3. To find the relation between oo and y, when the integral ydf -^ , taken between proper limits, is a minimum. which, integrated by parts, becomes ZydyHa? + ydy* iydfSx , p , /( j: _ rf ?y^<^+^J ^^ ^y^ 2jgy. ^^ ^ ^. in wbicb • db^ is put for dd(? + 6?y^, and C is tbe arbitrary constant. Because tbe equation must be satisfied so as to leave 5y and ^x under tbe sign / arbitrary, we must put their coeffi- cients equal to naught, and sball thence get % - d^yM^^p!^ = Q and dy^P^O; ds^ as* ■ as* consequently, the preceding variation reduces to Sydfd^^ + ydy* _ 2y^'dx ^ ^ ^^ ds* ^ ds* If the extremities of the integral are given points, we have 6y = 0, 6x = 0, and thence C — ; consequently, the con- ditions of the question are all satisfied. To find the relation of x and y, it will clearly be sufficient to take the integral of d ^-jj- = 0, which gives ^ J^ ^ = C = const, 15 3o8 TIME IN UNLIMITEDLY and to reject the other equation, or to put the 6y under the sign / , equal to naught The equation —fj— = C, gives — c' ^^* — c (^ + ^yy ^ ~ dy^dx~ dy^dx by putting '-£, = P- From y = ^^r^ = ^' (^"^ 2i?-^ +p\ we get c??/ = C (- 3j?-'* - 2^-' + 1) dp, and thence c?aj = — becomes whose integral gives - = o" + c-(|, + l. + h.i.4 in which 0" is the arbitrary constant, and h.L^ denotes the hyperbolic logarithm of j9. Supposing ^ to be eliminated from then, by putting in the resulting equation, the values of X and y at the given points at the extremities of the integral, we shall have two equations containing C and C as un- knowns, whose solutions will give the required values of the constants, as required; consequently, the required relation between x and y will be found. SMALL CIRCULAR ARCS. 839 Instead of supposing the extremities of the integral to be given, it will clearly be sufficient to use other conditions ; such as will enable us to find the constants C and C, and thence to get the values of x and y that ma}^ correspond to any assumed value of ^. Thus, if the limits of the variation of the integral are not given points ; then, if the variation is taken from the values of x and y represented by x' and y' to the values represented by x" and y'\ we shall have Zy"dy"Hx"'^ + y"dy"^ . ,, 2y"dy"Hx" ,//i 'y'^--^"-^^^'^" i Zy'dy'^dx'^^y'dy'' 2y'dy''dx' \ _ ~ V dl' y d^'^~ 00. j -i). If the co-ordinates at the extremities of the integral are independent of each other, it is manifest that this equatioa will be divided into the equations which representing -^ by p, are equivalent to ^__¥:_ and ^- 2^' 6x'' ~ 3 -\-2)"' dx' ~~ 3TF^' Since y = -^^' and . = C" + C (A + ^ + 1,L^), we may to these join the equatioas 340 TIME IN UNTilMITEDLY \9 C'a+y"y ,_ c'(i+y°)' a,"^C" + c'(J-,;5+^ + h.i.y'), and, representing the equations of the limiting curves by y-=cl>{x'') and y'='^{x'\ we shall have ^ = (p' {x/') and ^ = V* (»0) •which reduce the preceding equations to Hence, we have eight equations, which will enable us to find the eight unknowns, x'\ y'\ p'\ x\ y\ p\ C, and C ; consequently, the points in which the curve represented by the equations intersects the limiting curves y" = ^' {x") and y = -^ {x'\ may be supposed to have been found ; and since the con- stants C and Qi" may be supposed to have been found, it clearly follows that the curve represented by 01(1+^ and «, = C"+C'(|i + i5 + h.l.^.), may be supposed to be drawn, as required, between its limit- If for either limiting curve, as that whose co-ordinates are x' and y\ we take the point whose co-ordinates are x' and y\ then it is easy to perceive that our eight equations will be SMALL CIRCULAR ARCS. 341 reduced to six, whicli will enable us to find ttie six un- knowns, x'\ y'\ J)" ^ p\ C, and C^^, &c., as before. Remarks. — 1. It is easy to perceive, from tlie solution of Ex. 19, at p. 113, that / -j- / ^ represents the resistance of a solid of revolution around the axis of a?, moving in a fluid of uniform density, in the direction of the axis of x with its smaller end foremost, whose nature we have determined, so as to make the resistance a minimum. 2. The example is substantially the same as that solved by ISTewton, at p. 120, vol. II., of his "Principia."^ If, in the preceding equations, we put ^ = 1, and y' for the corre- sponding value of y and x = 0, then 2/' =40', or C' = |, = C" + ~C', or 0"=-^. From the substitution of the values of the constants, the equations become 2/ - 4 ^^, , ^nd X- ^\^^^, + ^, + hA.p ^ j , which clearly reduce to y^ and at the origin of the co-ordi- nates, since h. 1. 1 = 0. If we put ^ = 0.9, we readily get y = 1.1232/' ^^^ *' — 0.130 y' very nearly, and p — 0.8 gives y = 1.313?/' and x — 0.354?/' nearly, and so on. Hence, when the extremities of the integral are fixed points, as at p. 337, we easily perceive how the equation which connects y and x may be represented by linear de- scription. Thus, by putting 2/^=1, and assuming ox and oy for the positive directions of the rectangular co-ordinates, having o for their origin, we set 1 from o on the axis of y for a point 842 TIME IN UNLIMITEDLT in the curve, and then set 0.130 from o on the axis of a?, through which (point) we erect the perpendicular 1.123 to the axis of x for the corresponding value of ?/, and then having set 0.354 for x' from o^ as before, we draw the per- pendicular through the point to the axis of x equal to 1.313 for the corresponding value of y, and so on to any required extent; then, a curve drawn with a steady hand through the points thus found will be such, that by revolving around the axis of x it will generate a solid, which, moving in a fluid from X toward o, it will meet with less resistance than any other solid, whose end diameters and height are the same. It is manifest, that the preceding construction is substantially the same as that of Newton. 4. To find the curve surface, whose area between given limits is a minimum. Agreeably to what is shown at p. 302, the double integral when taken between the proposed limits, may be taken to represent the required surface. By taking the integral of the variation of the surface, we have SMALL CIRCULAR ARCS. 343 by using 'p and ^ for — and ~, and because z is regarded as being a function of x and y, considered as being independent variables. Since dp z= o — = —- and on ^= -=- ^ dx ax ^ dy on account of tbe constancy of dx and dy ; tben, if -g and Q ^' =^ff^^y -£ ^^ +/7q^^ ^ ^y- we sliall bave Hence, integrating by parts, we sball have 6h ^fvdySz +fQdxd3 -ff^^ ^zdxdy -ff^^ ^^dxdy ; and it is clear tbat tbe part of this integral wbicb is freed from one of the signs of integration, since it relates to tbe fixed limits, must be reduced to naught, since Sz at the limits = 0. Hence, we shall have ^'^-fJ%,^^^''^y-ffdy^^^''^^^ • which must equal naught, since s is to be a minimum ; con- sequently, since dz^ under the double sign of integration, is indeterminate or arbitrary, its factor -7-^ + -^ , under the double sign of integration, must be reduced to naught, which gives -7 [- -— z= 0. By restoring the values of P and Q, dx dy and taking the indicated differential coefi&cients, the preceding equation will be reduced to its equivalent, 344 TIME IN UNLIMITEDLY consequently, if' we put f^r, ^ = ^ = *, and ^ = t, ax ax dy ay we shall have (1 + q')r-2jpqs + {l+p')t = 0, for the equation of the partial differential coefficients of the sought surface, whose integral will, of course, be the surface. Remarks. — 1. This example has been taken from p. 753, vol. II., of Lacroix's work, where it is remarked, in a foot note, that the equation ^_. ^ = ves ^=: _^- dx dy ° dx dy ^ which is the condition of the immediate integrability of Vdy — Qfitx ; consequently, it is concluded that on the mini- mum surface, —TTT^- — t"- — 17 is an exact differential, as well .'.|/(l+y + ^-0 - as dz =pdx + qdy. Thus, all plane surfaces will be found to satisfy these conditions ; since and of course the preceding conditions are reduced to naught; consequently, since the differentials of constants equal nought, it is manifest that the preceding differentials may be regarded as having constants for their exact in- tegrals. If these tests are applied to the surface whose equation is az = xij, they will be found to give _dz _y , _ dz _x ■^ ~ dx~ a ^ ~ dy'' a"* which reduce them to SMALL CIRCULAR ARCS. 845 ydy — xdx j j _ 2/^^-^ + •'^^// _ ^^ {^V) . 4,/ (a? + x^ -\- y^) " a a ' consequently, since the first of tliese is not an exact differen tial, it follows that the proposed surface does not belong to the class of minimum surfaces. Nevertheless, if x and y are very small in comparison to a, it is clear that -yj-rf- — i 2\ ijcLij xdx does not sensibly differ from —- , which is an exact differential; consequently, if a is very great, the surface az — xy for finite values of x and y, will not greatly differ from a minimum surface. 2. Lacroix, at pages 806 and 875 of the volume cited, shows how to find the solid which, with a given capacity, contains the least surface. Thus, since J J zdxdy and JJ ^ (1 +i?^ + 2'') ^^^y express the capacity and surface, and that the first is given, it is manifest if stands for a constant, that when the sur- face is a minimum, ffCzdxdy ^ff V{\ +i?^ + ct) dxdy ■= fJ\C3 + V{^+p'-\-r)\dxdy will also be a minimum. Hence, using P and Q to stand for the same things as before, then taking x and y for the independent variables, we in like manner get the equation ax ay or its equivalent, (1 +y + ff - [(1 + ct) r - 2/?^^ + (1 +/) t\ = 0, for the equation of partial differential coefficients of the re- 15* 34:6 TIME IN UNLIMITEDLY quired body, whose integral will, of course, represent tlie body. Lacroix remarks, that the sphere and cylinder whose equations are represented by 2' -hf-\-x' = a'^, and z- + y- = a'\ will satisfy the preceding equations. Thus, in the sphere ^ = , and q z=i — ^^ which give |/(1 + p- -\- q^) = -^ z and thence P and Q equal and — - : which reduce ^ ^ a a 2 the first of the preceding equations to C + - = 0, and in the cylinder the same equation is reduced to C H — > == 0. 5. To draw the shortest line possible from one point to another, on any proposed surface. Let a?, y, 2, represent the rectangular co-ordinates of any point of the sought line ; then, because the point is on a surface, z may be considered as being — a function of x and y regarded as being independent variables, and we shall have T ch ., dz ^ ' Tx '^ dy ^y "^ ^ ^ ^ '^' From what is done at p. 240, we evidently have ds = x/{dx' + dy'' + dz') for the differential of the line, and s = f^{dx' + dy' + dz^) will represent the line, and its variation becomes SMALL CIRCULAR ARCS. 347 ds = df^/idx^ + dy^ + dz'') =z Cd .^{d'j? 4- dy" + d^) dx ^ dy p d^ , C being the constant Taking the integral from x\ y\ z\ to x'\ y'\ z"^ the constant will be removed, and we shall have Supposing the extremities . of the integral to be fixed points, the part of the integral without the sign / will vanish; and since (5s = 0, we must have since ^z = p6x + qdy. Because dx and dy^ under the sign / , are arbitrary, their factors must be put equal to naught, which give ^ dx ^(Iz - , , dii ^ dz - a -, — \r pa -z- ^ \) and a -^ 4- <7« -r- =^ ^ I ds ^ ds ds ^ ds 848 TIME IN UNLIMITEDLY wLich are the equations of the minimum line, and are tlie same as those given by Lacroix, at p. 270 of the volume befoi-e cited. Thus, to draw the shortest line possible from one point to another on the surface of a sphere whose equation is 2^= a^^ f+ z\ Here dz = ■ dx — - dy^ z z which gives ^ = and q = — -^^ z z which reduce the preceding equations of the minimum to whose integrals may be expressed by zdx + xdz = Ads^ and zdy — ydz = Bds. Multiplyftig the first of these by B and the second by A, we X 1/ readily get Bd - = Ad - ; z z whose integral gives ^ + C - B - = 0, or Ay -Bx + Cz = 0', z z ^ which is the equation of a plane passing through the center of the sphere, and of course the shorter of the arcs of a great circle which passes from one of the given points to the other, is the required minimum distance. Remark. — Besides the minimum thus determined, which may be called the absolute minimum on the spheric surface, there is what may be called the relative maximum. For the lesser arc of the great circle, between the points being a minimum, the remaining arc of the same great circle will be SMALL CIRCULAR ARCS. 849 the greatest distance on the surface between the points ; sup- posing the distance to be measured in planes passing through the points. (8.) We may now proceed to show how to distinguish be- tween the maxima and minima in examples, but shall refer for this to Art. 876, p. 807, of the " Calcul Integral" of Lacroix; noticing, that the maxima and minima can often be distinguished from each other by the nature of the case, as in the examples which have been given. As we do not profess, in what has been done, to have given any thing more than the first principles of the Calculus of Variations, we must, for more ample details, refer to larger works : such as "Woodhouse's " Treatise on the Calcu- lus of Variations," and the " Calcul Integral" of Lacroix, at p. 721. (See p. 614, Appendix.) SECTION III. INTEGRATION OF RATIONAL FUNCTIONS OF SINGLE VARIA- BLES, MULTIPLIED BY THE DIFFERENTIAL OF THE VARI- ABLE. (1.) It is clear that sucli differentials must be of one of tlie two forms (Aaj« + Baj* -f Cx' + &c.) dx, A.T" + Bx^ + Cx' + &c. , A'x"" + B V + G'x' + kc. ' in which tne indices of x are supposed to be positive in- tegers. Supposing the terms of these expressions to be arranged according to the descending or ascending powers of a?, we may suppose the index of the highest power of x in the numerator of the fractional form to be less than the index of the highest power of x in the denominator ; for if the index of x in the numerator is equal to or greater than in the denominator, it may be made less by arranging the terms of the numerator and denominator according to the descending powers of x, and then dividing the numerator by the denominator, when the fractional form will be reduced partly or wholly to the first of the preceding forms, accord- ingly as the numerator is not or is exactly divisible by the denominator. (2.) By proceeding as in (9.) at p. 26G, we may clearly suppose the integrals of all such differentials as the above to EATIONAL FUNCTIONS. 851 be found to any degree of exactness that may be required ; whicti is clear from the circumstance mentioned by Newton, tliat they have the sums of the inscribed and circumscribed rectangles for their less and greater limits. (3.) If we have differentials of the preceding forms, in which the indices of oc are some of them positive fractions, by reducing the indices to their least common denominator, and representing unity divided by the least common denomi- nator by - , and putting y = jtp , ox x = y^^ we shall have 1 dx—pyP-^dy\ consequently, putting y for a?^ and j9?/^~^58 • RATIONAL FUNCTIONS 2 1 2 2 1 9 {x — 2){x — 5) 27 (^ — 2) 27 » — 5 ' whicli must equal the remaining fraction, and, of course, Hence, from the substitution of the preceding values, the integral becomes J {x-^f {x- 5) "^ ~ 6F^' ~ 9 S=:2 "^ ^""^ ^ l;zj-6/ ' In like manner, we have (2-^y _ 1 B c (3 - a;)« (3 - ir)^ "^ (8 - a?/ "^ 3 - a;' which gives B = — 2 and C = 1 ; consequently, we shall have the integral / 6. To find the intepral of 7 -. . ° {x — ay Here we assume x^ A B C_ B_ (ic — of {x — ay (a? — af {x — af x — a or x'' = A-\-B(x — a) + 0(x — af + 'D{x — af, which must clearly be an identical equation ; consequently, putting a for x, we get A = a\ and, taking the differential of the members of the equation after dividing by dx, we have 3a;= = B + 2C (a; — a) + 3D (a; - af, which, by putting a for x, reduces to B = Sa\ By taking the differentials of the members of -the preceding equation, we have, after dividing by dx, 6aj = 2C + 6D (a? — a), which. FRACTIONAL FORMS 859 by putting a for x^ gives C =r 3a ; and taking the differen- tials again and proceeding as before, we bave D =: 1. Hence, we shall have Qi?clx /, {x — df a' 3a' 3a T . . ^ Eemark.— -The method here used for finding the value of A, B, &c., appears to be of remarkable simplicity, and can clearly, be applied in all analogous cases. Othervnse, and more simply. — Put x—a = z or a7=s + a ; then, since a? = s -|- a, we have dx = dz, and thence xhlx / > ' ' ., is reduced to (x — a^) r{z + dfdz _ r/(7z Sa^^^ Za-dz (^dz\ J z^ ~~ J \z ^'z""^^ '^ z^ "^ ^J 3a _ 3a' a^ , OU/ Oct u, ^ log 2 _ _- _ _ _-^ + C. (7.) To complete the integration of rational fractional differentials, it clearly follows from -what has been done, that it is necessary to reduce the integrals of differentials dz of tbe form j-^. j^—, in which m is a positive integral {z^ + h-f ^ ^ greater than unity, to that of like form in which m — 1 Thus, from dz z^dz Irdz {z^ + hy-' {z' + bY {s' + hy z _ dz 2 (?7i — 1) z^dz by eliminating ^^,^_^ly we get 860 FRACTIONAL FOllMS. (27/2- - 3) dz z_ 2{m~ \)lMz or dividing by 2 (?/i — 1) h^^ and taking the integrals of the quotients, we have z 2;m — 3 r dz 2 {ni - 1) <^-^ (3- + //)"'-^ "^ 2 (//2 - i)W (?T-^'r~' ' which reduces the proposed integral to that of dz f ^ and by changing 7n into m — 1, we may in like mannor re- duce the integral and so on to the integral of /-^ rr. , which equals -j tan "^ -7 ; z' -{- 0- ' * consequently, all the preceding integrals pan be found, as required. Thus, if J = 1 and m = 2, we shall have C being the arbitrary constant Also, if ^ = 1 and ?ji — 8 we shall have r_j}l__ _^ 3 r dz J {i- + I)-' ~\{z--\- Vf "^ iJ {z' + i)--' ' which, from /, FRACTIONAL FORMS. 861 dz Z " (^•2 + 1)- 2(s^- + 1)' 2 is reducible to + ^tan-^s, / dz z , 3s , S 0. 1 , n + TTT-^— .-^ + 77 taa-^5J 4- 0. {z' + Vf 4(^^ + 1)^ ' 8(^^ + 1) * Otherwise. — Supposing the integral dz f dz _ r J _ite„_,f -^n^ + p) to commence with z, then, by taking the differentials of the members of the equation, regarding h alone to be variable, we evidently get - ^^'^^fj¥T¥f = - ? *^""i + J '^ ^^'^^ I ; or smce d tan-^ n= 72--^-ll + pi, by substitution and dividing the members of the resulting equation by — %db^ we shall get, after adding a constant, for correction, dz 1 /, tan-'y + {z^-\-V'f~W h ' W{z'-\:¥) ' It is evident that, by taking the differentials of the members of this equation, regarding h alone as variable, we may, in /dz -r-2 — —j^i 5 s^nd so on to any extent that may be required. (8.) From what is said at p. 851, it is clear that if the dif- ferential of a variable contains terms which are affected with positive fractional exponents when the differential is of an integral form, or positive and negative exponents when the 16 3G2 FRACTIONAL FORMS. differential is of a fractional form, that the differentials m^y be changed into others in which the exponents shall be posi- tive integers, or, as is usually said, the expressions may be rationalized. Thus, the differential {ax^ + ^a? ) dx^ Avhich is of an in- tegral form, by reducing the indices of x to a common de- nominator, is equivalent to {ax^ + hx^) dx ; which, by put- ting X = s®, and dx = ^^^dz, is reduced to the integral form 63^ {az^ + hz^) dz, which is rationalized, or the exponents of z are integers. By taking the integral of the transformed differential, we shall have f{6az'^ + Qbz") dz = ^ az' +^hz' + G; or, putting for z its value a?^, we have J acc^ + 3 hx^ + C, for the integral; C being the arbitrary constant Also, the integral / -r-- — 7 ™^7 clearly be rationalized */ a;* — a?* by putting x = z^ and dx = 6z^dz^ which will give 1= 2z' + Bz' + 63 -f 6 log (s - 1) + C ; which, since z = x", is easily reduced to r /^^ = 2x/x + 3^^ 4- ei^x + log (.i^*- 1) + C. J X^ — TT /nx _j_ ^ — Y' dxj may be freed from the nega- x^ + X' -^ tive index of x in its numerator, by multiplying its numer- FEACTIONAL FORMS. 363 ator and denominator by a?*, wliicli reduces it to rax~^ + 1 , r a + x^ J J ~l ^- ^^ = / -^ 1 ^-^5 •^ a?* + a; ' =^ *^ x^ + x^ wliicli, by putting x — s^^, becomes 12 r^l, ."& = 12 f^ + 12 f^ = a [62^ - 125 + 12 log {z + 1)] ~T~~ — I ' ^y putting a? = 2^°, becomes a;^/+ a?' J ^ + z^ J z^ + 1 J \ z^ + 1} = 2^-5.^ + 10/^-^^3; noticing that this integral can be easily found by diverging series. (9.) If the surds which enter into the differential coeffi- cients of a given binomial form, contain the simple power of the variable, then it is clear that the differential may be rationalized in like manner as before. Thus the differential [3 {a + hxf + 2 (« + hxf] dx is rationalized by putting a + hx = z^, which gives 6Mz ax = — - — ; and thence the proposed differential becomes (33V 2^) x^^^ 364 FRACTIONAL FORMS. which is of a rational form, which reduces the integral of the proposed differential to the same result that the immediate integration of the pro- posed differential will give. Ai ^x. ' ^ 1 ^ (« + hx)^ + (a + hxf , . ., Also the mteffral of ^ H ~ ax, is easily {a-\-hxf ^ rationalized by putting a + 2>a7 = s^^, which gives Vlz'Hz ax = — Y — ; and thence the proposed differential is reduced to the rational differential 122^^ 1i2z''dz whose integral is 4z'' 1 23" 4 (a + hxf^ 1 2 (a + hxf^ ^ ~bh + 136" + ^- U + 135 + ^' adx The integral of — -t-y^ — j-r is rationalized by putting 2^dz a^ — hx = s^, which gives dx= ^^j— ; and thence the proposed differential is reduced to the rational differential /adz _^ r dz 1 r dz z^ — a^ ~ 2J z~-^ ~~ 2J z~+a ' whose integral is log4/iZ« + c, or fJ.^-l-^ = logCC-Zl^t ^ ^ z + a ' J z^ — a^ ^ \z + aj' as required. RATIONALIZING INTEGRALS. 365 icdis The integral of ' - — r is rationalized by putting 1 -f- a? = s^, which gives dx = ^zdz and x ^=- z^ — 1, which reduces the proposed differential to 2 {f — 1) dz^ whose inte- gral is xdij The differential ^ — -^ , by putting 1 + a? = s^, is re- (1 + a?)^ duced to the rational differential Zz^dz — Szdz, whose in- tegrai is -^ ^ — [- kj, as required. (10.) We now propose to show how to rationalize differ- entials whose coefficients involve the square root of an ex- pression of the form a -{- hx + cx^, or an expression that may be supposed to be comprehended by this form or come under it. dx Thus, to rationalize the differential ' , ^ , ^ we Va -\- bx + cx^ assume a + hx -{- cx^ = (x -{- zfc = x^c + 2xzg + z% which gives a -{-Ix = 2xzg -\- z^c, and gives j/^ + bx -\- cx^ Va + hx + a? = - \/G and thence x = ^ ; 2gz — 0- by adding z to a?, we have {2gz — h) and by taking the differential of the value of x we also have _ —'^o{a — bz + ez^)dB '^'^ - {2gz - hf • 366 RATIONALIZING FORMS. Hence, from the substitution of these values in the given differential, it becomes 2 {a— hz -h C2^) i^cdz ^ " {2cz-h) ia-hz + cz-y or by reduction we have the differential -^ — , , which is reduced to — -r-; jr --■ yc, which, by integration, gives 2 — \- z for the mteerral / — -, ^ ^ , as re- quired. If c is negative, or the proposed differential of the form dx . , ^= , we may find the factors of a ■\-hx— ex- by r (I -j- OX — CXi solving the quadratic equation a -\-l}x -- cx' =. 0, or its equivalent x^ x = -] whose roots will be found to be c c X- ^^ and X- 2^ the first being positive, and the second negative when a is positive. Hence, if a' and V stand for the first and second of these roots, we shall evidently, from well-known principles, have CL OX {a! — x) (x — h') equal to — h x^\ consequently, the G dx proposed differential is reduced to — = V{a' -x){x-b') To rationalize this differential, we may assume (a' — x) (x — h') = (x — h'fz" or a' — x = (x — Z>') z', , . , . a' + h'z' which gives x ~ — ^ ; RATIONALIZIJS^G FORMS. 867 wKose differential gives _2(h' — a') zdz "^'^ - {z' + If ' Hence, since Via' — x) {x — V) = l-r-r^) ^> dx _ 2(73 ^^ ^^"y S*^* VeV {a'-^){x-b' ) = (? + "r)' '"'""'' '' a rational differential, since z is not affected by the surd sign. Remark. — If tlie proposed differential is of the form j^{a -\- Ix + car) dx, bj multiplying and dividing hj \/{a + hx i- car) we have {a + bx -{- cify) dx \/{a + hx -{- cx^) ' wbicli is equivalent to adx hxdx cx'^dx + -77—1 : ?: + |/(a -{-hx -\- cx^) \/{a + hx + car) \/{a + bx + cx^y in wbicb, as in the preceding examples, the irrationality is brought into the denominator of a fraction ; which we may clearly always suppose to be done in practice. To illustrate what has been done, take the following EXAMPLES. dx 1. To find the integral of the differential — — - — - . Va -\- Gx^ Because the iirst of the preceding general forms, or the general form in (10), by putting J = 0, is reduced to the pro- posed example ; it clearly follows, that by putting Z> = in the results in (10), we shall get the corresponding results in 368 RATIONALIZING FORMS. the preceding example. Hence, we shall get -— log Cz for the integral^ .— --^ > as required. dx 2. To integrate the differential a -\- hx — u^' By putting 1 for c in the second of the preceding general forms (see p. 366), we shall have x = — ^ r^ , a' and ¥ being the roots of the equation a^ — hx — a=0; and /v(a/L-a^) = -^*'^""'^ + ^' or since (from p. 366), z = y - — j-, , we shall have /dx _ p o -1 /^' i/(a 4-hx — x^)~ '^ ' X - — X \/{a -^-hx — x^) ~ ^ *^ ''"" ^ X — b'' dx 3. To integrate the differential — -j-^ — ^~¥. • It is manifest from the nature of a differential, that the dx integral of the differential — in 3, must be ex- Va^ + (rx" pressed by a logarithm, and be of the form divided by = — ^ we have - — - = — 1 an integer, and we have z '* + C = ^ + ^- 4/(1 /x'dx Because the integral is equivalent to I af {oc^ + a^)~^ dx^ we have m = 5, /i = 2, and j9 = — 1, and =r - = 3 an integer, 7i 2 and we have fa^ {x'' + a')-' dx = ^ f{z - ajz-^dz = - - a-^ + - log s + C = -^ ^ + — log (aHiC') + C ; which may also be found by converting the fraction -^ 5 into a series, arranged according to the descending powers of X, and then taking the integral of the quotient Since m = 5, 71 = 2, and j? = —-^ the equation /I r "l±l_i x"" (a + hxydx = — „^^ J iz — a) » z^dz becomes fa^ {a + Wy^dx = ^ f{z - afz-^dz = 376 INTEGRALa 7. To find the integral / ar^ (a -f hjr^fdx. This integral can clearly be easily found, since m+l —1+1 0, 1 ^ m +1 1 and ^dz 1 /» m +1 _^ is reduced to 7: I {z — a)-^z^dz = 7: I SJ ^ ^ S*/ z — a By putting s = y^ we have dz = 8?/^cZy, and s*c?s = 3yWy, and thence we have By putting y = va^, the integral / - is reduced to / -3 -■ ; whose integral can be found from the principles at page 371, «fec. 8. To find the integral / x~'^{a -\- hx^) dx. /hx^ x~'^ (a + hx'^) dx equals a log aj + — -f- C, as re- quired. SECTION IV. EEDUCTIONS OF BINOMIAL DIFFERENTIALS TO OTHERS OF MORE SIMPLE FORMS. (1.) These reductions generally result from tlie differen- tial dxy = ydx + xdy^ wliich gives ydx = dxy — xdy and / ydx = xy — I xdy^ or / xdy := xy — I ydx, whicli is called integyvition hy parts ; and reduces the in- tegral / ydx to the integral / xdy^ or the integral / xdy to / ydx. Thus, if we represent {a + hx'^y by 2^, we shall have {a + hx^'Y = zP, and thence {a + hx^'Yx'^dx = z^d I x^ dx = z^d m + V which gives fx'^zHx = zP — ^^ ^^~ fx"^ + ''zP-'^dx ....(«); J m + 1 m+lJ ^ ^' which reduces the integral / x"^z^dx to the integral / g,m + n^p-i^^^ jjj which p is diminished by unity, while 7fi is increased by 71. Also, from x'^z^dx = x^'-^'^^d l{a + hx^^yx^'-^dx, 378 INTEGRATING. we shall, as before, get wliich shows that the integral / x'^z^dx is reduced to the integi-al jx^-''z^ + ^dx. Because the integrals in the right members of (a) and {h) admit of like changes, it clearly follows, if j9 is a positive integer greater than 1, while m + 1, w + n + l, m-}-2;i-f 1, &c., are finite, that the exponent /> will finally, by successive applications of (a), be reduced to unity, and thence the in- tegral I x"* {a + hx'ydx will be determined ; and in like manner, from (^), if jt? is a negativ^e integer numerically greater than 1, while h, p -\- 1, p -\- 2, &;c., are finite, it is manifest that the integral will be reduced, by successive applications of (5), to an integral in which a + hx'' will enter in the form (a + 5a?") -^ = ^— ; consequently, agreeably to what has heretofore been shown, the integral will be re- duced to the integral of a fractional differential, having a rational denominator, and is to be, according to what has been shown, regarded as known. (2.) From z = a + Ja?", we get a = z — hx'' and h = ^ar"" — oa?"", and thence afx^'z^dx = J x'^'zf'^^dx — ifx^^^^'z^dx, and ifxTz^dx = J x'^-^'z^ + \Jx — ajx^-'^z^dx. Since, by putting p + 1 for p in (a), it reduces faTz^^'dx to gp^i-^!!^ __ iP + 1) ^^-^ fx^^-z^dx, J m-f-1 m-\-\ J J aim-^l^ a (m -hi) J INTEGRATmG. 879 tlie first of the preceding equations gives a.m+1 {pn-^n 4- m. + 1)?; a (??2 + l) a (m -\- 1) ; ("); and, since by changing m into m — n^ and p into p + 1, in (rt), it gives /m— n + i (n-\-V\nl) r ?7i — n + 1 m — /i -|- 1./ which being substituted for / a?"^~"s^"^ V^^ in the second of the same equations, we readily get dx ,P+i {in — n -f- 1) a jx'^''sP(lx...{d). {pn + m + 1) h {pa + m + 1) ^ ■ It will be perceived that in (c), the proposed integral is reduced to an integral in which rn is changed into 7)i + ^, while in (d) it is changed into in — n. It is also clear that integrals which can not be reduced by {a) or {h\ or with difficulty, can often be easily reduced by {c) or {ti). Thus, the integral I x-- (a^ -i- o[r)-'^dx, in which m = — 2, n = 2, jp = — 1, a=za^^ and h = 1, is by (c) immediately reduced to Cx-^{a/-\-x-)-^dx zzz ar-2 + i _24-2-24-l /.— .; X / x-''^'-27hlx a^(-2 + l) aH-2 + 1) 1 i r dx 1 1 ^ , aj ^ a^'a? a' J a- -\- x^ a-x a* a In like manner, by {d) the integral / cc'^ (a- + x'^)-'^dx is easily reduced to fx' (p} + x')-'dx = '^~^ log (a^ + x') + C. 880 INTEGRATING. (3.) Multiplying the members of ;? = r/ 4- Jjf by x^'z^-'^dx, and taking the integrals of the equal products, we have J^xTz^diii} — aJx^'z'P-^dx + ljx'^'^''z^-^dx. To the products of the members of this by — — - , adding the corresponding members of (a), we get \ m±V J m + l m + lJ ' or its equivalent fx'^z^dx = z^ — —- — r + ^!L- [x'^'z^-^dx. ..(e)', which reduces the integral / x^z^dx to the integral / x'^z^'^dx, in which z^ is changed into 2^"^ Thus, if 2 = a^ + a?^ we have j9 = 1, a = ar, and n = 2, and thence get a?"*2^a? = z ~ H — / x'^dx m + 3 m-\-^J ^ '^ 7/z + 3 m 4- 3 m + 1 ' which is clearly the same result, that the immediate integral of the proposed differential will give. If ^ stands for a positive integer, it is clear that successive applications of the above formula will reduce / a?'" z^ dx to Jx^'z^-^d.e, fx'^z^-^dx fx'^dx. Changing^ in (c) into 2> + 1? multiplying its members by (/> + l)7i + m+l, transposing, &c., we have ^'^' -7^—7-.- + INTEGEATING. 381 wliich reduces ix'^z^dx to fx'^zP+'^dXj and Cx'^'z^^hlx to fx^'z^+'^dx, and so on. Thus, if jp = — 8 we have y .'".-c^^ .. — ^_-_y .'^.-.z.; and then x'^z-^dx = — ^— / x^z-^dx. 7ia na J Hence, if z=za^ -\- a? and 7n is a positive integer, the integral J X" -[- or x^ which, by converting ,, — ^ into a series arranged accord- ing to the descending powers of x^ can now be easily fonnd by the common methods of integration. We will now, for convenience in what is to folk>w, collect the preceding formulas into a (4.) Table of Formulas for the Redtiction of the Litegral Cx'^ia + hxydx = fx"'zHx, I. fx'^zHx := Z^-—^ - ^^~ fx'^^^zP-^dx. J m + 1 m ■\- \ J 382 INTEGRATING. 4 II. III. •^ aim + 1) « (m +1) J a{fa + 1) a {m + 1) IV. V. futTz^dx = ;sP ?^^^^ + ^^ — -, fx^'z^-'dx. *> vn + m + 1 jm + m + I*' jpn + m + 1 2^n VI. ^ a(^ + l)?i a(^H-l)n J This table, under a different arrangement, is substantially the same as that of Mr. Young, at page 42 of his " Integral Calculus ;" noticing, that our formula I. takes the place of his formula II., which is incorrect. (5.) To perceive the use of the formulas, take the follow- ing EXAMPLES. * 1. To find the integral Jx-^{l—x'fdx, Q Since m = — 4, n = 2, ^ = - , a = 1, and J = — 1, it is clear that formula I. reduces the integral to INTEGRATING. 383 fx-'z^dx = Z^^~fx-'^'3^~'dx and another application of L, reduces I x~^ (1 — x^ydx to /-. oA^ C dx 4/(1 — 07^) . , ^ ^ ^ X J \/{^ — x-) X hence, we have fx-' (1 - x^fdx = - ^,^' + S^-^' + sin-^aj + C. •/ ^ 3,r^ a? 2. To find the integral fx^ (1 — .'»')-^(/.z; == Jx's-hix. Since m rz: 5, n = 2, ^ = — 3, a = 1, and Z> == — 1, from II., we have / x^2~^dx= — J / xh~^dx; and another application of ii. reduces / x^2~'dx to / x^2--dx = — ~ / X2~^dx. Hence, / x^2-^dx = —^ ~ + / xz-^dx' since J xz'Hx =^j J^J^^. = - ^ ^^S (1 - ^) + C, the integral becomes /x^dx _ a?* x^ log (1 — a?^) ^ (1 - x'Y ~" 4 (1 - £C^ ~ 2 (1 - ar) 2 " "^ * 3. To find the integral Jx-^{a^ + a^2)-icZ;r. From m = — 4, m. = 2, ^ = — 1, a = a-, and J = 1, we get, from iii., 384 INTEGRATING. fx-' {a? + x^)-\lx =fx-'3-'dx = ^^^ - \fx-h-hix and from another application of ill., we have /x-'^z-\ix = — —^ / ~^z-^dx = 7.— I -Trrv-J,' Hence, we have and since dx \_CJ^__\ /*IZI-ltan-i-- aVa^ + ar^~a^' J ^ ^ ~ a^ a' a" consequently, we shall have 4. To find the integral J x^ {a^ + x^)-^dx. From m = 5, 71 = 2, ^ = — 1, a = a-, and J = 1, we shall, from IV., get fa^iw" + aP)-'dx =fx'2-hlx x^ aV y_ r^. dy e" /sin v \ 1 / e° sm V -4 = -rrr^ I ^ — cos ?/ - + const, ILLUSTRATED BY EXAMPLES. 401 r ^- dy e" /cos y . \ 1 and / e"" cos y ~ z=z — — I ^ -f sm ?/ 1 - + const. J ^ a- /ly .2 ^ « '^1 a By re-substituting the value of ?/, we shall have e-^ sin ascdx = 5 (sin ax — a cos ax) + const., and / 6 ^ cos aa?c?aj = 5 (cos ax -h a sin aa?) + const. J 1 + a^ 4. To find / e^ sin'^ xdx, and / e^ cos^ xdx. From the tables given at pp. 77 and 78, we have .0 8 sin a? — sin 3a? , „ 3 cos a? + cos 3a? sm.-^ X = , and cos^ x = ^ , 4 ' 4 ' which reduce the integrals to / e^ sin'^ xdx = - / 6^ sin xdx — j I ^^ sin Sxdx^ and / e^ cos'' xdx z=z - I e^ cos xdx + t I ^^ cos 3a'cZa?. • By taking the integrals by (a) and {h), agreeably to what has just been shown, we have / e^ si sin'^ xdx = -Q- (sin x — cos ^) ~ -jK (sin 8a? — 3 cos 8a;) + const, /. and / e^ cos^ xdx — 8e^ . e^ -^ (cos X + sin a?) + -— (cos 8aj + 3 sin Sa?) + const. (10.) We will now show how to find the integrals of dif- ferentials into whose differential coefficients enter arcs with 402 INTEGRALS OF VARIOUS FORMS. algebraic functions of the arcs. Thus, to integrate / X (sin~^ xY djc and / X (cos~^ ic)" o?.r, X being an algebraic function of the arc, it is clear that we may put / X.dx = Xi , and thence, integrating by parts, get fx (sin-^ xf dx = (sin-^ xf X,- nf{sm-' xy-' X, /^ .. , , . , , . r x,dx which, by putting y -—-—^ gives {sm-'xy-'X,-{?i-l)f{sm-'xV-'X,^^,; and so on to any required extent in this, and such forms as JX (cos -^ a-)" dx, Jx (tan-^ xf dx, JX (cot"' xf dx . . . (A). EXAMPLES. 1. To find I X sin~'^ xdx and I x cos~^xdx. /x- xdx = -, which gives X" Xi = — ; consequently, dx , 1 /* ij^dx f^'^Ai^) ^""^^^"' If |/(l-ar^) 2 J 1/(1 -a^')' and thence fx sin - > Xo, = «_i!il.^ _ ip (1 _ ,^) - ij, 4/(1 -fJ) ar' sin-^'» 1 ,, ^, 1 /* dx 2aj2 1 1 — - — sin~^ ^ + J a?4/(l — ar') + const. ; ILLUSTRATED BY EXAMPLES. 403 and in like manner, /2a!^ + 1 1 X 0,0^-^ xdx — V — cos ~^ a? — j x\/{l — x'^) -h const. 2. To find / a?"^ - ^ sin - ^ xdx and / x "' - ^ cos - ^ xdx. x"^~'^dx = — , we shall, by integrating by parts, from (A) get r , . , y sin - ' a:'!^?"' 1 T « dx I x"' ~'- sm "" ^ xdx = / a?"' -—, — -— ^ , J m mJ |/( — X-) ■X r . A 17 co^-^x^x"^ 1 r „. dx and / a?"'-^ cos~^ xdxz=^ -\ \ x^ — ^ J m, tnJ 4/ (;1 — X' we also Lav dx \/{l — ar)x'^-'^ m — lr o dx r..m dx _ 4/(1 - x^) X--' m-l r J 4/(1 -^'')~ 'i^' m J^ |/(1-^'')' and by tbe same process dx _l^{\—Xr)X'^~^ ^^^'-^ /* ,n-4 ^^ C^..-. dx ^ v{i-xr) x-~^ _ m-s r ,__ and so on to any required extent. It is hence clear, that if m is an odd positive integer, the complete integrals of the proposed integrals will be algebraic ; while if r/i is an even positive integer, they will be reducible to circular arcs or be dependent on them. Remark. — It is easy to perceive that like conclusions are applicable to the integrals -'^ ian-'^ xdx and I x''-^ tan.-^ xdx. I x"'-'^ tan-'^xdx and / ; 404 INTEGRALS OF VARIOUS FORMS. 3. To find / — ;;;^in— = / sin - ^ ica;-"^^ dx and / cos"^ xar"'-'^ dx , /x~^ x-""-^ dx = = Xi, these integrals become /, ^ . , sin-^«,aj-"* 1 /• dx sm-'xx-"'-^dx= i + - / -i;r-^7T ^^ m mJ x"' 4/ (1 — ar) ' , /*cos ~ ^ xdx _ cos ~^x 1 r J a."*+"^'~ ~ irix"" mJ x' dx If m is a positive integer not less than 2, we shall have m — 1 m— w ^ ^ bj putting for ?i its value 2. By changing — m into — 771+2 we shall, in the same way, have m — 3 7)%—ZJ ^ ' and so on. Hence, if m is an odd positive integer, we shall C dx T 1 + i'(l — •'»') n which will enable us to find the integrals corresponding to any other odd positive integer, while it is manifest from what is done above, that when m is an even positive integer, the integral is algebraic, and can be exactly found by the preceding process. Thus ILLUSTRATED BY EXAMPLES. 405 - 2 ^ .„ - 2 sm-^x 1 (1 -«-)^ ,2 c, „„ + const; ^x" 2 X and r 1 w COS-^'2? 1 (1 — 9?^)"^ , , / QO^-^x^x-^dx= — — ^-.^ \- ^ + const. Remark. — It is clear that we may, in mucli the same way, « , f* isin~^xdx , rGoi~^xdx fl°d y — ™-T^ and j-~„-iT-. 4. To find I {sm-'^x)'dx and I x-{co8-'^xydx. Since X = 1 we have / Xc/^ = a?, and thence, from (A), I (sin-^ xf dx = (sin-^ «)^ a? — 2 / sin-^ a?, ' - — ^ , and in like manner from X = a?-, we have / x^ dx = ~ ^ and thence from (A), J ar (cos-^ xfdx = —^ h o / cos"' a?— yr- We also have / sin-^ ^ ^ V (1 — i^ ) xdx „ ....,,, .., dx sm dn-i a? |/(1 -x") + f ^{\-a?) ~ — sin-^ a? y (1 — a?^) + cc ; consequently, we shall have J (sin-i xj dx — (sin-^ aj)"a? + 2 sin-^ x |/(1 — ar^- 2a; + const 406 INTEGRALS OF VARIOUS FORMS. From IV., at page 382, we have "" 3 3 * and thence / cos"^ a?.2r' — t^^ :r, = - cos-^ X ^^^^—-—^- + - |/(1 _ x")! - consequently, <> £P* 2.'?/ g cos-^ x[^{l-x')x'-2 v'(l - ^') ] - 36 + 36 + ^^^s*- 5. To find / X tan"^ xdx and / a? cot~^ xdx, (Tfic = - , we have r . , ., , (tan-»aj)»aj= A _i ^'^^^ / a; (tan-'a?)^ dx =^ ^-- / tan^a; -^ and r , , ,, ^ (cot-' ajVa;^ r , , irWa? If at IV., at p. 382, we put a = 5 = 1, n = 2, and 7n — 2, it will give ILLUSTRATED BY EXAMPLES. 407 r xHx = fx' (1 4- xY '^ dx ^=^ X — r dx or taking the differentials, x^dx , r— 2 — dx — 1 + «^ dx which can be found more simply bj actual division. Hence, by substitution, we have / X (tan~^ a?)^ dx = (tan-^a^y^^ P i / , A i ^^'^ •^^ —-^ / tan-^ xdx + / tan"^ x z ^ , and / a7(cot- ^ x)-dx = — — + / cot- ^ xdx — / cot" ' x rr— — 2 ; and since / i^n-^xdx = tan-^ Xi^x — - log (1 + x"^), dx and r~7~^ ~ ^^ (t^^~^ x) = — d (cot-^ x). we shall have finally I a?(tan-^ xydx = -— ^ h ^^ — ^ tan-^ XiX + - log (1 + ar^) + tan"^ a? -f- 2 + const, and r / . 1 NO 7 (cot-*a?Viz;'^ (cot-^ ojV y a; (cot-i xy-diB = ^ ^-^— -f ^^ — ^—^ + cot-^ XiX + - log (1 + x") + const. (See Lacroix " Calcul Integral," pp. 95 and 06.) 408 INTEGRALS OF VARIOUS FORMS. Eemarks. — It is manifest, from what has been done, that to find an integral of the form \ a' + h' cos z) dz {a -{-h cos zY ' we ought to represent it by the form A sin z r{B + C cos i) dz r- J (a {a + b cos zy-'^ J {a + b cos 2)"-^ For by taking the differentials of these equals we have {a' + 1/ cos z) dz _ {a -\-b cos 2)" ~~ A cos zdz ('^~ ^) A J sin^ zdz (B + C cos 2) dz ^ (a + b cos s)"-^ {a ^- b cos z^ (oTTcosT)"^ ' or, by omitting the common factor dz and a simple reduc- tion, we have a' + Z>' cos s = A cos 2 (a + ^ cos s) + {n — 1) A5 (1 — cos'' 2) + (B + C cos z){(i-^b cos 2), or a' — (;i — 1) A5 — Ba + {b' — Aa + Bb — Ca) cos s - [Ab +(n-l)Ab+ Cb] cos'z = 0, which must clearly be an identical equation, and be satisfied so as to leave cos z and cos^ z arbitrary ; consequently, we must have a' -(n- 1) Ab-Ba = 0, b' -Aa-Bh- Ca = 0, M-{n-l)Ab-h Cb=0. From the last of these equations we immediately get C = (;i — 2) A, which reduces the second J'_ Aa-B5- {fi-2) Aa = 0, or b'-~Bb- (n-l) A=0, which gives B = j ; r ILLUSTRATED BY EXAMPLES. 409 and thence from the first equation we have . _ ah' — ha' {n-l){a}-hy and of course B=«4^ and c= (;-?,<;^'-y .. ar — 0^ {n — I) (<2- — h^) Hence, from the substitution of these values of A, B, 0, in the assumed integral equation, we shall have *{a' + h' cos z) dz _ {ah' — ha') sin 2 {a-\-h cos zy ^ (fi — 1) («■' — b') (a -{- h cos zf "^ ^ 1 r i{n-l){aa'-hh')-^{n-2){ah' -ha') cos z] {n-l){a'-¥)J ^ {a + h cos zY~' ^ so that the complete integral is reduced to that of another in which n is represented hj n — 1; consequently, if 7i is a positive integer greater than 1, we shall, by successive rep- etitions of the process, finally reduce the integral to that of an integral in which n is equal to unity, or to the form \p + 5' cos z) dz a -\-h cos z (see Lacroix, p. 109) ; noticing that this integral is reducible by division to the more simple form / q^ , hp — aq (* dz OS J "^ ^ h J ^TFT cos z If (with Lacroix, at p. 106) we put cos z = ^ , we shall X ~\~ OS get f f '=2f- ° J a -\- cos z J a dx + J + {a-h)x'' whose right member is clearly of an integrable form. Since cos z — cos^ - — sin^ ^ , we shall have 18 410 INTEGRALS OF VARIOUS FORMS. / dz _ r dz a + J COS 2 "~ / , , / 7^ . .z\ %/ a + I cos' - — sin'^ -1 _ r dz J cos^ I Ua -\.h) + {a-h) tan' |) Va — b di 2 f_±±]Ll /a — h dz .2 — — o- -^ cos' - a + 5 2 in wliich a and h are supposed to be positive, a being greater than h. Since o + 1 a/^ "~ ^ 4. ^ ^ 1 sin s 4/ («' — J") 2 tan-^ y y tan - = tan"^ — ^ ^ ^ ^ , a + J 2 5 + acos2' we liave r dz 1 ^ , sins 4/ (a' — J') / — r-T = -77-1 F\ *^"~ r . ^ +const.: J a + cos s 4/ (a^ — 6-) 6 + « cos s ' noticing, that the same integral may also be expressed by either of the forms / dz 1 . sins 4/ (a' -Z>') , 1 = -m> 7^ sm - f-^ ^ + const., a -\- cos z \/ {a' — 0') a -\- cos z dz _ 1 _ J J + a cos z 7 = — r-s fT, cos ~ ^ y h const. a + b cosz \/or —0^ a + cos z (11.) We have shown, at p. 262, that every differential expression containing a single variable, admits of an integral of either a diverging or converging form, by integrating by parts, as in John JSei'tiouillis Theorein. We have also applied, at the same page, the Theorem of Maclaurin to ob- tain series of more rapid converge ncy than can often be done ILLUSTRATED BY EXAMPLES. ^^ 4:11 by tlie aid of the Theorem of Bernouilli ; and from the problem at p. 266 we have obtained formulas for the compu- tation of such integrals by series of any degree of con- vcrgency that may be required. Because, in what has been done, the series have been supposed to be arranged according to the ascending powers of the independent variable, we now propose to show how to apply series to find integrals when the series are arranged either according to the ascending or descending powers of the independent variable. 1st. To find the integral of a proposed differential by a se- ries, it is manifestly necessary to convert the differential co- efficient of the differential of the independent variable, according to the known methods, into a series arranged either according to the ascending or descending powers of the in- dependent variable ; then, to multiply the terms of the series by the differential of the (independent) variable, and to add an arbitrary constant to the sum of the integrals of the pro- ducts, for the integral of the proposed differential. It is manifest that the sum or generating function of the series thus found will be the finite integral of the proposed differential EXAMPLES. /dx A~~ — ^ ^y ^ series, arranged either according to the ascending or descending powers of x. By dividing dx by 1 -|- a?^, when 1 is taken for the first term of the divisor, we have / :j — '- — 2 = / {dx — x'dx + xdx^ — xdx^ + &c. + a^ ^ X "3"^ 5"~ 7 -^ + -F — TT + &c. 4- const, 412 INTEGRALS OF VARIOUS FORMS. for the development when the series is arranged according to the ascending powers of the variable. And by taking ^ for the first term of the divisor, we have / dx _ Cidx dx dx dx \ for the form of the integral, when it is arranged according to the descending powers of x. To find the constant, we re- mark, that X being the tangent of an arc whose radius = 1, it is clear that, supposing the arc and tangent to begin to- gether, the constant in the first integral must equal naught, and the integral becomes dx ^ x' x^ while, by supposing x to be unlimitedly great in the second integral, it will clearly be reduced to the constant in its right member, since the terms which involve x must clearly be rejected on account of the infinite value of a?, and at the same time / -^ r must equal ^ , the length of the arc of the quadrant of a circle whose radius == 1 ; consequently, the constant in the second integral equals '- , and the integral becomes /i / dx TT 1 1 11 - a.-2 + 1 ~ 2 x^ Zx^ bx'^ Ix' ' /dx — -,■:, s^ = si \/{l-x-) sm-^ic. in a series arranged according to the ascending powers of a?, 1 , 1.3 , = ^+2:3^ + 2A5+'^^-' ILLUSTRATED BY EXAMPLES. 413 wHcli needs no correction, supposing the arc and sine to commence together. Since the binomial theorem gives ^ .-. o.-i , a?-' 1 3 , 1 3 5 6 (1-^^) ^ = 1+ +x' +-.-.- x'+,kc., y{l-x') ^^ ^' ' 2 ' 2'4 '246 we shall have 1 , 1.3 , 1.3.5 , sm-^o. = ^ + O ^ + 27475 ^ + 2747 677 ^ +' ^''' which needs no correction, supposing the arc and sine to begin together. /dx —7rrT-^\ "=■ ^^g ['^ + 1/(1 + «^')] + CJ' in a series. Because 1 . 1.1.3, 1.3.5 , , , 4/(H-a5'^) 2 ' 2.4 2.4.6 we shall have log [x+V(\ + '. + i-^^ a^- i||- + , &c, which needs no correction, supposing the integral to com- mence with X ; which we clearly may do, since ic = gives log 1 == 0, as it ought to do. /dx Ti7~i~Zrr\ ~ ^^^ '-^ "^ \/{x^ — 1)] + a series. Since \/ {x^— 1) = X y ll A^ we clearly have 4/ (»2 - 1) x\ xV 1 _L 1-^ 1.3.5 aj'^2c^"^2.47?+ 274.6^^"^' ' 414 INTEGRALS OF VARIOUS FORMS. consequently, we shall have and by putting a? = 1 in this, since log 1 = 0, we have 1 J.^3 hJjL^ - Xr ^-~2.2 2.4.4 2.4.6.6 '*''•' and thence i/^^rz.i\— ^ _ , 2.2 ^ 2.4.4 ' 2.4.6.6 log {x+V^u^- 1)= A + o^i~T + o^^#A + &«• 1 1.3 1.3.5 + ^og aj - 2^-, - ^^^-4 - 274:6. 6^« +' ^^- = loD^ (a + a?) into a series. a + a? ° ^ ^ dx /I 37 a? ar \ Since = ( b H — <, ^ + &c. ) dx, we shall a + a? \a a- a^ a* / lyt /y»2 rti° /yi^ have log (a + x)^'--~ + ^,-^, + &c. + C, by putting a? == in this, we have log a = C; consequently, we have log (« + ^) = log a 4- - - 2-. + 3«-3 - 4-7. +. &=• Remarks. — It is easy to perceive that this deYclopmont can be immediately obtained from log a, by changing a into a + a?, and then developing log (a + a?) according to the ascending powers of ^, by Taylor's theorem. 6. To find the integTaiy|/'lz:_^Z: .j^ ^ f^~=^ ^^^ J. — a? V f L — X' m a series. ILLUSTRATED BY EXAMPLES. 415 Bj converting |/(1 — e-x^) into a series arranged according to tlie ascending powers of x, we have //v /dx ——, r- = sin-^a?, and that formula IV., at |/(1 — a?-) ' p. 382, gives r xHx r 2,. 2x-i^ a?4,/(l— a?-) 1 . . and /x' 1 3 \ ,,, ,-13. , = - (4 + 2 • i ^/ ^(^ ~ ""') "^ 2 • 4 ^'''~ ^' and fx'{l-a^)-^dx = \6 +4-6^ + 2*4:-6^r ^ "^ +2-4-6 and so on ; by collecting these results, we shall get 5 . , J V Y^'aT "^ sin-^a? + ^\^2^^'^ ^^ ^~ 2^^^~ j"^ e* f/l , 1 3 \ ,,, ., 1 3 . , 1 1.3«« 416 INTEGRALS OF VARIOUS FORMS. for the integral ; whicli needs no correction, supposing it to commence witti a*. Remarks — It is easy to show that the j -receding integral represents an arc of an ellipse, reckoned from the extremity of its minor axis. For let y = - |/ (a* — x^) represent the common equation of an ellipse, then if e equals the ratio of the distance of the focus from the center to the semi-greater axis, we shall have J = a |/(1 — ^) for the half minor axis, and the equation of the ellipse reduces to y = |/(1 — ^) i/(«^ — ar*) ; 'whose differential gives dy = — .^^ — ^ ox - • Hence dy' + d^ = d^^= (^=^4^^ +d^. or dz =l/^5'^ wliich reduce the differential equation to dz ^=^ a ^\ — ^ sin^ <^ x ci? ; if we put a = 1, the half major axis = 1,. and we have dz ^ y \ — ^ sin^ <^. c?0 or representmg rl— Vsin^ ^J ^j we shall have which is an elliptic function of the second kind, according to the notation of Legendre. (See p. 19 of his Exercises, "DeCalcul Integral.") ILLUSTRATED BY EXAMPLES. 417 Q. . <, , 1 — COS 20 , Dince sm- 9 = , we have or putting 20 = 6^, we have c?0 = — ; consequently, we sliall have dz=z |/(1- 6= sin^ 0) d<^ = r -y-- X |/(l + ^372 cos ^^t^^, or putting ^ = ^1 w® ^^^^^ -^^^'^ /o ^2 6^2 = y — — - 1/1 + c cos d (Id. o Bj tlie binomial theorem, 4/ (1 + c cos 0) = 1 + ^ c cos — ^ c^ cos^ ^ "^ Tft ^^ ^^^^ ^ — consequently, multiplying the terms of this by d6^ and taking the integrals from = to == tt, we shall have /V(i+.oos«)d.=.(i- y-^,0*- Ssr'- ^'=-)' which gives the quadrantal arc of the ellipse, reckoned jfrom the extremity of the minor axis, *^ 2 2 A_l_il__Jl^ ^5 _^L-_&e) \ 16(2-«7 1024 (2 - 'x + C; also J e^dx^ = fdx fdx {e" + C) = fdx (e« + Ca? + CO = ^" + -^ + C cc + Q'\ DEVELOPMENTS IN SERIES. 421 3. Mr. Young, at p. 91 of his "Integral Calculus," gives (/"- Xc&»-) ^^^ + . . . . (/X.&) j£lU + ^^r27...7i "^ \dx) 1.2. ..." + 1"^ \^ji:2 -^' '^'*''-' in whicli tlie development is. made according to the ascending integral powers of a?, by Maclaurin's theorem; ( / X(ia?"j denoting the last of the arbitrary constants according to the preceding methods of development, I / XcZa?"~M denot- ing the last constant but one, and so on until there are no arbitrary constants; noticing, that the terms within the parentheses stand for the values of the corresponding quan- tities, when a?, in them, is put equal to naught EXAMPLES. —7(-T- K according to the ascending powers of x. Since the binomial theorem gives ^ =l-^.^^ + ^a^^- &c., Vl+x-' 2 2.4 the development is 422 DEVELOPMENTS IN SERIES. 2.3.4.5.6 ' 2.8.4.5.6.7.8 x^ ^ X as given by Mr. Young. 2. To develop / sin xdx^ according to the ascending powers of a?. sin xdoi^ = C3 + CiX -{- Ci ^ -f cos a? = cos i» + Ci 2 + Cgaj + C3, as in Young. d*v 3. To develop -r^ according to tlie ascending powers of x. Here, we have y=:C,+ C,x + C,"^ + C,^', which, in the language of curves, denotes a parabola of the third order. 4. To develop / e^'d.x^. Here /3 Qfi e'di^ = C3 -h C.x + Ci 2 + e\ (13.) We now propose to show the use of arbitrary con- stants in finding definite integrals by series or otherwise. According to what is shown at p. 265, the notation dx n /; i/il-x") 2 DEVELOPMENTS IN SERIES. 423 signifies tliat the integral being taken from x = to £c = 1, gives ^ = one-fourtli of the circumference of a circle whose radius — i, for the result or value of the integral contained between the preceding limits ; and a like notation is to be used in all analogous cases of definite integrals. If we take the integrals indicated in example 7, at p. 386, from X = to x =z 1^ when m stands for an odd or even positive integer ; then, for 7n odd, we have the results r^ xdx _ r^ xHx _ 2 r^ xMx _ 2.4 J o|/(l-a.'^) "~ ' y o|/(L-^ ~V J 0,^/(1 -a;-) "" 875 ' r^ xHx _ 2.4.6 r^ x-'^'^Hx _ 2.4.6... %i J ^'^~x^)~ZJ7l J o|/a-aj-)"~ 3.5.7... (2m + 1)' by using 2n + 1 for m, and by proceeding in like manner for m even, we shall have r^ dx _ n r^ x'dx _ 1 ^ J o7(r-^"^ ~ 2' c/ o^{l-x-) ""2*2' /»! x'^d x ^ ij TT r^ Mx 1.3.5 n J VT^^' ~ 274:' 2' J oTTa / |/(l-aj'-) 2.4.6*2' ^ x'"dx _ 1.8.5.7 ... (2/1 — 1) TT o|/(l-a;2) 2.4.6.8 ... 2;j. '2' by using 2n for wi. It is easy to perceive, from a comparison of the preceding values, that if ?i is large we shall have J oTTi—^^ = J oTXT^^) '''^'^^' 1.3.5.7. . . . {2n - 1) TT _ 2.4.6 2?i '''' 2.4.6.8 . . .72/1 • 2 - 3A7 . . . .7(2rrhT) ^^^^^' , ,,' n 2.2.4.4.6.6 2^.2/?. or we shall have ^ - i.3.3.5.5.7...(2. - 1).(2.+ T) ""'"'^^'^ 424 DEVELOPMENTS IN SERIES. and by supposing n to be uulimitedly great, or infinitely great, we must evidently have r _ 2.2.4.4 2;i.2??. , 2 - 1.3.3Ao....(2;i-iy(2;rrT) ^^^""^^^^ for the length of the quadrantal arc of a circle whose radius = 1 : where it may be noticed that this expression seems to have been first discovered by Dr. Wallis. (See Young, pp. 97 and 98, and Lacroix, vol. iii., p. 415.) Kemarks. — Mr. Young, although he has with reason objected to the manner in which the formula of Wallis is frequently written by English authors, yet, at p. 97 of his work, he has written 2.2.4.4.6.6.8.8 . "" - ^ a ^ 2.2.4.4.6.6.8.8 J — 7z~i-r~^ — f' I-, i-r r^ t^ lor ^ , instead Cn. z — „ „ ^ — ^ — — ,. , 1.3.3.5.5.7.7.9.9 2' 1.3.3.0.6.7.7.9' which is its proper form when the numerator and denominator each consists of eight factors ; noticing, that the numerator and denominator of the fractional forms of - must each consist of the same number of factors as the preceding forma If we write the successive approximate values of - , after the factors common to their numerators and denominators are rejected, we shall have 2^ _ 4 _ 2^A4 _ 64 _ 19 2.2.4.4.2.2 _ 81_ — _ - _ l.d, ^ g g^ - 45 "" "^ 45' i.l. 1.5.5.7 ~ "^ 175 ' 2.2.4 .4.2.2.8.8 _ -, . n , 2. 2.4.4.2.2 .8.8.2.2 _ -, ^^ , 1.1.1.5.5.7.7.9 ~ "^' 1.1.1.1.1.7.7.9.9.11 ~ * '^' and so on. From these results it is clear that the successive terms approximate very slowly to the IcDgth of the quad- rantal arc, the last result being correct only to one place of decimala DEVELOPMENTS IN SERIES. 425 For anotlier example, we will show how to find the de- —rr^ IT* Because l — x^ = (1 — a^) (1 + x^\ we shall clearly have r^ dx _ f^ d x ^ 1 • f r' dx /, x" ISx' 1.3.5 , , \ "" ./ V (1 - a?^) 2 y ^(1 - a?"') "^ 2.3 */ oy(l - a;-) 1.3.5 /* x'dx 2.4.6 y i/(l-a^) '^' Because, from what is shown in the preceding examples we have p dx _ TT p /: Oy^dx 1 TT o|/(l-ar^) 2'^ o|/(l-a;'^-) ~ 2*2' x'^dx 1 3 TT |/(l-a^^) 2' 4* 2' and so on, we get bj substitution r ^^ fi /IV /1-3V /1.3.5\2 „ ] T — . oVx{i-x^) By putting y — 2^x we get dy = —, ay^ = U\ ; consequently, 426 DEVELOPMENTS IK SERIES. A^-m V(l-3') by putting | = ^. Hence, from the preceding example, we shall have which is twice the integral found in the preceding example. (14.) We will terminate this section by showing how to sum series, or to find their generating functions, by means of thie preceding principles. The processes here proposed seem to depend, for the most part, on transforming, by means of the integral or differen- tial calculus, the proposed series into a new series, or in find- ing a new series, such that its sum or generating function can be found, so that the proposed series may be supposed to have been derived from it. EXAMPLES. 1. To find the sum of the series ^ = a? + 2^ + 8aj^ 4- . . . . ^-nx\ Multiplying the members of this equation by — and taking the integrals of the products, we have I 8^- ^= I {dx + ^xdx -\- Scc^dx + &;c.) = X + a^ -{- ar^ + 4- a?**, DEVELOPMENTS IN SERIES. 427 which is clearly a geometrical progression ; whose sum, by the common rule, clearly equals "-— , and thence we /dx X ic" "^ ^ s — 1= ^ — . To find s, the sum of the pro- X 1 —X posed series, from this, we must remove the sign of integra- tion / , by taking the differentials of its members, which dx dx — (ri + 1) a?" dx + nx'' + ^ dx. or, by a simple reduction, we have _x — (/?, + l)r6'" + 1 + nx"" + 2 {1-xf 2. To find the generating function of the same series con- tinued indefinitely. It is manifest from development, that we shall here have s = -pr-- — ^2 ; which is clearly the same as to suppose the definite parts, or those that depend on 7?, in the preceding X sum, to destroy each other, and to put s = 3. To find the generating function of the series JJ.2 ^3 ^4 ^ + '^ + 2 + 2:3 + 2:3:4 + -^■■- Denote the sum by y, and we shall have y = 1 + X + ~ + Ig+j&c; whose differential coefficients give di/ x^ x^ ;s = ^ + *+2 +2r3 + *''- = ^' 428 DEVELOPMENTS IN SERIES. consequently, we shall thence get -— = dx. By taking the integrals of the members of this equation, we have log y — x^ which needs no correction, supposing it to commence with a?, since a; = gives 3/ = 1, whose log = ; consequently, put- ting e for the hyperbolic base, we shall, by the nature of logarithms, have y = e"^ and thence 6' = l+a,+ | +_+,&c., as required. 4. To find the sum of /;pn +11 aj" + ^ aj" ■•■ ^ * ^ ^TTT "^ 2J{n~^~^) "^ 2.3.5 (71 +T) "*"' ^^ From (J''), at p. 51, we have ^=1 + * + 2 +2:3 + 2X4.5 +'*"■' and putting — a? for a? in this, we also have e~^ = 1 — X ■ whose half diiference gives 2 - T = "' + O + rsTlis +"^°-' and multiplying the members of this by a?"-^ dx^ we have „n + l ' O Q/zK. I Q\ "T" O o ./I r: /., , k\ +» *^^-> n + 1 ' 2.3(w + 3) 2.3.4.5(^1 + 5) since the method of integration, explained at pp. 391 to S93, reduces the equation to ,9 =: ^ j e^ ic" ~'^dx~ I e^ a?" DEVELOPMENTS IN SERIES. 429 dx _ 1 X [-^n-i _ ^.^ _ 1) aj"-2+ (7i — !)(?! — 2)a;"-^— &c.] + ig-^ [cc"-^+ {n - l)aj"-2 + (7^ - l)(;i - 2) a!"-^ + &c.] ; consequently, from equating tlie values of 5, we sliall have ^ e^ [ajn-l _ (^^^ _ 1) ^^n-2 _^ (^^ _ I) (^ _ 2) «"-3 _ fc] + g-x ["a^n-i + (^ _ 1) ^n_2 _^ ^^ _ 1>^ (^^ _ 2) a?"-3 + &c.] = 71 + 1 ' 2.3 (/i + 3) ' 2.3.4.5(^ + 6) whicli needs no correction, supposing its members to com- mence with X. If we put n = 2 and a? = 1 in this equation, we have e-^=l:=l + ^ + -^_ +, &c.; which is the same result that Mr. Young has found at p. 100 of his "Integral Calculus," from which the example has been taken. 5. To find the generating function of the series whose •general term may be expressed by the term 1 (j) + qn) (r + sn) &c. ' in which n stands for the number or place of the term in the series. Because ^ _^ = x^ ± xi + «« ±, &c., 1 + a; -i-) ' if we multiply the members of this equation by dx and take the integrals of the products, we shall have 480 DEVELOPMENTS IN SERIES. 1 I XI . X'i , X'i XI , . - / :; (^^ ~ ± ?r -\ TT ±) ^^- 5 consequently, if Xi represents the sought function, we shall have \ r ^ ^^i ^+5 ^+3 Xi = - / :r-^-r- dx = -— ± --r- H -^ ±, &C., which, since its right member vanishes when a? = 0, we shall suppose its members to be so taken that thej both commence with £c, and extend to a? = 1. Thus, if p — 0, we have Xi=— -/ zj = log(l — a?) = - -f- — + — +,&c., qJ 1 — X q ^^ ^ q 2q Zq ' ' when we take 1 — x for 1 =F a? ; and if we put a? = in the members of this they both vanish, while if we put a? = 1 in the members they reduce to i log = infinity =:i(l + | + |+ &c.), a well-known result Again, if we take 1 -\- x for 1 =F aj we shall, as before, by putting ^ = 0, get or log 2 = 1 - -^ -f - _ - +, &c. Kesuming, X 1 r -^ -^' -"' -+3 1 I x'-f-Ttial. Thus, - ^ — du ydx xdu -~yi ^, , y ydx + xdy = dyx, -^ - ^- = a^ = "^ ^' are complete or exact differentials, while ydx, xdy — ydx^ are incomplete or inexact differentials, provided there is no assigned relationship between x and y ; other examples of exact differentials will be obtained by reversing the exam- ples at pp. 7 to 12. (2.) It is easy to perceive that if 'M.dx + ^dy is an exact differential of two variables x and y^ that its integral may be found by the following RULE. 1. Take the integral / M.dx on the supposition that y is •constant or invariable, and add to the result the integral of all the terms in l^dy which are independant of x or do not 440 DIFFERENTIAL EXPRESSIONS contain x ; then the result, increased bj an arbitrary con- stant, will be the complete or exact integral. 2. Or we may take the integral / Nc^y, on the supposition of the constancy of a?, and increase the result by the integral of that part of Mci.r which is independent of y, and an ar- bitrary constant lor the same integral as before. Bemarks. — It clearly results from the rule, that when Mdx -f- '^dj/ is an exact or complete differential of a func- tion (M and N being functions of x and y) we must have -7— = -7- ; which is called Euler^s Criterion or Condition ay dx of Integrahility of the differential McZa? + Nc?y (see p. 22) Hence, since / M<^fe — I Ndy^ we have d I Mdx dy -^» and from -7- = -7— , we have aN = -7- dx. which gives dx dy ^ dy ^ ° N = I -r- dx\ consequently, we must have dy d I Mdx rdU , dy J dy which is agreeable to Leibnitz's rule for differentiating under the sign / ; noticing, that the right member of this equation is independent of the Urst integral, or that with respect to x. (3.) To illustrate the rule, take the following with two 01^ more variables. 441 e:xamples. 1. To find the integral of {Qxy - tf) dx + (S.^^ - 2xij) dy. Since x enters into every term of the coefficient of dy^ it is clear, if the proposed differential is exact, it will be suffi- cient to find the integral / (Qxy — y") dx^ supposing y con- stant ; consequently, o.-c- y — y- x -\- C must be the integral, which is evidently true, since it equals the integral f{U^-2xy)dy, ' regarding x as constant. Remark. — ^Because, in this example, M and N are repre- dM. sented by Qxy — ?/" and 3a?^ - 2s? i/, which give -j- =6x — 2y , f/N , ^ , . . ^ . , .,. ^M c/N and -J- =: tx — 2y, the criterion oi Integra bility, -j~ — -y- , cix dy cix is satisfied. 2. To find the integral of i^x'' + 2axy) dx + {ax'' + Sy^) dy. Here / (3.'' + 2axy) dx = x^ + ax^y, to which adding the integral of Sy'dy, the part of(ax''-{-Sy^)dy which is independent of a?, and we have ar^ + ax'y -\- y^ + C, after adding the constant C, for the exact integral. The same integral is also found from the integral Jiac^ + 3y-) dy = axy + y\ by adding the integral 3 / a?'-^ dx ~ x^ + G, the integral of the part of (3a3^ + 2axy) dy which is independent of y, to it The criterion is also satisfied. 19* 412 DIFFERENTIAL EXPRESSIONS 3. To fmd the integral of ^^^ = -J^"-, Here the integral ^ + y' "" V' + '/r y- + ss^ d.v /^-^^ r_J^_^ = tan-^ + C; ^ y' + X- J x^ y y and the integral dy f--^±y-^= ri^Zl..tan-? + C, J y^ ^- 9r J ^ ^x^ y y- the same as before. 4. To find the int-cgral of —jr-r- — s; + V^V' Here M = —rr-^ -^r aiid N = y. which, since they do not contain y and a?, give -j- = and y- = 0, which, being naught, may be regarded as satisfying the criterion of in- tegrability. Hence, the proposed differential maybe regarded as having an exact integral, which is also evident from principles here- tofore given, since each term of the proposed dilFerential is clearly the function of a single variable. Indeed, the integral r x'dx __ X j/ (cz- 4- X-) _ cr r d .c J j^{a^ + aj2) ~" 'I '2 J 4/(rrT" log {x + \^X' + (/-) , and since / ydy = ~ , the integral of the proposed differ- ential is of course found, after the addition of the arbitrary constant. i/ia" + x') 2 '2 J ^{rr -^x') X {/{a^ + X-) a- 2 2 WITH TWO OR MORE VARIABLES.' 443 5. To find the integral of i^ay? -{- ^dcfy) dx + hMy, Here we have fudx = [{^aa? + 2lxy) dx — ax" ■\- hx^y + Y, in whichi Y stands for the arbitrary constant in tlie inte- gration with regard to x while it may be a function of y^ since y has been supposed to be constant in the integration with reference to x. To determine Y, we take the differential of the preceding equation, regarding x as constant, and thence get d I M-dx j^ r. d I M.dx consequently, since d I U.dx 1^ — haP and j = haP. dy d I Mdx we have N -r == 0. dy Hence, the sought integral is reduced to dy I ^ which might also liave been expressed by the form J Ndy + y (M -—-^ j dx =bx'y + «a^ + C, the same as before. 444 DIFKEKEXriAL EXPliESSIONS. Remark:. — We Lave performed this solution according to the common metliods, in order to show that they are sub- stiiiitv'! ;y tiic s:une as our rule. (4.) It is easy to perceive that our rule may be extended to lind the integral of a differential consisting of any number of terms, like ^Ldx + l^dij + P^^^ +, &c., by adding to the integral / Isldx taken relatively to a?, the integral of all the terms in l^dy which are independent of a?, and then adding the integral of all the terms of Ydz that are independent of either x or y (or both of them), and so on to any required extent. Thus, the integral of ydx _^ (x + 2ay) dy _ {x y + af) ^^ z z z- /ydx y r -, V^ - — = - I dx = ^—, z z J z f/2.Wy = «-f, and („y + a,^)/-J = ?^^, whose sum, corrected by the addition of an arbitrary con- XtJ -f" O- XT stant, is ' ~ + 0, which expresses the integral as re- z quired. If ^Idx H- NcZ?/ + Ydz +, &c., is the differential of some function of .r, y, 2, &c., of u^ we shall have ^g du T,-r du _, du . which give c?M d-u c/N c^?/ ^M d"-u dy dxdy^ dx dydx^ dz dxdz'' , dV d}u dN dhi dF d'u „ dx azdx dz dydz ' dy dzdy WITH TWO OR MORE VARIABLES. 445 Because -, — - = -7-7- , daxly dydx and so on (see p. 22), we shall have ^M _ d^ d}l dy ~ d^ ' d3 d/u d'^o djidz dzdji' I have d^ dji ' cZN "dz dY -dy kc, for the Criteria of Inteyrahility of a diflerential of the pre- ceding form ; which, being supposed to contain n different variables, will give — - equations, like the preceding, in the criteria of integrability, since by the known principles of combinations, — - shows how often two may be taken out of n different things. It is hence clear that any differential which satisfies all the 71 ( ?h T I - ^ - — - criteria, can be integrated by the preceding method, and its integral wall be exact ; but if the criteria are not all satisfied, the integral can not be found, and must be incom- plete or inexact; hence the importance of examining the conditions of integrability before we proceed to integTate the equation, becomes too evident to require any further notice. (5.) Supposing Adx + Bdy + Cds +, &c., to be an exact differential, or one that satisfies all the criteria of integra- bility, and, at the same time, suppose each of its coefficients, A, B, C, &c., to be of 71 dimensions in terms of its variables, a?, y, 2, &c., or, which is the same, suppose the equation to be homogeneous, the degree of homogeneity being 71 ; then, we , , ,1 , ., • , 1 A;?? + By + Cz + &c. propose to show that its integral — , ... n + 1 ' provided n is different from — 1. Thus, since y, 2, &c., may clearly be expressed by xy\ X3\ 446 DIFFERENTIAL EXPRESSIONS &C., because the differential may evidently be supposed to have been obtained by regarding x alone as variable, it must be expressed by the form K.dx -h V»y'dx + (^z'dx +, &c. Because, from the nature of homogeneity, each term of this differential must be supposed to contain the factor x"" dx^ X n +1 which, integrated by the rule at p. 254, gives for the common variable factor of the terms of the integral ; conse- quently, the integral must evidently be expressed by Ax + Bxy' + Oxz' + &c. n -\- 1 + const, or its equivalent, ; h C, ^ n -\- I C being the constant. It may be noticed that if /i = — 1, the integral I x'^dx = I — = log X ; consequently, when n =: ~ 1, it results that log x must be a factor of the integral of Adx -f- By'dx -f Cz'dz +, &c. Hence, when n, called the index of homo'jeiieity^ is dif- ferent from — 1, change dx^ dy^ dz, &c., severally into X, y, 2, &c., in the differential Adx ■{■ Bdy + Cdz +, &c., divide the result by the index of homogeneity, increased by unity, and add an arbitrary constant to the quotient for the integral EXAMPLES. 1. To find the integral of {Sx^ + 2a.Ty) dx + {ax^ + 3y-) d,j. Here the index of homogeneity is clearly 2, being the sura of the indices of x and y in each term of the differential ; WITH TWO OR MOllE VARIABLES. 447 consequently, since the differential is clearly integrable, by changing the differentials dx and Jy into x and y, we have (3.-t''^ + 2aa?y) x -f- {ax^ 4- 3y-) y. Performing the requisite multiplications, and uniting like terms of the products, we have Zx^ -f- Zaxhj + 3y^, which, divided by 2 + 1 — 3, gives Zx^ + Zax-y + 3y' , o and adding the constant C to this, we have x^-^ax^y-\-y'-\-0 for the integral of the proposed differential, 2. To integrate {^x'-\-1lxy-^]r)dx^ {hx^—^xy-V ^g/) dy. This being both integrable and homogeneous, we have, as before, Zx^ + Ihx-y — Zxy- + hxhj — Qxy'' + Zcf ^ ^— — + C := x" + hxhj - Zxf -f cy^ + C for the integral. 3. To integrate (2/^ 4- Zf) dx + i^x-y + 9.^y- -f- 8/) dy. The answer is yV + 3y''^ + 2?/'* + C. 4. To mtegrate -^ f- ~—- + — — t^ — . ° z z z^ Since the indices of x and y are positive, while those of ^, in the denominators, are to be considered as negative, it is manifest that the index of homogeneity is naught. Hence, it is easy to perceive that the integral is expressed by ''jLzJ: + c. z 6. To integrate the integrable and homogeneous differential dx I ^ X \ dy + 1 ^j putting y — xy\ the differential is readily reduced to 4AS DIFFERENTIAL EXPRESSIONS dx dc du) _ dx whose integral may be expressed by log X + log C = log Cx. If C = C [1 + V (1 + y'% we have log C^' = log C' [x + X |,/(1 + 2/'^)] = log C [.r + |/ (0.^+2/=)], wbicb is tlie well-known form of the integral as determined by the ordinary process of integration ; noticing, that the integral appears under quite an undetermined form, on account of the terms that have destroyed each other, agree- ably to what is said at pp. 445 and 446, the index of homo- geneity in this example being — 1. 6. To integrate the integrable and homogeneous differential xdy ydx. x^ + y^ ar + 2/^ * Here, the index of homogeneity is — 1, and the differ- ential is readily reduced to y'dx y'dx xiyVy^') ~ ^i + y'') ' whose terms destroy each other, and have the differential dx — - for a common factor; consequently, it is clear that the X integral is here under a more undetermined form than in the preceding example. It is hence clear that such integrals as these ought to be avoided as much as possible. (6.) We will now show how, according to the preceding principles, to integrate a differential expression of the form Q^dx^ + 'Rd.cdy + ^df, in which dx and dy are supposed to be constant or invariable, and X and y are regarded as independent variables ; then, be- WITH TWO OR MORE VARIABLES. 449 cause eacli term of the expression contains two dimensions of the differentials, it is said to be of the second order of differentials, or of the same order as that of the differential of a differential, and in dimensions the expression is said to be of the second degree, or of two dimensions. (See Lacroix's " Calcul IntegraV p. 232.) It is easy to perceive that we may consider Q^dx^ and S ///- to have been derived from / Q^a? and / Sdy by taking their differentials, regarding x and y as separately variable in the expressions ; consequently, the proposed differential may be supposed to have been obtained from taking the differentials of the differential dx j Qdx ■\- dy I Sdy, on the supposition of the constancy of dx and dy^ while the first integral is taken on the supposition of the constancy of y, and the second supposing x to be constant. Hence, by taking the differential of this assumed expres- sion by considering x and y both to vary, and by diffei'en- tiating under the sign / , according to the rale of Leibnitz, given on page 440, we shall have Qdx^ + dxdy I -j-' dx ~^ dydx I — dy + Sdy^ ; which must clearly be identical with the proposed differen- tial, and thence I -^ dx -{- I -j- dy = B,: Differentiating the members of this equation with regard to X alone as variable, and differentiating the second term under the sign / , by the rule of Leibnitz, we shall have ^dx + dxj^^dy=^,dx, 450 DIFFERENTIAL EXPRESSIONS or we nave -7-^ + / -rr dy = -y-: ay J dor ^ dx and removing the sign / , by differentiating the members of this, regarding y alone as variable, we have drq d'S ^R dy'^ dj? dxdy ' r7-R which is the same as -7-— for the condition of integrabilitj of the proposed differential. (7.) We now propose to show how to find the integj*al of a differential expression of the form Vd-y -\- Qdr, given by Lacroix at p. 234 of his work, in which x is the independent variable, and y is regarded as being a function of a*, and P and Q are supposed to be functions of x, y, dx, dy. Putting dy = i^dx, and taking their differentials, regarding dx as being invariable, which we clearly may do, we have d'^y = dj)dx; which, substituted for d^y, reduces the given differential to the form {Fdj? + Qdx) dx, which may evi- dently, as in Lacroix, be represented by the more general form (M.dj? -\- No?a?) dx", whose integral ought evidently to be of the form udx-^ ; or we must have icz= I Mdp + Y, sup- posing the integral to be taken with reference to^, regarding x and y as being constants, and Y as being a function of them. Differentiating the members of this equation, regard- ing M as being a function of x and y, observing the rule of Leibnitz for differentiating under the sign / , we shall lu ave , rdu , , rdu , dv , dv , du = mip + dxj ^ dj> + dyj ^ dj>-^ ^ dx+ ^ r/y, which, compared to Mdj? + 'Ndx, gives WITH TWO OR MORE VARIABLES. 451 ^^ rd^i ^ rdu , dY dV which must be an identical equation. To remove the sign / , we differentiate this twice suc- cessively with reference to ^, and thence, since V does not contain J?, get d^ __d}l rdU ^ dU dV dp ~ dx J dif ^ dy ^ dy ' , cV-^ d}\i ^d\l d-W. and -j-r — -j—r + 2 -,- + -^-j- p (1), dp- dxdp dy dydp ^ an equation freed from V, that must be satisfied. Hence, dy ~ dp dx dy ^ J dy ^' , dY _^ fZN ^M dVi. , f ^M , ^"^^ Tx--^-^ -dp^^ -dx^-^ -dy^^' J IFx^^'^ and since the dilferential of the first of these with reference to X equals that of the second with reference to y^ we have dy dpdx dpdy^^ djT '^'-'dxdy^'^ dif ^'''^ ":^''^' When a proposed differential satisfies (1) and (2), bj sub- stituting the values of -r- and -7- in dx dy du^^dp^dxj-^^ dp + dyj -^ dp^- ^ ^+ ^ %, we shall get du =. Mdp + (^ - ^ + _ ^ + ^-^j ax + dp -^ dx^ dy dU _ dU dp dx dy /dN _ dU _ dU \ \dp dx dy -^l '^' 452 DIFFERENTIAL EXPRESSIONS which is freed from / and under the form of a differential of p^ X, and y, whose integral can clearly be found by the rule in (4), at p. 444. Thus, to find the integral of (^.vydy + a^ydx) d/y + xd]f + (y + a?^) dy'^djx + (2 + 3y) xydyd^ + y^d'j^ from what has been done, we put jpdx and djpdx for dy and d^y^ and thence get M = 2.^yj? + x-y, N = £cy + (y + d?)f^ (2 -\-.Zy) xyp + y', which give _=6^ + 2(y + .^)^^ = 2y-^ = ^^^ + ^•^'11^ = 2^^' which will satisfy (2) ; consequently, the expression is an exact differential, which is reducible to the form du = i^xyp + x^y) d^j + {yp^ f ^typ + y^ dx -\- {xp/ H- x^p + Sxy^) d-y. The integrals of the first term of this relative top, and those of the two last terms relative to x and ?/, by omitting the terms containing p in them, when added,, give xyp^ + x^yp + xy^ + C for the value of 'w ; consequentlj^, since the sought integral evidently has the integral =: udj?^ we shall have udn^ r= xydy^ + xnjdydx + xifdj? + Qdd^ 'for the required integral. It is easy to see that we may, in much the same way, proceed to determine the integral of any WITH TAVO OR MORE VARIABLES. 453 diiTerential expression between x and ?/, when it is of any order of differentials greater than the first. Concluding Eemarks. — Because the differential dx A {r — X) dx A 7 (^" ~~ ^^''^ "^ ^")^ {r'-2rx+¥f ~ ^'' {f- — 2r.'»+ Ir') ^rdx — — — ^(t dr we thence get A r ir-x)dx __ .^ r{7»-2rx + 'b^)~^'rdx ^ {f - 2nx + &n^ " 1 L dr (/■^ — 2ra; + Vf = Ad ^ + dr for the integral. It is hence easy to perceive how the forms of differentials may be sometimes changed, so as greatly to facilitate their integration, by taking the differential of them with refer- ence to a constant in them. SECTION VI. INTEGRATION OF DIFFERENTIAL EQUATIONS OF THE FIRST OKDKR AND DEGREE, BETWEEN TWO VARIABLES. (1.) It is manifest that a differential equation between any n\imber of variables, when the variables are separated from each other, is such that the integral can always be found ; and if the terms of an equation are of an integrable form, it may evidently be integrated by the methods given in Section Y. (2.) If we have an equation of the form ^idy + Ydx = 0, between x and y, such that X is a function of x alone, and Y a function of y alone, then, dividing the equation by XY the product of the differential coefficients, it is reduced to ut/ dx =^ + :^ = 0, which is clearly an integrable form, or such X X that the integral / ^- + I -- = can be found. Thus, the particular differential equation {x + Ifdy = (y -f Ifdx dy dx is reducible to {y + 1)^ {x + 1)' 2> whose integral is ^r-, -— , = zr + 0. 2 (y + 1)- X + 1 (3.) Similarly, the differential form XY^y -I- X,Y,d^ = 0, EQUATIONS OF THE FIRST ORDER. 455 divided by the partial product XY, of the differential coeffi- cients, becomes ■ + ^ -L ' ■ =i 0, Yj A. in which the variables are separated, and it is clearly an integrable form, or such that the integral can be found. Thus, the particular differential equation {x + l)i/dx - {y"' + \)xdy = 0, divided by xy"^^ becomes (i + l)^, = (i + y,y. which is clearly an integrable form, the integral being X + log X ^= y h C. J (4.) The equation dy + Vydx — Qdx, sometimes called a linear eqxiatio7i^ can have its variables separated by assuming dy + Yydx ■=■ 0, which gives -^ r= — Ydx^ whose integral may evidently be expressed by log 2/ - log C = log I = - Jvdx, or using e for the hyperbolic base, y = Qe~f^'^^. To adapt this to the proposed question, we may suppose to vary ; consequently, by taking the differential of y on this suppo- sition, we sball have ^y = - Qe-P^^dx + dQe-A'^^ 456 EQUATIONS OF THE FIRST ORDER Bj substituting y and dy in the proposed equation, and erasing the terms that destroy each other, we have dQe-P^^ ^ ^dx, or dQ = eP^'(idx, whose integral gives C = / ef^^'^-'Q^dx + C. Hence, from the substitution of this value of C in that of ^, it becomes y = e-/^'^(feA^'Qdx + C') for the integral of the proposed equation. Remarks. — Hence, the integral obtained from a very sim- ple case of the proposed diiferential equation, by the varia- tion of the arbitrary constant, has enabled us to find the integral when taken in its utmost extension. OtJierwise, — By assuming y = Xs, we shall get dy = zd'^ -f Xc?s, which values of y and dy^ substituted in the proposed equa- tion, reduce it to zd^ + Xo?0 -h VXzdx = Qo?a?, in which X being arbitrary, we may assume dz -f Vzdx = or z = e-P'^'', and thence get ^ ^ Qd» ^ ^-^Q.7aj = ef^'^^Q^dx, z whose integral is X = Cef^'^''Q,dx + C. Hence, from the substitution of these values of X and ^, we shall have y =: X,2 r= 6-/1'^- (feA'^Qlx + c) for the integral, the same as found by the preceding method. B:r:wE:-:N two variables. 457 Eemark. — This method of integration has been taken from p. 254 of Lacroix's "Calcal Integral." (5.) The more general differential equation dy + ^ydx = Qy'^ + ' dx can readily be reduced to the preceding form. For by multiplying its terms by ^^-^^ , it becomes ndy nVdx ^ , -which, by putting ^ = — , becomes dz — riFzdx = — 7iQdXj whicb is of like form to the differential equation in (4). Hence, by putting — nP and — 71 Q for P and Q in the integral in (4), we shall have n/Fdx /_ ^ A-"/p<^^ Qdx +cA for the integral of the preceding equation, and thence we get y. (See p. 192 of Young's ''Integral Calculus.") To illustrate the preceding formulas, take the following EXAMPLES. 1. To find the integral of dy + ydx = ax^ dx. Comparing the equation to that in (4), we have P = 1 and Q = ax^^ and thence / Vdx = a?, which reduces e/^"^^ to e'', and feA^^ Qdx + C reduces to afe^afdx + C ; 458 EQUATIONS OF THE FIRST ORDER whose integral, being found by integrating by parts, gives a fe'a^dx + C = ae" {x" - 2x + 2) + 0'. Hence, from y = eS''^^ (Jef^^^ Qdx + c) , we get y = a {x" — 2a? + 2) -f C'e-' for the sought integral. 2. To find the integral of dy + ydx = aa?" dx. Here we have P = 1, Q = «a?", and thence e/^^"^ = e'- feA^"" Qdx + C = a fe^x'^dx + C, which, integrated by parts, as before, becomes e'a [a?" — nx''-'^ + n (ti — 1) aj"-^ — &;c.] + C; consequently, we shall have y = e-A'- {eP^^ Qdx + CO = Q [a?" — Tia;"-^ + n {n — 1) x""-^ — &c.] + C'e"* for the required integral. xdx CLX 3. To find the integral oi dy -{- y ^ = ^ dx. Here we have ^~ l+aj^' ^~ 1 + a^' TPc/aj = log (1 + aj-)^ ^/P'^^ = ^logvd + x')^ and y^A'^^ Qria? + C^ = Q fe''^^^^' ^ -'> ^^^ + C = Q^iogva + x') + C'; consequently, y = e-A^-'x IfeA^^Qdx + c) == Q + 0^6-^^^^^^ + ^ is the required integral. BETWEEN TWO VARIABLES. 459 4. To find the integral of di/ -{- ydx = y^ xdx. Here, from the formula in (5), we shall have and J~ ne-^f^^- Qdx + C^ =J*- 2e - '- xdx + 0' = .-- {x + l^+0^. consequently, we shall have ' = ? = (^ + '^) + CV' for the required integral. 5. To find the integral of dy -f- 7/---— - ,/ ^^^^.. thence we have y Pdx = — log ^/(l _ x'') and 6'^/'^'^^ = e-i^g i/a-*«)^ Hence, we shall liave which gives z ~ - = 1 + C' 6^°2^^^-^^> for the right integral. 6. To find the integral of dy - ^^^ - __^^ ^ 1 + a^' l+ar'- Here ' Y = ~ _^_ o — —-. yp^^=-iog|/(i+a^), J ~7(r-r^)' ^»d thence 460 EQUATIONS OF THE FIRST ORDER consequently, y = ax-\-G'Vl + x^ is the integral. 7. To find the integral of dv + ^^- = y^xdx. ° ^ 1 — a?- ^ Here p = _^_, Q = a!, n = - -, fWx = — log (1 - x')^, n fvdx = log (1 - x')\ and ^V^*^^ = e '«« ^' - ^'^*^ = (1 - x'')^i from the nature of numbers and their hyperbolic logarithms. We also have consequently, from s = — ^ r= — ^-j^ = 2/% 2/ y ■' since s = ^"A'^^ (— w J e-''f^^^ Q,dx + C) we shall have y^ =: C (1 - ar^)^ - ^ (1 - ar*) for the required integral. (6.) If M^a? 4- Nc?y = is a homogeneous function of X and y of the degree n, its variables a? and y may be sepa- rated. For if we divide M and N by a?", it is manifest that the equation will be reduced to the form /(l)'^-+/'(l)'^^ = «' BETWEE^^ TWO VARIABLES. 461 since it is clear that the dimensions of y in the numerators of the quotients equal those of a? in the corresponding de- nominators. If we put - = 2 or y = a?^, we have dy = zdx + xdz^ and thence our equation is reduced to f{z) dx +f {£) {zdx + xdz) = 0, or [f{z) + zf{z)-\dx^-xf{z)dz, dx f'{2)dz or its equivalent — — — ' . , /., tt? X f{z)-\-zf{z) in which the variables are separated ; consequently, we shall have log x = — I - f{z)+zf'i^) EXAMPLES. 1. To find the integral of (af^ + yx) dy = {xy + y^) dx. Dividing by x^, we have , ydx or dy = - — , or , ydx dij dx III = ^ — . or -^ — — , y X which gives log y = log cx^ 'y) - \ log (a^ - h^ y) + C. b' b' BETWEEN TWO VARIABLES. 465 2. If m is different from naught, we mav put y = z^^ and thence get dy = kz^~^d2 ; consequently, from the substitu- tion of the values of y and dy in the proposed equation, we have kzd2^~'^ + h2"''dx=.ax'"\ix. To make this a homogeneous equation, we must equate the exponents of z and a*, and we shall have k — l=2^=w, or ^ = — 1 and 7?^ = — 2 ; consequently, the equation dy -\- h/dx = ax^dx becomes integrable when we put z ~ ^ for y, and — z for m, and is reduced to — z~-dz + 'hz~^-dx:=^ ax~^dx^ . T ^ dz hdx adx or its equivalent ^ H s- = — s-« ^ Z^ Z' X- 3. If, withLacroix, at p. 256 of his "Calcul Integral," we put y = '^_j + J- , we shall have x" ox and thence we shall have or, since c??/ + hy^^x = ax"'dx, we shall have -~ + ^ ^ = ax^'-dx. x^ x^ ' or c// + 5/2-^ = aa;'"+2^; Si/ which, by putting a; = ~ becomes dy' - ly'Hx' = 20* 466 EQUATIONS OF THE FIRST ORDER or putting — y' for y\ we shall have dy' + hy"-d,Xi' = ax'-"'-^ which is an equation of the form dy + hy-dx — wx^dx ; and becomes integrable, as before, when 7/1 -f- 4 = 0, or when m = — 4, and is obtained immediately from dy + hyHx = ax'^^dx, by putting y=-|,+ ^ or y = -y'x" + j, whenx' hi put for - . It is hence clear that the equations dy + hifdx — ax-'''- ^dx and dy' -f h/"dx' = ax' -"'- ^dx\ are of such a nature, that if in the first we put a? = — and y z= — y'x'^ + -r , it will be changed into the second ; and that if in the second 1 x we put x' = -- and y' =z — yx^ + ^ , X it will be changed into the first ; consequently, either of the equations is a transformation of the other. 4. Eesuming the equation dy + hy'^ dx = ax^ dx, and putting y = ± — , we have dy = ^^ -^ ; and thence we get T ^ H ^ = ax^dx, or T dy' + Idx = ay'^x'^dx. If we put a?"* + 1 = a?', we have BETWEEN TWO VARIABLES. 467 — — dx' X — x' "'^K x"'dx — — ^— r; and thence the preceding equation is easily reduced to 7 — m ^ ^ 7/1 + 1 m + 1 T — Wl or to dy' ± . y"" dx' = ± x'"' + ^ dx'. It is manifest that if m = — 4 in the equation of Riccati, that it will be integrable, and thence 7 — wi dy' ± -% y"dx' = ± — — , x' "^ ^' dx' ^ m + l^ 7?^+l derived from it, and having the same form, by putting V = ± -7 and a? "' ^ ^ = x', must also be integrable ; that is to say, the equation dy + hy'dx = ax~^dx being integrable, it follows that dy' ± 3-3 y"dx = ± -"3 r/'' ^a?' must also be integrable, and thence, by putting — m — 4 = — - or 7?2 = g— 4= — g, is the value of m for another integrable case ; and putting — - for 7/1 in the equation T y -dx = ih , 7/i + 1 ^ m + 1 C^^' ± — -^ 7/^Va?^ = ± -— --r ^^»* +~1 dx', g we have ??i == — - for the value of m in another intearrable o ° 468 EQUATIONS OF THE FIRST ORDER. case of the equation of Kiccati, and so on ; noticing, that the general form of the exponent m, when and — 2 are not included, is m = — - — --zr . which is called the CHterion of Integrahility of BiccaWs Eiiuation^ q being any number in the series 1, 2, 3, 4, &c. * It may be noticed, that all the terms that result from taking — for ± in the denominator of the criterion, must Jdc considered as resulting from the equation dy -\- hy^ dx = ax-"^ ~ '^ \ while those terms that result from taking -|- for ± in the denominator of the criterion, must be supposed to have re- sulted from the equation dy' ± -z y'^^dx' = ± x'^ + 1 . 5. To perceive the use of what has been done, take the following EXAMPLES. 1. To find the integral of dy + y'^dx = a^x-^dx. Here, by putting g = 1 m the criterion, and usiDg — for ± 1 in its denominator, it becomes -4 . ""'■ ^ 2"=ri = - ^' which agrees with the exponent of x in the right member of the proposed equation, and of course shows the equation to be integrable. To perform the integration we proceed, as at p. 466, by putting X — ~ and y = — y'x'^ + x\ since 5 = 1, X and thence get dy' -\- y'Hx' — a^x'-'^dx^ BETWEEN TWO VARIABLES. 469 as at p. 4G6 ; where, by regarding — 4, the exponent of x in the right member of the proposed equation, as being equal to — m — 4 the exponent of a? in the equation dy + htfdx — a-x~"^~'^dx, we shall have m = 0; and thence the preceding equation reduces to dy' + y^'^dx' = a^dx', which gives &' = -^'-^^ = (^, + JI-) ^ 2a. a^ — y ^ \a -\- y a — y I Integrating this equation, we have "lax' =log C ^^^, or e^^^' x ^-^ == C ^ const From X' =. - and y' =l — yx^ + a?, we get X / x(xy-l )-a \ ^ ^ \a? i—xy— 1) — a/ for the required integral. 2. To find the integral of dy + y'^dx— — a'x-Hx, Putting a? = - and y — — y'x'^ + a?', we get, as in the X preceding question, dy' + y'^'-dx' = — c^dx' . cly^ Hence dx' = — — ■ ^ a^ + y'' '■Mfl .-1 y . whose integral gives aa^^ + C = cot' , or,, since x' = - and y' — — yy^- + a?, we have X X a 470 EQUATIONS OF THE FIRST ORDER 3. To find tlie integral of dy + ifdx = 2x~ *dx. Here, by putting $' = 1, and using + for ± in the de- 4 nominator of the criterion, we have m = — - , and of course we must compare the proposed equation to the equation a h —-- dy' ± y^'^dx' = ± ^ x'"' + ^ dx\ given on p. 467 ; consequeatly, we shall have 1, = 2, and m + l~ ' m -f 1 "" ' 7/1 + 1 ~ 3 ' agreeably to what is said at p. 468 ; hence, Srn = 4:m +4 or m = — 4, a = 77i + 1 = — S, h = 2?n + 2 = — 6, and thence we get dy" - ^"\ dx" = - Zx"-'dx'\ Hence, from the formulas at p. 466, we have dy'" -^y"'Hx"' = -Ux'" ', since — m — 4, the exponent of a? is here — 4, and of course 7n = 0. From this equation we have whose integral is 1 y'" + A Because x'" = ^ and y'" = - y"x"' - ^ , and from the formulas at p. 466, BETWEEN TWO VARIABLES. 4:71 1 _i_ 2/ = ± -, , »■'"» + 1 = x'\ we here have x'" = 4, y" =± \ , and x" ^x'^K Hence, since ??^ + 1 = — 3, we have -^ and x"'z=.\^-l- or, since ij? = -, , we have a? ^ =: , and thence x'" =ix ^ ; and from y"' = -y'V'^-y, 2/"=^,, and y' = -ya==+c., we have ,,, _ _ ^'"^ _ ^ — _ ^ + a?^(l — ya ?) Hence, from the substitution of the values of x'" and y'" in 1 lo^C y- + i/^ 6i/2 ° .,,„ ./I' we shall get y"'-V'--. 6 |/2 6 + a;^ (3 4/2 + o^'O (1 - yx) — L or 6.^/2^) ' = log 7/1 + 1 ' 7/i + 1 ' 7;<. + 1 ~ 3 ' whicli give 7/i = — 4, a = 3, and 5 = — 6, and thence we have dy" - Qy"'dx" = %x"-Hx". Hence, from the formulas at p. ^^^^ we have dy"' - ^y""dx"' = Zdx'", which gives whose integral gives 3-v/2aj'''= tan-V''V2 + 0. From a?''' = —r. and a;'' = aj'^^^n = a.'~3 _ — x" ^ ^-V we have x'" = x* ; also ,"-_,'v4and," = ^ = -^, we have 4 JL 2. ,„ _ X* x^ __ 6 x^ (yx — 1) y - " -y^ + x 6 "" e^^ (yaj - 1) ~ 6a;^(y^~l) X* — ya?* + 6 Hence, we shall have 6a? * {yx — 1) if- = tan- ^^-r^^ + ^ + aj' 3 \/2x' {yx — 1) for the required integral. BETWEEN" TWO VAETABLES. 473 (9.) It may be added, that differential equations may often, by the introduction of new variables and particular processes, be reduced to integrable forms. 1. Thus, to find the integral of jpdx rdy x^dx X y ~ «^" ' since the integral of the terms ^ + ~- IS log xHf. X y by putting x^y^ = 2 we have y"- = — ov y = ^— j , n thence 2/" = (-^1 ; consequently, the proposed equati( 1 and mr + 7ip reduced to d log x^y'' = d log z = , n Z^dz ?»>• + riTp Vl — \ >nr + np or = a? "" dx, ov z'' dz = x '" dx, z ' in which the variables are separated. Integrating this, we have const., n Zr mr + np + r X r n 7rir + np -\- r T n r mr -^ np -y r IZr X r or = h const n mr -\- np -\- r Kestoring the value of z^ we have m.r + np + 2 ■np nx au'x ^ = , ^ mr -\- np + r which needs no correction, supposing y and x to commence 474 EQUATIONS OF THE FIRST ORDER np together ; dividing the members of this equation by ic »• , it is immediately reduced to aif nx m + 1 7nr + np + r Remarks. — The preceding method of finding the integral is analogous to that of Lacroix, at p. 259 of his " Calcul Integral." The integral can also be immediately found by mulliplyii .g its members by - a? »• y'\ which gives — y^x'' dx-\-7ix "- y"'-^ dy = d ix *• y"^] = — x' '•" dx; whose integral, as above, is np mr + np + r m -f- 1 nx ^ nx »■ ax ^ y"= or ay" = , mr -\- np -\r T 'rnr -\- njp ■\- r supposing the integral to commence with x. 2. To integrate the equation dy dx _ x^dx y X ay \/n ' we multiply its members by - , and thence get dy ydx _ x"^~'^dx . X x^ ^ a |//i ' an exact differential. Taking the integral, we have y a^ g^m + i . - = — or y = — ; X ma \/n ^ ma \/n which needs no correction, supposing the integral to com- mence with X. BETWEEN TWO VARIABLES. 475 8. To integrate tlie equation , . dx dy we may clearly, from what is shown at pp. 3-i and 35, take the differential of its members by regarding dx as being constant, and shall thence get , d-ydx . N 7 , 7 1 d'^y d^ ^"^ ^' y) = dx + dy~dy- -^ X] or, by reduction, we shall have dx , , X dx- dip- dif^ c/; ^y^j ^ a -\- y' in which the variables are separated. Hence, we shall have dy dx {a + yY X' whose integral gives (a + 2/)^ = ± a?^ + c ; or, by squaring, y -{- a — x ±_ 2cx^ + c^, which can be further reduced to iy + a — x^ — a-f zsz 4:o\ which represents the integral of the proposed equation, taken in its most general sense. 4. To find the integral of ady — ydx — xdx. By assuming y == a 4- v + a?, we have dy =: do + dx^ and thence by substitution the equation becomes adv -f adx = adx -f- vdx + xdx — xdx ; or, by erasmg the terms that destroy each other, we have — =z dx whose intep^ral is a? = a looj cv ; or, since 476 EQUATIONS OF THE FIRST ORDER V ^= y — a — x^ -we sliall have x = a log c (y — a — x). (See Vince's "Fluxions," p. 181.) (10.) We will now show that if we have a differential equation of ^idx + ^dy = 0, of the first order, between two variables x and ?/, in which the condition -r~ = — r- of ^ dy dx integrability is not satisfied, tliat the condition may still be satisfied after it has been multiplied by a suitable factor; and of course the integral can be found. For since ^dx + ^dy = is not considered as being im- mediately integrable, it may be supposed to have been obtained by eliminating a constant from an equation of the form F (a?, y) — and its first differential. Hence, if C stands for the constant, by solving the equation with reference to C, we shall obtain an equation of the form C ^^f{x^ y) ; consequently, by taking the differential of this, we shall, without reduction, get the differential equation M'c^^ + Wdy = 0, in which -/ or -y- must clearly be the same as in dx ay ^ Mdx + ^dy = 0, since the two equations result from the elimination of the constant C, from the equation F (x, y) = in two different ways ; the proposed equation resulting from the elimination of C from F {x, y) = hj means of its differential equation, and the equation M'dx + N'c?y = resulting from the im- mediate differentiation of the equation G =zf (ce, y). Hence, eliminating ~ from the preceding equations, we V n ^ dy M - dy W Bhallget £ = -^ and -£ = -^^ BETWEEN TWO VARIABLES. 477 M M^ consequently, we get ^ = — -, such, tliat M^ and N' must clearly be like multiples of M and N. Eemarks. — 1. Having found M' and N', it is manifest that the integral of Wdx + Wdy — will give C =/(«, y\ in which C represents the arbitrary constant, and which rep- resents nearly a transformation of the equation F (a?, y) = 0. 2. Since M'dx + ^^dy = is an exact differential, it follows, from Euler's Criterion of Integrability (see p. 440), , ,, , ^M' dN' that we shall have —7- = -7-. ay ax Hence, if z represents the factor of M and N, which gives Ms = M' and N.s = N', the condition of integrability dUi dm becomes -^- = —j- , ay ax which gives (Mr/s + zdM) -^dy — (Nc^s + ^dl^) -v- dx, dM. (iN\ __-^ d3 ^dz dy ~dx I dx dy"* which z must satisfy. Having found M.zdx + '^zdy = dic^ it is manifest that the members of this multiplied by any function of u will also be an exact differential ; consequently, there will he an unlimited nmnber of factors that will make the proposed differential an exact differential. EXAMPLES. 1. To find the factor which will reduce ydx — xdy = to an exact differential. Here we have M. =: y and K = — a?, and thence or z ( ^ — -~ 1 = N ^ — M ^ _ ^\ - N ^ - M — dy dx) ~ dx dy"* 478 EQUATIONS OF THE FIRST ORDER which can clearly be satisfied by putting 2 = -3, and gives -^ = ^, , an identical equation ; or, by writing the form if J ^ = 0, and intesratinff X y ' ydx — xdy _ .in dx dy ^ ,- — ^ =z 0, or the form -, y^ a? y ' X we have - = C = const. y X X £C X Kemarks. — - (^ - = , — cZ - r= 0, and, generally, y y r y i> V/ y are also exact differentials of the proposed equation, agree- ably to what has been done. 2. To find the factor that reduces ydx — mxdy = to an integrable form. Here, as in the preceding example, we get 1 , 1 ydx — mxdu , x ^ = -^-n 1 and thence zr—r-^ = d —- ytn + 1 7 ytn + 1 ytn is the transformed differential, whose integral is -— = C = const. 8. To find the factor that makes dy + Vydx = Qdx inte- grable. Here M = Py — Q and N = 1, and thence we have -J 7- = P and 2P == N -7- = -T- » dy clx ax ax supposing 2 to be independent of y ; consequently, we have dz /* — = "PdXj whose integral is log 2 := I Fdx, supposing the constant to be included under the sign of integration / . BETWEEN" TWO VARIABLES. 479 Multiplying the proposed equation by s = ^/ ^"^^^ which gives log B =J ^'^^ we have whose integral is eJ ^^""y = I cJ^'^^Q/ix^ and thence y = e-f^'^4 CeA'^^Qdxj, supposing the constant to be indicated by the preceding sign of integration, or the integral may be expressed, as at p. 456, by y = e '/^"^(^feA'^Qdx + C'\ , 4. To find the factor that makes aryy + l4:aPy ==^J ^^ = integrable. Here M =z 4a?-y -:, ^, and N =: a?^, and thence we shall have ay ax consequently, supposing s to be a function of x only, we shall have zx^ = x^ -p^ or ~ = — , ax 2 X which is clearly satisfied by putting 2 = x. Hence, multi- plying the proposed equation by a?, we have whose integral is a?'*y + |/ (1 — x^) — C. 6. To find the factor that will make aydy + (px — hy-) dx=^ an exact difierential. 480 EQUATIONS OF THE FIRST ORDER Ilere 'M. = ex — h/ and N = ay, and thence —^ ^ = — 2hy. dy dx ^' dz wtiicli gives — s x 2hy = ay ^ ^ by supposing z to be independent of y ; which gives dz 2hdx -=^" z a for the sought factor. Hence the transformed differential becomes 26a; \_aydy + {ex — hy-) dx'] e~ " = ; whose integral, sometimes called the prir/iitive, is (See Young, p. 210, &c.) (1 1.) We now propose to show how to integrate any homo- geneous differential equation consisting of any number of variables. Thus, let Udx + Nf7y -f Fdz + &c. = be a homogeneous differential equation, consisting of any number of variables ; then, if the equation is not integi-a- ble, it is clear from what is shown at p. 445, that it must be on account of the omission of a homogeneous factor, com- mon to its terms. Hence, if ic stands for the omitted factor, we shall have uMdx + uNdy -f uTdz -\- ko. = du' = 0, the differential being exact. If n denotes the degree of homogeneity of u\ we have, from what is shown at pp. 445 and 446, wM^ -f i/Nt/ + wP^ + &c. = qui'] BETWEEN TWO VARIABLES. 481 consequently, dividing the members of yMdx + wN% + &c. = du' by the members of the preceding equation, we shall have yLdx + l^dy + &c. _ du' ^ M.X + Ny + &c. nu' ' consequently, since the right member of this is an exact differential (its integral being - log w'), it is plain that l^dx + l^dy + &c. Ma? + Ny + koT must also be an exact differential. It hence follows, that the factor which makes the proposed, differential Mc?a? + Nc^v + &c. = exact, is ^^^ -^r-f — ; — -. — ; ^ ' Ma? + JN y -f &c. and thence^ if lA.dx + l^dy = is the proposed equation^ tlie requisite factor is y^ :^. Remarks. — 1. It is clear, from pp. 445 and 446, that the degree of homogeneity of Ma? + Ny +, &c., when the pre- ceding process is applicable, must be different from naught ; and Mii? + Ny +, &;c., must also be different fi'om naught, 2. If Isidx + Nc?y = 0, and, at the same time, Ma?+Ny=:0, then, eliminating N from the first of these by means of the second, we shall have which shows that if My is a function of - , the integial can if be immediately found in its most general form. 21 482 EQUATIONS OF THE FIRST ORDER EXAMPLES. 1. To find the factor that makes {xy — i/^dx -\- {yx + ^) dy = 0, an exact dififerential. Since Ma; + Ny =.2d^y^ bj dividing the given equation by ar* y, we have whose integral is log xy ■\- - =^ Q,. X 2. To integrate {^ — y'^) dx -\- {xy + oF) dy = 0. Here Ma? + Ny = aP {x -{- y), and thence, dividing the given equation by this, we have \x X^/ X whose integral is log x + ~ = G. X 3. To integrate ydx — xdy = 0. Here M = y and N = — a;, and thence Ma; + Ny = ; consequently, from what is shown above, we shall have Myd- = 0, or fd- = 0; " if 1 / 7'\ ( ir\ X and this multiplied by -^

,T+ 86^ X 12-a-Z'^,^'2 + 12VJV a 4fr 3^ X 12'a'b which clearly can not be reduced to the integral ay^ + cxy — hx^ — 0, or the proposed equation. If we integrate the equation dy'^ — a-dx^ = 0, supposing x and y to commence together, by either of the preceding methods, they will be found to give y^ = aV ; while dy"^ — axdx^ = 0, integrated by the first 486 INTEGRATION OF DIFFERENTIAL EQUATIONS method, gives y- = — - , and integrated by tlie second method, o 4 . gives y' = Q aar*, which does not agree with the preceding integral (2.) The common method of finding the integrals of equa- tions of the form dy"" + Vdy^'-'dx + Qdy^'-^da^ + .....+ Uc^" = 0, or its equivalent consists in solving it like an equation of the nth degree, by- regarding -- as the unknown quantity, and of course there will result ?i equations of the forms dx ^ ^' dx ^ -^' dx -^ -^' and so on, to n equations ; _p, j?', p^\ &c., being the roots of the equation. From these equations we get y— J V^^ = 0, y — J jp'dx =0, y — J 2y"dx = 0, and so on. Hence we shall have \y - Jpdx) X (y - fp'dxj X [y- fp"dyj = 0, which may be taken to represent the integral of the pro- posed equation ; noticing, that each of the factors may be supposed to be corrected by the addition of the same con- stant. For the method of integration here proposed, the reader OF THE FIRST ORDER AND HIGHER DEGREES. 487 is referred to Lacroix, " Calcul Integral," p. 279, &c. ; Young, "Integral Calculus," p. 224 ; and Lardner, p. 318. EXAMPLES. 1. To find tlie integral of 2/ ^-^ + 2;:e ^ -y = 0. Eeducing tlie equation to the form ydif" + 2xdydx — ydx^ = 0, and taking tlie integral, regarding x and y as independent variables, we liave 1/ {7,7/ of '7' ^—^ + 2yxdx-\-iK?dy — ydxx=0 and |- + ^y—y^ =0, found on tlie supposition that x and y commence together, and that y in the last term is constant ; but, since y is not constant in the last term, it is clear that the equation has not been obtained on the supposition of x and y being inde- pendent variables ; noticing, if the last term of the equation had been x or any function of it, the proposed equation might have been obtained on the supposition of x and y being independent variables, and of course a doubt as to the true origin of the proposed equation would have been the result. Hence, solving the equation on the supposition that x and y are not both independent variables, we have dy _—x -\- Vy'^ -\- x"^ , dy __ —x — \/ {if -f- x^) dx y dx~ y ^ which may be put under the forms and by taking the product of these factors, we have 488 INTEGRATION OF DIFFERENTUL EQUATIONS whose integral is ± iY + ic" = a? + C, or y"" =: 2cx -\- CK 2. To find the integral of dy^ ± da^ = ^dxdy. Integrating on the supposition of the independence of x and y, we have ^^ =xy + C, or f'±x'= 2xy + C. Remarks. — If we take + for ± in the proposed equa- tion, we have dy^ + da^ = 2dxdy, or dy'^ — 2dydx -\- da^ = 0, or dy — dx = 0, whose integral is j/ — a? = C ; the same as by the preceding method. If the proposed differential is dy^ — dix? = 2dxdy^ it is^clear that the integral found on the principles of the independence of x and y, and their dependence, as in the common method of integration, will not agree ; consequently, the origin of the proposed differential is doubtful. 3. To integrate a? -^ + a? — 1 = 0, or -,—„ = 1. ° dx^ ' dx^ X Multiplying by da?^ and integrating on the supposition that X and y are independent variables, we have da? dy^ = ~ dx^^ and thence ydy = dx log x — xdx, X which integrated again, gives y^ x^ ^ = X log X — X — — + const, OF THE FIRST ORDER AND HIGHER DEGREES. 489 or if- — 2x log X — 2x — X' -\- C ] consequently, the origin of the differential is doubtful. Eemarks. — Mr. Young, at p. 226 of his work, finds y =: \^x — x^ — tan-^y — X for the integral ; a result very different from the preceding. (3.) When only one of the variables x ov y enters the pro- posed equation, and the value of the variable in a function dy oi -~ = jp can be found ; or if ^ can be found in a function of the variable ; then, in solving the equation in the com- mon way, the other variable can be found. Thus, having found aj = F(», or y=f{p\ ^ , (!¥{])) , , . df{p) , we get ax — — r^^ dp, and dy = ~j ^P j consequently, from dy =z pdx, or dx = —, we shall get y = fpdx = fp '^ dp, OV x= /J = /^|> dp, whose integrals will determine the value of y ov x. For integrating by parts, we shall have y=p'F{p)- fF{p)dp; so that if F (p) = ^ , we shall get = 2 . -t — / T---^ = -ir-—^ — tan-^ p : p^ + 1 J 1 +y y + 1 -^ ' y 21* 490 INTEGRATION OF DIFFERENTIAL EQUATIONS 1 /f^~i "which, from x = -5 , gives « = y , and thence J9r+ 1 ^ ^ X y = Vx-x''- tan-^ |/1— - + G X the result quoted from Mr. Young in the preceding example. If the equation involves such high powers of x or y that it can not readily be solved, we make such a substitution for du dx 1 ^ -J- =J7, or -- = - =p ax -^ dy p as will reduce the degree of the equation, so that x ox y may be found by the common methods of solving equations. Thus, to integrate we put J = ,T^, and thence, by substitution, get a^ + x^z -\- a?'2^ — 0, or x= — :j 5 , which gives _ _ dz Bz'ds _ _ (1 - 22') dz '"^ ~ 1 + 2^ "^ (1 + s^j' ~ "(i - ^J ' From -^ 1= xz, ccx * by substituting the value of x, we have dy z^ , ^ 1 and, substituting the value of dx, thence _ __ (1- 22')z'dz __ __ Sz'd z 2zHz ^ ~ (1 -f zj ~" (1 + ^y "^ (1 + ^T OF THE FIKST ORDER AND HIGHER DEGREES. 491 whose integral gives 1 1 p ^ ~ 2 (1 + ^J 3 (1 + s^ "^ Solving this equation bj quadratics, regarding —^ as being the unknown quantity, we shall get ^3 in a function of y, whose reciprocal gives 1 -f 2^ in a function of y, which, diminished by 1, gives s^, whose cube root gives the value of z. Hence, x is easily found in a function of y, as required ; noticing, that we may clearly proceed in like manner in all analogous cases. (4.) If the equation involves both variables, in such a way as to make its terms homogeneous relatively to the variables, then, putting ?/ = a?s in the equation, if ^i denotes the degree of homogeneity of the equation, its terms will be divisible by a?** ; and we shall have an equation in terms of z and p, whose highest power in s will be z"^. Hence, if the equation can be solved with reference to z, we shall have z=:F (p) ; or, if the equation can be solved relativ:ely toj?, we shall get p =f{z). Since y = xz, we have dy = xdz + zdx = xdF {p>) + F {p) dx^ dy dF(p) ^, . , . , . dx d¥ (p) which gives — X p-F{py whose integral can be found in a function of p ; and thence from y = xz = x¥ {p), by eliminating p^ we get y in a function of x^ as required. 492 INTEGRATION OF DIFFERENTIAL EQUATIONS In mucli the same way, from p =z -^=f{2\ and from dy = xdz -f- zdx^ xdz , J,, . xdz „. we have -^- + z =/(4 or -^- =f{z) - z, which gives a dx dz which, integrated, expresses a? in a function of 2; and thence, from y = xz, we express y in a function of z ; consequently, eliminating z from the values of x and y, we shall get y in a function of x. EXAMPLES. 1. Given y — xj) = x Vl + j^Ho find the integral By putting y = xz, the equation reduces to z - p - Virrp, or z^p + Vl + pS which gives dz =: dp -\ ^ ^ : 1/1 +y and from dy = pdx = xdz + zdx^ T <^aj c?3 dp pdp we have — = = — ^ — ^-^^ , a? p-z Vl-hp^ l+i> whose integral clearly gives log ;» = - log {p + vrry) - log i/rrp + log o,- and thence we have G X = Vl+p'{p+ Vl+p')' and since z = p + Vj^+p\ we have xz = y = G Vl +y OF THE FIEST ORDER AND HIGHER DEGREES. 493 consequently, we hence get X — ^ c + Vc- — 2/^ for one form of tlie proposed integral. Otherwise. — By squaring tlie members of tlie proposed equation, we liave y^ — ^xyp + x-^^ =z a^ -\- x"p^ ; or, erasing the terms that destroy each other, we have r - x" V \ or, since y = xz^ we shall have z^ -\ , , c?a? dz , , dx 2zdz and thence — = -ttt- reduces to — = /(«); ^ /C^) ~ ^ ^ 1 -{• z^ whose integral is log ^ = log ^-^, or <^ = j^-., 1/ . , which, since = - , is immediately reducible to X X {x^ 4- 7/-) :=z x^ (?, or x^ + 2/"^ == c'x, Eemarks. — It is easy to show that this integral agrees with the integral found by the first method, since it can be put under the form G + Vg" - f = ^\ or VT^f = - - c, a? ^ X which, by squaring and an obvious reduction, becomes a;2 + 2/" = 2gx^ which agrees with a?" + ?/« z= g'x^ when we put g' for 2g. For the first of these solutions, the reader is referred to p. 229 of Mr. Young's work. IIM-^)') 494 INTEGRATION" OF DIFFERENTIAL EQUATIONS 2. To find the integral of y^ — ji?ar = jpy^. Putting xz for y, the equation immediately reduces to z^ z^ —p =pz^^ wliicli gives p = T'Jr^ —f(^)' L -\- Z ^ , dx dz dz (1 + 2^) Hence, we have — = ttt- = — -^ — r— ^ . X f{z) — z z^ — z- -{■ z Smce 5 5 = 5 -, we snail have z^ — z^ + z z z^ — z ■\- V -dz dx _ dz dz _ _ ^ ^ aj"~ z z^ — z + 1 ~ z wliose integral gives , A - 4A log K = — log Z — g^-j = — log C3 g- , in which A is an arc of a circle, whose radius = ~-~ and 1 1/ y tangent = z — -. Since s = -, if we put - for z m ^ XX this equation, we shall have the required integral. (5.) Supposing ~ = j9, we will now proceed to show how to integrate the equation y = xp + F (j)\ on the supposition that F {/j) is independent of x and y ; noticing, that this equation is called Olairaufs form. Bj differentiating the members of the equation, we have dx ^ ^ dx dp dx'' or, erasing the terms that destroy each other, we have OF THE FIRST ORDER AND HIGHER DEGREES. 495 Tliis equation is satisfied hy putting -J- = ov dp = 0, whose integral is j9 = C = const, and of course y = Cx is the proposed equation. The preceding equation can also be satisfied by putting its other factor equal to naught, which gives consequently, if - is a function of^;>, by finding^ from this, and substituting its value in the proposed equation, we shall get an equation between x and y, which does not contain any arbitrary constant, and is hence called the sin- gidar solution of the proposed equation. Thus, if y — xp + af>^, we have y — Cx -\- aOP' X for the integral, or j? = — ^ = is the singular solution. Similarly, if y = px -\- a{\ + p^\ we shall have y = Cx + a{l + C-) X for the integral, and — — is the singular solution. Eemarks. — 1. If we have the equation y =^Vx -\- Q, in which P and Q are functions of j9, then, by differentiating, we shall get dy ^ xdV dQ . d'P dQ dx dx dx'' P — j9 P~j9* Taking the integral of this, by the form given in (4) at p. 455, we have 496 SINGULAR SOLUTIONS OF by changing y and dy into x and dx\ then, eliminating p from this and the proposed equation, we shall get the integral between x and y. 2. It is manifest from what has been done in the first part of this section, that in the a23plication of the Differential and Integral Calculus to estimate the changes of position of bodies, which result from the violent and sudden actions of powerful forces, we ought generally to take the diflferentials of the variables on the supposition that they are independent of each other, since the tendency of the actions of the forces is plainly to introduce multiple points or cusps into the motions of the bodies. Keciprocally, in finding the integrals of differentials thus found, we ought to proceed on the sup- position of the independence of the variables, as explained at p. 484, &c. (6.) From what has been done, we are naturally led to the consideration of what are called the Singular Solutions of Differential Equations of the First Order. 1. If F (a?, y, c) =: 0, in which c represents a constant, and we differentiate the equation, regarding c alone as varia- ble, we shall have ^^ ^^' ^' ""^ ^ = ; dc , then, if, as in the example at p. 187, we eliminate c from the ■n / N /^ ^ d¥ (x. y, c) equation F {x, y, c) = and \J-MlU. — o, when its dimensions exceed the first degree, the result will (generally) be what is called a singular solution of the pro- posed equation. 2. If we regard x and y as being functions of c, then, by differentiating F (a?, y,c)=z with reference to c, we shall have EQUATIONS OF THE FIRST ORDER. 497 dx do dy do do ' d^ (x, V, c) or, since V ^' ^ = 0, we liave d'E (a?, y, c) dx ^ , ^F (x, y. c) dy , ^ dx dc dy do Because this equation must evidently be satisfied so as to leave ^-_l1^), or ^'-Ii£l dx ^ dy arbitrary, we must have either dx __ dy _ do ~~ '' do ~~ "* wbicli may clearly be used instead of d^ (i^, y> c) ^ Q do dv Similarly, if p = -— , and we have the differential equa- tion f {x, y,j)) = such, that F {x, 3/, c) = represents its singular solution ; then, solving/" (x, y^ p) = relatively to c, we get the form c ^^ (,t, y, p), which reduces F {x, y,c) = to the form F {x, y, &) = 0, by using to stand for the function d (a?, ?/, p), or its equiva- lent c. If, for brevity, we represent the first member of this equa- tion by -w, then, since the function may contain x and y, by taking the differential of t^ = 0, we shall have (dti. du do dp\ Idu du dO dp\ , _ ^ dx'^ Td'd^'dx}'^'''^ \dy + Wd^'Tyj'^y-^' 498 SINGULAR SOLUTIONS OP which must clearly be satisfied so as -to leave dx and dy arbitrary. Hence, we may clearly put the coefficient of dx or dy equal to naught, and shall thence get dxi du dd dp _ du du dd dp _ dx'^ dd'd^'dx" ' ^^ dy^ dd''dp'd~y~ ' ich give dp _ du ^ du dd dx~~ dx ' dd' dp'' dp or -f = dy du dy du dd "" dd'dp Because d is used for d (aj, y, p), or its equivalent c, it is {77/ fl^J clear that ~Ta^=^ ~t- ^ which (by 1), for the singular sol ution, must equal naught ; consequently, because — = 0, we must have -J- = infinity, or -^ = infinity, and thence — = 0, di/ or ^ =0, which are the conditions for finding the singular solutions of differential equations of the first order. Hence, if p is eliminated from the pyroposed differential equation hy either of these conditions^ and if the result satisfies the proposed differential equation^ it will he the shigular solu- tion of it 3. To simplify t'le applications of what has been done as much as possible, we shall represent the proposed differ- ential equation f{x^ y^p) — hj v^ which gives, supposing ^ to be a function of x and y, dv ^ dv , dv dp ^ dv dp ^ . dx dy ^ dp dx dp dy ^ and thence we get dv _ (dv dv dy\ ^ /dp dp dy\ dp \dx dy'^dxl ' \dx dy' dxr EQUATIO]SrS OF THE FIRST ORDER. 499 which reduces -y- to nausrht, since -y- or — in the divisor d_p ° ax ay is infinite. Hence, — == gives the values of p^ that give the singular solution. (See Young, p. 232, &c.) It may be noticed if i^ = F (a?, y, 0) = does not contain ?/, tliat we must here regard x as being a function of y, regarded as being the independent variable, and put p' = -r- for p. (See Young, p. 237.) EXAMPLES. 1. To find the singular solution of V = (x -\- y)p — xp^ — (a -{- y) = 0. dv Here -y- = becomes x -\- y — 2x]) = 0, which gives ^ ~ 2x '' which, being put for^ in the given equation, gives Hence, we easily get a? — y = 2 \'ax^ or y = a? — 2 Vaa?, which gives -^ — p = 1 — ^ . ax Yax From y = X — 2 Vax we get X -[- y = 2x — 2 Vax and a + y = a + x — 2 Vax ; and since p =^ 1 -nr , we thence get Vax 2{x-V^) (l %^-xil - -%)^- (a + aj- V^) = ^ Vax^ ^ Vax' 2 (i/a? — 4/a)2 — 2 (|/aj — ^/aj = ; 500 SINGULAR SOLUTIONS OF consequently, since y =z x —2 Vax satisfies the proposed equation, it must be its singular solution. 2. To find the singular solution of V =z a? + 2xyj[> + {a? — utr) p'^ = 0. From — = we get xy -f- {a^ — x^)p = ] which, multiplied by p and subtracted from the proposed equation, gives x^ -\- xyp = or ^ = ; if which, substituted in the proposed equation, gives a.2 + y2 _ ^2 ^ ; consequently, since this satisfies the proposed equation, it must be its singular solution. 3. To find the singular solution oi xp — y = \/{x^ + y'), or, more properly, of {xp — y)^ =z x^ + y\ Here v — x^p^ — %xyp — a?^ = 0, gives -3- = 0, or its equivalent x'p — xy = 0, or p = x' which reduces the equation to — f — xi — 0^ or — 2/- as required. 4. To find the singular solution of = ^, a;y — Ixyp -f- y"" - — x^ — p^ x'' = 0. Here -7- = becomes dp ^x'p — 2xy — "Ix^p — 0, or — 2ajy = ^> which clearly shows that the question does not admit of a singular solution. E:.r^'ATI0>r3 OF THE FIRST ORDER. 5. To find the sinsfular solution of 601 V = (x' — 2if) iP- — ^xyp df 0. dv Here — = becomes {oi? — 2y^)^;> — 2a?y = 0, which, multiplied by j?, and the product subtracted from the given equation, gives — ^xyp — cc- == 0, or ^ = — 2y' consequently, substituting this in the proposed equation, we have (a?^ + 2y") d^ = 0, which evidently gives x^ + 2y" = for the singular solution. 6. To find the value of c, which gives the singular solu- tion of y = X + (g — If {c — xf. Differentiating the equation by regarding c alone as variable, we have 2{G-l){c-xf + 2{c-iy(c-x) = 0, or c-x + g-1 = 0, which gives c — — ^ — for the particular solution, since the values (? = 1 and c = a^, which also satisfy the differential equation^ clearly correspond to particular integrals. T. To find a curve such, that the perpendiculars drawn from a given point on its tangents shall be constant, or equal to each other. 502 SINGULAR SOLUTIONS OF Let be the given point, taken for the origin of the co- ordinates, and OP for the perpendicular from the given point or origin to the given line AB ; then, we may clearly suppose Y — y = -^ (X — a?), or its equivalent Y=:^X + y —px^ to be the equation of AB, regarded as touching a circle having OP = R for its radius, center O, and Ojc = x, Oij = y, for the rectangular co-ordinates of the point of contact P of the tangent and circle. Supposing y to decrease when x increases, we shall clearly have — -^ =z — p for the tangent of the angle yOV, and y —px equals the part of the axis of y between AB and the origin 0. Because \]p^ -f 1 equals the secant of the angle 2/OP, it is manifest from known principles of trigonometry that we shall have ^ ~^ ^ = R = const ^f + 1 for the invariable expression of the perpendicular from the origin to the tangents to the sought curve, which must hence clearly be a circle, having O for its center and OP = R for its radius. It is manifest that the preceding equation may be written in the form y = j9a7 -1- R rj5>'^ + 1, an equation that agrees witb Olairaut's form of differential equations given at p. 494, whose integral is there shown to be y — ex ■\- ^V& -^-1. The singular solution of this gives c = i/(R-^ - af)' EQUATIONS OF THE FIRST ORDER, wliicli, being put for c, we have 503 y = + 4/(R^-n and thence x^ -\- qf = W is the singular solution of the proposed question. 8. To find a curve such, that the product of the two per- pendiculars from two given points on any tangent shall be constant or invariable. "^^^^^--^ y -^ r - ^ D \ c ) F 1 3 X Let A and B represent the given points through which the axis of x is supposed to pass, the origin of the co-ordi- nates being at 0, the middle point between A and B ; then we may suppose that the sought curve touches one of the tangents CD at the point E to the right of Oy, the axis of y, and that OF and FE represent the x and y which corre- spond to the point of contact of the curve and tangent. Representing AO = OB by a we shall have AF =: a -\- X and FB = a — a? ; then, as in the preceding question, Supposing y to decrease when X increases, we shall have — jp — the tangent of the angle of inclination of CD to the axis of a?, and thence AC = y — (a + a?) J? and BD ^ y -\- {a — x)^] 504: SINGULAR SOLUTIONS OF consequently, by dividing these by ^^{p^ + 1)> we, as in the preceding example, get y -{a + x )p y + {a - x) p V{P' +1) V{f + 1) for the perpendiculars from the points A and B to the tangent CD ; and thence, if h^ denotes their product, we shall have y - (^ + ^) P ^ y -^ {^- x)p _ T2. V{f +1) v'(i/ + i) ~ ' or, by performing the indicated multiplication, we have y^ — 2pxy — y {a^ — 0?°) _ y + 1 ~^ for one form of the equation of the sought curve. Solving the equation for y, we readily get y =px ± \/(J/ 4- fn'p% in which m^ = a^ + h^; this being a differential equation of Clairaut's form, we shall, as heretofore, get y = Ox ± Vb' + m'G^ for its integral, C being the arbitrary constant. By taking the differential of this equation, supposing C alone to be variable, we readily get for the singular solution. Taking — hx for ± hx in C, and •using + for ± in y, we get from what has been done my + h'^x^ = m^h'^ for the equation of the sought curve, when the perpendicu- lars to the tangents do not fall on opposite sides of the tangents, and it is manifest that the curve is an ellipse ; EQUATIONS OF THE FIRST ORDER. 505 noticing, if the perpendiculars are drawn on opposite sides of the tangents, that the singular solution will clearly be an hyperbola, 9. To find a curve such, that the length of the normal shall be a (given) function of the distance of its foot from the origin of the abscissas. (See Lacroix, p. 4:6Q.) Supposing X and y to represent the co-ordinates of any point of the proposed curve, it is easy to perceive that represent the lengths of the normal and the distance of its foot from the origin of the co-ordinates ; consequently, ./m- !:=/(.+, I) will express the differential equation of the question. It is easy to perceive that the equation {x — a)^+ y^ = 0, in which C is the arbitrary constant, by putting C = /* (of will satisfy the question, and be the complete integral, since it contains the arbitrary constant C. For by taking the dif- ferential of {x — of + y^ =: C, we have {x — a) dx -\- ydy = 0, which gives a^= x -\- y —^ and x — a ^= — y -~^ so that f'£ + f=fiaf=f(. + yf^; and taking the square root of the members of this, we have 22 506 SINGULAR SOLUTIONS OF agreeing with tlie assumed differential equation. It is manifest that (a? — a)- + 3/^ = c is the equation of a circle, the axis of x passing through its center, a being the abscissa of its center, and c =f{af is the square of its radius. If we take the differential of {x-af + f = c=f{af, regarding a alone as variable, we shall have then, by eliminating a from {:c-df + f=f{aY and - {x- a) = f{a)f' {a), the result will be the singular solution (called by Lacroix, the particular solution) of the proposed differential equation. Kemarks. — 1st. If we put c=^ka in (a? — of -\- 1/ = c^ it will become {x — of + y^ = ka, whose differential being taken by regarding a alone as variable, is -{X-. «) = 2' which gives a = X *l. and thence the equation {x -af + f = lea reduces to 7c' = k(x + 2)' or 2/2 = k(x-\- 1). the equation of a parabola, the origin of the co-ordinates being at the focus of the parabola ; noticing, that this is a EQUATI02f3 OF THE FIRST ORDER. 507 singular solution, compreb ended under the general singular solution given above. 2d. The equations (x-df + f=f{a:f and _ (a, - a) ^ /(«)/' (a), when a is eliminated, or supposed to be eliminated, from them, give, when taken together, a result which is some- times called the general integral of the differential equation while {x — of + if —f{(if^ which involves the arbitrary function f{af, is called the complete integral of the same equation. Thus, xz^ — yz + a — and 2x3 — y = 0, given at p. 187, may be supposed to correspond to (.^_„)^ + y= =/(«)= and -(x-a)=f{a)f{a); and y"' = 4aa?, at p. 187, resulting from the elimination of s, corresponds to the elimination of a from the preceding equations. 10. To find the equation of a curve which cuts a curve having a variable parameter, at any proposed angle. Let OB represent the proposed curve when referred to its 608 SINGULAR SOLUTIONS OF rectangular axes as in the figure, and ach tlie cutting curve, when referred to the same axes ; then, if ~- •= V' and -T- = p dx -^ ax -^ in the proposed and sought curves, stand for the tangents which the tangents to their arcs at their point c of inter- section make with the axis of a?, we shall, from a well- known formula of trigonometry, get tan

J (7J,+ 1) {n + 2)' a^d thence ^ = ^_^1__ + C^ + C^ 2. To integrate a differential equation of the form d^'yy ='('.S) = «- Here we have -^-^ = Y, which, by putting -^ = ^ becomes -^ = Y, whose members multiplied by P=.£, become p£ = Y£, whose members multiplied by dx and then integi'ated, give ^ = fYdy-{-C of the form "I'^Y'+C. SECOND AND HIGHER ORDERS. 517 From this we immediately get —^ = KjyT'Tl'u i ^^^ thence 1 dx 1 r dy — - or a?— ' =/ p~dy~dy~ ^"iij' + Q) J |/(2Y^ + 2C)* dx which is of an integrable form. Thus, if Y = — -„ , we shall have ^ --J -a^---2a^^^^ and thence we readily get the form /ady _ ^ r ^y noticing, that C^ has been used for the first arbitrary con- stant. 3. To find the integral of the form ^ (^, ^) = 0, or of its equivalent F f^, -^ J =0. The equation solved for ~ , gives -~ =Y = £l function of CvX ccx p, and then dx = -~-j an integrable form, whose integral will be expressed hy x= / p 5 ^^^j since ;t^ =i>, we also have 'Ddj) dy = _pdx =-—- J an integrable form, whose integral will be expressed by the form y — f-^ • Hence, eliminating p 518 DIFFERENTIAL EQUATIONS OF THE from the equations . = /f ana y = f^l, we shall get the required relation between x and y. Thus, if we easily get a-~ = dx^ whose integral gives C — ajp-'^zzzx or p = and thence, since dy = pdx, we have y^=(i log C (C — a?) C and a log C being the arbitrary constants. 4. To integrate the form F U ^ , 2) =0, or its equivalent, F lx,2?, -j-\=0. This being an equation between two variables, x and p, by regarding p as being a function of x taken for the inde- pendent variable, may be integrated by the methods given in Section V., which will give the form F (a?, p^c) = 0; in which c denotes the arbitrary constant. If we can from this equation find j9, we shall get ^ = X = a function of a?, and thence^ = -^ gives dy = 'X.dx or y = I X.dx, whose integral gives the relation between x and y. But if we can not find j9 in a function of x, or can find x more readily, then we shall get the form a? = P = a func- tion of p, whose differential gives dx = dF ; and thence from dy = pdx we shall get dy = pdPj whose integral SECOND AND HIGHER ORDERS. 519 y = j pdF gives the relation between y and ^. Hence, from the elimination of ^ from a? = P and y = I pdF, we shall get the relation between x and y. If F (a?, j9, c) can not be readily solved for^ or x, we may put -~ for^, and then try to integrate the result, by the methods for integrating differential equations bstween two variables. Thus ^^^(l+^T ^' ,. 2xdx' V ^ dxV ' or its equivalent 2xdx = ^-— t» {i+rf lias . «= + C = "'-^ , for its integral ; and by subtracting the squares of the mem- bers of this from a*, we have «^-(^+C/ = ^-„ or Va^_(^ + 0)---jl^^, consequently, dividing the members of ^= + C = --^1= by the corresponding members of the preceding equation, we get p = , and thence {a^ 4- 0/ an integrable form. or 2/ 620 DIFFERENTIAL EQUATIONS OF THE For another example, let there be taken the equation or its equivalent Multiplying the members of this by dx and dividing the products by 1 + jr^ we readily get dx 4- -, , 2 xdp = -— ^ dr>, which is* a linear equation, agreeing with the equation at p. 455, when we put x for y, ai^d^ for x, and change P and Q into rr-^ — « and Hence, from the integral given at p. 455, we have X = 6~-^iTp^ (a / ^^r^^ ^- — + C) \ ^ VlH-y / Because 1 the value of x is clearly equivalent to By taking the differential of this, we have which, since dy =^dx^ gives SECOND AND HIGHER ORDERS. 521 , _ cijpdp Gp'^dp _ apdp dip Qdp whose integral gives y = • A == — C loo;^^^^ -pr-, — =^ Op — a ^ , p + Vi + p^ noticing, that we here use C log ;^, for the arbitrary constant. By eliminating p from a? and y, the equation between x and 2/ will be found as required. For another example, we will find the integral of 2 la' "^ ^J\^ -x^ or its equivalent 2 {a^p' + i»") <^ = xpdXy which is clearly homogeneous in p and x. By putting x =pz^ the equation is easily reduced to 2{a'' + z')dp=^B{zdp-{-pd3\ or -£ = ^~-^, whose integral is log^ = log V(2a2 ^ ^2)^ Q^ ^^ =r C |/(2a' + s') ; consequently, from x=zp3 we have From x=pz we have and thence, from cZ?/ = fdx^ we have 622 DIFFERENTIAL EQUATIONS OF THE and from x = Cs ^ (2a^ + z'), which gives o^a? = C -/ (2a' + ^) dz + .^ "^^ "" ^. , we get (^y = C (2a' 6^2 + 2^- dz\ 2 whose integral is y = - C^ z{Za? + 2-) + Q'. o If we now eliminate z from the values of x and y, we shall get the sought equation between x and y, as required. 5. To integrate an equation of the form ^ V' dx' dx^I ~ ^' or its equivalent F |y, ^, -j-j Because dx = — , we have the form reduced to F(..,f) = 0; dy and it is clear that we may now proceed in much the same way as before. Thus, to integrate or its equivalent By putting ^- for -^ , the equation is easily reduced to {^ + yP) £. = ^ + V\ or {a + yp) ^^ = (1 + f) dy, which gives the linear equation ^y - YT^y^P = xTf^P- SECOND AKD HIGHEE ORDERS. 523 Comparing this to tlie linear equation at p. 455, we have and for dx we have dj) ; consequently, we shall, from the formula at p. 455, get r Pdp / /» /• —pdp -^ \ or since 4/(1 +y) ='6'°g ^^ + ^'^ and ^^ = ^-^°^ '^'^^ + P% r 1 + jp^ we shall have j/ = aj? + C ^/(l + i>^) ; and from dx = ~ we shall get J9 1/(1 +i>')' whose integral is = a log^ 4- C log Q' {p + Vl + y) =: log [^« (C> + C' ^1 + i>0 ^• Eliminating^ from a? and y, we shall get the sought equa- tion between x and y. 6. When a differential equation between a? and y\s of the second order, and x is taken for the independent variable, then, regarding dx^ dy^ and d'y each as being variables of one dimension, when the proposed equation is homogeneous in terms of its variables and their differentials, it can be re- duced to a differential of the first order, which does not contain », by putting y = -ua? and — ( = ^ in it. For if n dx sa 524 DIFFERENTIAL EQUATIONS OF THE denotes the degree of homogeneity of the equation, it is plain, since -t4 is clearly of — 1 dimensions, that it must have a factor of ?i 4- 1 dimensions ; and since vx is put for y, it is evident that wherever y is, a?'* must enter as a factor ; it is also evident, since -^ is of no dimensions, that it must have a?" for a factor. Because the remaining terms of the equation are of n dimensions, it is manifest that after the preceding substi- tutions we may divide the equation by a?**, and thus free it of X. Thus, to find the integral of xd^y = dydx^ we divide its members by dx^^ and thence get da? dx^ in wliich we put -~ = - and -^ = p. ^ ditr X dx -^^ and thence get q = j[)-^ and since -^ = ^ is the same as -f- = ^ by putting dx- X dx x^ '' ^ ° p for ^, we thence get dp _p dp _ dx ^ dx~ x^ p ~ X ^ and from y = vx we have , , , , dx dv dy = pdx = vdx + xdv, or — = , ^ ^ ' X p — v^ and hence get dp dv , ^ , '^'=JZr-^^ ^^ Fdp = pdv + vdp', SECOND AND HIGHER ORDERS. 525 and taking the integral of this, we have -|^=^?) + C, or / = 2p?; + 2C. cloa (I'D Also, by taking the integrals of — = — , we have log X = log pG\ or a? = ^C\ which gives i? == p7 . Substituting this value of p in ^^ = 2jpv + 20, we get x' ^ 2G'xv + 2CC^^ which, since y = xv, may clearly be represented by x' =: 2Cy + C; when C and C^ represent the arbitrary constants. Otherwise. — Since the equation a? -r^, = -^ IS reducible to — = -7— , ddir ax X dy ' dx by taking the integral we have log a; = log C ~ , or x = C ~, or xdx = Qdy^ whose integral '— = Cy + const, gives o? ~ 2Qy -f C, the same as by the preceding process. Again, if xd?y = ady^ + hdx^j or x -— = a -—, + 5, then, putting ^ and^ for — ^ and -^, we have < ^' jind thence, from the substitution of the value of a?, we get and integrating this, we get y in a function of ^, with a new arbitrary constant. Hence, eliminating jp from the values of X and y, the equation between x and y will be determined as required. 7. Finally, it may be added that there are two classes of equations of the forms in which n is supposed to be a positive integer greater than 2 ; such that they can be reduced to the forms of equations SECOND AND HIGHER ORDERS. 527 of the first and second degrees, which we have heretofore seen how to integrate. For, bj putting -i-~i — 'W, in the first of the preceding equations, it reduces to F l-y- , wl =: 0, which is clearly of the form of a differential equation of the first order be- tween u and a?, which gives w = X = a function of a?, and /« — 1 Xc?a?" - ^ To integrate the second of the preceding equations, we put ^- = «, and thence get ;^„ = ^. ; and, by substitution, the equation becomes "©.•l-o. -2 an equation of the second order, which we have seen how to integrate at p. 516. Hence, we shall have 'w = X = a function of x ; or, restoring the value of ?/, we have ^n — 2,w /»« — 2 -j-~i = X, which gives y = I X-dx""- Thus, to integrate -7^.-7^3 — 1, we put -^ =: r, and thence dr get r -J- — 1, or rdr = dx, for the transformed equation. By taking the integral of the preceding equation, we ffi have — + C — a', or r = r 2 (-« — C). Denoting the differential coefficient of the next inferior order to r by q, we shall clearly have dq = rdx = r^dr^ or q = — -{- Cij 528 DIFFERENTIAL EQUATIONS OF THE and since dp = qdx = l~ + CA rdr^ or p = consequently, from ^ = 3:5 + + ^- dy = pdx = (~^ + 172 + ^) ^^^» ^^g^* ^ = 3:5:7 + 0:1 + 1:2 + ^ ■ for the complete integral, in which r = V2 {x — C). For an example of an integral of the second order, we will take -7^ = -7^. ax* dx' Puttmg q = ^, we have ^ = ^„ and thence the equation is reduced to d'q d^ = ^'^ whose members, multiplied by dq^ give % = q^ + 0\ or dx=-M=^,, dxr ^ 4/^2 + C« whose integral is x = log ^- a~, , in which log p^, is 1 integration. From in which log p> is the arbitrary constant introduced by this q = i~7, and dx = — — ^ — ? dif ^^-2 ^ 0* we easily get dp = qrljo — ^ ^ — , and by taking the integral i> = V^r -i C"- + C and dy - jxle — dq ^ O'dx, SECOND AND HIGHER ORDERS. 529 whose integral is y z=: q -\- Q'x -f C ; e being the hyperbolic base, aj = log ^^±i^-l+— is equivalent to Q'e' = q-\-V^G\ which gives (CV-^y = / + C^ or CV^-2CV^=:C^ and thence q = ^r^^j e-\ From the substitution of ^ in y = q ■\- Q,' x '\- Q,'\ we have which may more readily be represented by y = Ce^ + Ci6-^ + C^a? + Cs, C, Ci , Ca , and Cg being the arbitrary constants. (6.) We will now show how. to lind the integral of a linear eqtiation of the nth order, represented by ;s^+^^ + B^ + + Mj + % + x = o, in which the coefficients A, B, C, &c., may either be constants, or they may contain the independent variable x without y. We shall determine the general form of the sought integral by integrating the more simple equation which is independent of X, and the coefficients are supposed to be constants. Supposing e to be the base of hyperbolic logarithms, C and m constants to be determined; if we represent y by Q^mx ^Q^ ^j-ig j^g ^^ p^ gg^ ^g g]^g^21 have 23 630 DIFFEREKTIAL EQUATIONS OF THE dy = Cde"''' = mCe"'''dx, or -/ = mCe""', CIS} and in like manner dor ^ dsc^ ' and so on, to any required extent. Hence, by substituting the values of g, J;^, g:^, &c., in the preceding equation, and rejecting the factors C and ^mx^ whicb will be common to all the terms of the resulting equation, we shall get the algebraic equation of the nth. degree, m" + Am"-^ + Bm""-^ + + Mm + N = 0, which, from the well-known theory of equations, must have n roots. Solving the equation, we shall have the n roots, which may be denoted by mj, m^, Wg, and so on, to 7?i„ inclusive ; and since each of these roots satisfies the equation, if Ci, Ca, Cg, and so on, to C„ inclusive, denote the const., then, if the roots are unequal, we shall clearly have for the complete integral of the proposed equation ; as is manifest from the circumstance that each of its terms satis- fies it, and of course all its terms, conjunctly, will satisfy it; and, since y contains n constants, or a number equal to the order of the proposed differential equation, it evidently re- sults that the preceding equation between x and y must be the complete integral required. It may be added, that, so long as the roots of the equation are unequal, the integral will be of the preceding form, whether the roots are all real, or some (an even number) of them are imaginary. If two SECOi^D AND HIGIIEB ORDEKS. 631 of the roots of the equation, as m^ and ^/i^, are imaginary, they may be expressed by the forms a + h V"—l and a — h V~\ since (see p. 440 of my Algebra) the imaginary roots of equations are known to occur in pairs of the preceding forms ; consequently, for the terms Ci6"'i'" + Csc'^a^ of y, we may write and since (see p. 58), ^bxV-i _ ^Qg J)x-\- V — 1 sin hx and g-6x V _ 1 _ ^^g })x— V —1 sin hx, this is easily reduced to e^^ [(Ci + C2) cos bx + (Ci - C2) 4/"^! sin hx']. Since we may clearly assume such values for the constants Ci and Co, that we shall have Ci + 0.2=^ sin q and (C^ — C^) V —1 =p cos ^, we shall thence change the preceding expression into jpe"'"' sin (l)x -\- q). Hence it results that we have reduced the terms Ci6'"i=" + C,6^^^ of y to the form pe""" sin (bx + q\ and it is clear that every other corresponding pair of im- aginary roots will admit of like reductions. It may be noticed, that if our equation has two or more equal roots, our method of finding the equation between x and y will fail for the equal roots, and will be applicable only to the unequal roots. Thus, if in the equation between 632 DIFFERENTIAL EQUATIONS OF THE X and y we suppose the root mj equals the root 7/23, the equation will become which clearly shows that the two constants Ci and Co are actually only equivalent to a single constant represented by Ci + C2, so that the equation, will be defective in not having a sufficient number of constants, which ought to equal n^ the order of the proposed differential equation. The defect may easily be remedied by writing the terms containing the equal roots in the form if three roots, as m^, m^, mg, are equal, they must be written in the form and so on ; observing, that the index of x in the last of the terms of this kind is less by a unit than the number of equal roots. To be satisfied as to the correctness of what has been said, it will be sufficient to apply it to the integral of the equation g + A g + B | + Cy = 0, which is clearly a particular case of the general differential equation, obtained from it by putting 3 for n in it ; conse- quently, the algebraic equation at p. 530 will here become m' + A?n" + Bw, + C = 0, which, we shall suppose, has a pair of equal roots represented by mj = r/?2 ; then, shall y = Ci^) cos Ix + (Cs + C4a?) sin lx\ ; and it is clear that we may proceed in much the same way, when the equation contains any number of pairs of equal imaginary roots. Hence, having found y = Ce""^' + 0,6"'='^ + Ca^'^^a^ + 4- C"^^n^ = C,2/i + C,y, + 03^3 + + C'^y", for the complete integral of dry . . d^'-^y ^ d^-hi ,^ dy whether A, B, &c., are functions of x or not, yx^y.^^ &,o., being called particular values of y, since each of them is supposed to satisfy the above equations ; then, we may assume y = Ciyi + C,y, + + C"y for the complete integral of in which X is supposed to be a function of a? ; by supposing SECOND AND HIGHER ORDERS. 535 the arbitrary constants Ci, C^, C3, and so on, to vary, by subjecting them to the following conditions, which we will illustrate by the case of 71 = 3. Thus, the equation to be integrated is whose integral we suppose to be represented by y = Ci?/i + 02^2 -f Caya, supposed to be subjected to the following conditions : — 1st. "We have dx ^ dx ^ dx dx ^ by assuming yidCi + y^^^O.^ + y^dCs = 0. 2d. We have d^' -^''M ^ ^' dx^ "^ ^' 1^' by assuming the equation dCidi/i + dC^dy.2 + dC-^di/i = 0. 3d. We have d'y _ d% d^y., dy^ d^'-^''d^ ^^''d^'^ ^' "(kf ~ ^' . by assuming dC^d/ y^ dG,d% dCsd% ^ _ ^ dx" "^ dx' "^ dx' + -^ - ^• Hence, if we determine the constants from these three conditions, we shall have y = Ciyi + Co2/2 + C32/3 for the complete integral, since it will (generally) contain three constants, as it ought to do. It is manifest that we may 536 DIFFERENTIAL EQUATIONS OF THE proceed in the same way to find the integral, when the pro- posed differential equation consists of any number of terms, or is of any order. Kemarks. — 1. For the preceding beautiful method of finding the integral of ^ + A^ + 5^1-:^ + +m|^ + % + x = o duf" cbf-^ ^o?.zj"-2 dx ^ from that of the same equation, when X is omitted, by means of the variations of its arbitrary constants^ we must refer the reader to p. 323, vol. i., of the "Mecanique Analytique" of Lagrange, the inventor of the process ; reference may also be made to p. 326, vol. ii., of Lacroix ; to Young, and to most of the late writers on physical astronomy. 2. On account of its importance in determining the lunar motions, we propose to give a different and very simple method of finding the integral of g + ,«=« + P = 0, which is an equation of the second order, in which P may involve v or be independent of it, according to the nature of the case. By putting sin mv = s and cos mv = s\ we have 6-^ -f s'^ =z 1, which gives sds + s'ds' = ; and by taking the second differentials of the equations sin mv = s and cos rnv = s\ "we ^Iso have the equations ■=-7; + ms = and -^^-o + ^ * = ^* dv^ dv^ + m^s'u -\- I Yds' = mB ■= const. SECOND AND HIGHER ORDERS. 637 Multiplying the proposed equation bj ds^ and the first of these by cZw, and adding the products, we have whose integral gives dsdu „ /*_, , . — j-g- -j- nv8U + / rds = mA. = const ; and in like manner, from the proposed and the second of the preceding equations, we have ds^du Multiplying the first of these by 5, and the second by 5', and adding the products, since sds + s'ds' = and s^ + s'' = 1, we get m^u + sjYds + s'jYds' = m {As + Bs'), or mu = A sin mv -f B cos mv — sin 7nv I P cos 7nvdv + cos mv I P sin mvdv, for the sought integral. If p _ M +^ _ 1 such that M and m represent the masses of the sun and a planet revolving round each other at the distance r, C' rep- resenting the square of twice the area they describe around each other in the unit of time, and v the angle r makes with a fixed line, then, if 7n = 1 , our integral becomes 1a- -r, M + m u = - = A sm V -\- B cos v + - — 7^, — . r (j- If in this we put 23* 63S DIFFEREXTTAL EQUATIONS OF THE C' = (M + 7n)p^ Ap = e cos w, and B/; = e sin Wj "we shall pret r = :i ——, r ° 1 -\- e cos (v — to) for tlie equation of the curve described by m in its revolu- tion around M, regarded as being at rest, which is clearly an ellipse when e is less than unity. (See Whewell's " Dy- namics," p. 27.) (7.) If a differential equation between x and y involves only the simple powers of y and dy in separate terms, and has other terms that are independent of y and dy, which do not involve fractional or negative powers of a?, then the proposed equation may be greatly simplified by differentiating it successively on the supposition that y is a function of x regarded as the independent variable. For, since (according to what is shown at pp. 11 to 13) dx is constant, the terras that do not contain y and dy will disappear from the equation in consequence of its successive differentiations. It is hencQ manifest, that if we integrate the first of the differential equations that is freed from the preceding terms,. we shall (often) readily find the integral of the proposed equation. Thus, taking 7 ady = ydx + Cx'dx, or its equivalent -—- — y = Gz^ (from p. 217 of Simpson's "Fluxions"), by taking its suc- cessive differential coefficients, regarding x as being the independent variable, or dx as constant, we have ""dx^ dx - -^"^^ ""d? ~d?- ^^' ^""^ "" dx^ dx^ - ^' thence to find the integral of the proposed equation, we must find y, such that it shall satisfy SECOND AND HIGHER ORDERS. 539 If e represents tlie base of hyperbolic logaritbins, and D an arbitrary constant, then y = Dd"*^, in which m is constant, reduces the preceding equation. to 1 - which gives tn = -; and thence y = De^ , is the first d integral of the preceding equation, and of course a part of the integral of the proposed equation.. Since «^_^ = 20 dx"^ dx- is clearly the first direct integral of having 20 for the constant, it is manifest that y = Be**, having D for its constant, must represent what is called an indirect integral of the same equation. To find the remaining terms of y, or the proposed integral, since it is clearly to be regarded as the integration of an equation analogous to d^y dry _ ^^ dx^ dx^ ~ " ' which being of the third order of differentials, its complete integral must contain three arbitrary constants; conse- quently, we may represent the sought terms by y = Ax' + Ba; 4- C, in which A, B, C, are the constants, and the integral is evi- dently of the proper form, since it must be supposed to have vanished from the equation 540 DIFFERENTIAL EQUATIONS OF THE in consequence of the successive differentiations of ttie pro- posed equation. To find A, B, C, we substitute the values of y and -~ for them in the equation and thence get (A + c) ar* + (B - 2aA) cc + C - aB = 0, which must clearly be an identical equation ; consequently, we must have A + c = 0, B - 2a A = 0, C - tiB = ; which give A = - c, B = 2aA = - 2ac, Q = aB = - 2o?c. Hence, from the substitution of these values, we have y = Ax- + Bx + C = — c{x^ + 2ax + 2a^) ; consequently, adding this to the preceding value of ?/, we X have y = De"" — c {x^ + 2ax + 2a-) for the complete integral of the proposed equation, which is the same as found by Simpson. By a like reasoning, the integral of the differential equa- tion ady = ydx + cx'^dx, will be found to be expressed by X y = De'' — c [i»"+naaj"-i4-n(^-l)aV-H7i(7i-l) (n-2)aV-5+ &c.], which agrees with Simpson's integral; noticing, that the SECOND AND HIGHER ORDERS. 541 integral will consist of an unlimited number of terms, when n is not a positive whole number. (8.) The integral of a differential equation of the second order between x and y is often readily effected by inter- changing the dependent and independent variables, or, which is the same (see p. 36), since -—- and -y- are equivalent to ^(l)=- dyd?x d^ and d \dy)- dxdJ^y dy^ by writing -'-1? ^- ^'y or — dxd^y dy for d'n Thus, by taking dxdy - - xd?y ■ -ad?y xdy"- h ■ = (from p. 183 of Yince's "Fluxions"), and writing^ ^4 — for d-y in it, we get dx ' dx h -, , xdyd-x adiidfx xdiP- or its equivalent Because c^a?^ + xd^x =: d{xdx), and that dy is constant or invariable, by taking the integral of the preceding equation, after dividing by dy, we get xdx adx ^ _ r\ dy dy 2b ~ "^ an arbitrary constant. Since this equation gives 7 2hxdx 2abdx 2ho + x^ 21)c + ar*' 542 DIFFERENTIAL EQUATIONS OF THE by taking the integrals of this, and using G' for the arbitrary constant, we have y + C'=log(2JC+ar^)* + a|/M X &rc(rd=l and tan -^L.) which agrees with Yince's integral after the constant C is added to his value of y. For another example, we will take the equation X(Px — dydx = 0, which is such that the method of integration does not readily present itself to the mind. By substituting -— for d^x in this equation, it becomes dxd^y 7 7 /^ 7 d^y dx r-^ X — dydx =0 or c?aj V^ H = 0, dy ^ dy^ X ^ in which dx is constant or invariable. Since this equation is the same as by integrating and using C for the arbitrary constant, we have dx , ^ dx — -7- + log x= C or dy = dy ° ^ log a? — C ' in which the variables are separated, and /dx kg^'^^^ indicates the integral. Putting dx log x — C = 3, we get — = d2 or c?aj = xdz, X which, since log x= C + z gives x =6^'^' = e^e% and re- duces the preceding integral to SECOND AND HIGHER ORDERS. 543 d3 which (see p. 51), gives ^ f*/d2 dz zdz z^dz „ \ .( ^°"" + 1 + 1- S + 1- lis + i- lAi + ^') + ^"' in which C^ is the arbitrary constant. If ?/ = when z = \^ and we have C = — a^*^, and thence which will clearly give all the values of y that correspond to those values of z or log x — that do not differ greatly from unity, or that are positive and not very small. (See Lacroix, p. 512, vol. iii.) (9.) Sometimes a differential equation can be integrated more easily by eliminating an arbitrary constant from it, by means of its differential equation, particularly when it con- tains two variables, as x and y, and higher powers of -~- ox -X- than the first power. ax dy ^ Thus, by taking the differential of ^^y% + {x'-Af-B)^-xi, = 0, we have g(.A..| H- ^-A^-b) -. (Ag -m)(.|-.)=0, 544 DIFFERENTIAL EQUATIONS OF THE in which x is taken for the independent variable. From the proposed equation we get aj2 _ Ay'* - B = xy -^ , dx which being substituted in the preceding differential, and re- dip- jecting the useless factor A -j-^ + 1, gives the differential equation xy^y + dy {xdy — ydx) = 0, or - d^y — dyd - = 0. The first integral of this clearly is dx y^ and the integral of this is y' = Cx' + C\ From the substitution of the values of — and y^ in the pro- posed equation, after a slight reduction, we get ACC + C = - BC, and thence C = AC + 1' consequently, from the substitution of this, we shall get for the integral of the proposed equation. Eemark. — For the substance of what has here been done, we shall refer to p. 123, &c., of Monge's " Application de r Analyse a la Geometrie," and to Lacroix, p. 370, vol. ii. (10.) The integral of a differential, or differential equation, between x and y, may sometimes be found by assuming an expression for the integral, or for the relation between x and SECOND AlTD HIGHER ORDERS. 545 y^ in ■undetermined coefficients and exponents of x and y if required, such, that by putting the differential of the as- sumed integral equal to that proposed, tliey may be made identical, so as to determine the indices and coefficients. Thus, to find the integral of the differential adx + hxdx ex + x^ (given by Yince at p. 184 of his " Fluxions "), it is evident that it may be represented by the form A log {ox'' + «?'■ + ^), of like form to the integral assumed by Yince, in which A and r are to be found. By taking the differential of the assumed integral, we have . Tcx^ ~ hlx -f- (/* + 1) x'dx ^ ^^ + x^-^' ' which must be made identical to the proposed differential which serves to find the unknown A and '/'. If we put r = 5 + 1, and multiply the numerator and denominator of the proposed differential by x% its denominator becomes identical to. that of the assumed differential; consequently, we must have the identical equation {ax" + Ix^ ^^) dx —. \_A:rcxf + A{r + 1) cc^ + ^] dx^ or, rejecting the useless factor x^dx^ we shall have the identical equation a + hx = Atg -\- A{r -\- 1) a?, which, by the method of undetermined coefficients, gives b T -\- \ a = Arc and h = A (r + 1\ and thence - = , ^ ^' a TO which gives a , . a 1)0 — a r = f and A = — = . OG — a TO G 616 DIFFEREN"TIAL EQUATIONS OF THE From the substitution of the values of r and A in the assumed integral, and using log -^ to stand for the arbitrary constant, we get he I adx + l)X(lx he — a ^ ca?*<'-« + a/'' — log ex -\- a^ Q for the correct integral. Putting a bo in which x' represents some particular value of a?, then be 1 6c — a /adx + Ixdx , Icx^"- 6c- 6c l^^/6c-a ^ aj^6c-.j which equals naught when x = x\ and agrees with Vince's integral when it is properly corrected. (11.) Sometimes the integral of a differential equation of the higher orders, between x and y, may be simplified in form by taking a function of the independent variable for a new independent variable. Thus, if we take dx'^ X dx ^^2/-^, and put 1^ r= a?" + 2 = a?*™, we may evidently take w for the independent variable. Because y is supposed to be a function of a?, which equals a function of u^ by taking the differentials of these equal functions, we have ^dx=.'kdu or ^^^/i!f. dx du ' dx du dx^ SECOND AND HIGHER ORDERS. 547 consequently, since u — x""- gives ax we shall ffet ^-^ -— ?n~ x^'-K ° ax dti Again, since u is to be taken for the independent variable, in we must, for — in the proposed equation (see p. 36), write CL'X" d its equivalent J"'^— ; consequently, since dx dy dy ^ ~ =z m ^ x' ax du d'u and that v^ is a function of x. du ^^ -m(;7i \) ^^x +^^..yj ai^ air Hence, from the substitutions of these values of di'^ dy \dxj ' dx'' dx in the proposed equation, after an obvious reduction, it will become -^^ + (1 + ' — ) -f 5- = 0; which, by putting 1 -\ = g and —^ = h. becomes du^ xidu u ' 548 DIFFERENTIAL EQUATIONS OF THE which is the sought transformation, and it is clearly of a much simpler form than the proposed equation. For another example we will take the equation and shall take u = e""' for the independent variable. T, du dy du , dy dy „, From -/ = -/ ^ we have -f =n^ e"* ax du ax ax du ^ Jdy \dx) _ ^_^ dx ~ du^ " ' '" du and thence — -. = n^ Vi. ^^"^ + ^' ^y.2nx . .2^2/ consequently, from the substitutions of these values, the proposed equation becomes du" "^ which, by putting du^ \ n) udu n^ u ' 1 H = c and -^ =: h becomes -x-^ + c — r h-=zO, dw udu u which is the same as the transformation in the preceding example; consequently, the solution of one of the given equations must be given by that of the other. Eemark. — The preceding equations are due to Professor Peirce ; they appear to have been first published at p. 399 of Professor Gill's "Mathematical Miscellany." (12.) When a differential equation between x and y is such, that its integral, in finite terms, can not easily be found, then we express the dependent variable in a series of terms of the independent variable, having undetermined exponents SECOND AND HIGHER ORDERS. 649 and coefficients ; and then, substituting the assumed series for the dependent variable in the differential equation, we deter- mine the exponents and coeffieients, so that the indices of the independent variable shall increase from left to right for an ascending series, and shall decrease from left to right for a descending series. Thus, to integrate dv^ udu u ' the transformation found in (11), it is easy to perceive that "we may assume y = Au'' + Bi^« + 1 + C2^« + 2 +, &c., which, being put for y and its differentials for those of d^y and dy in the equation, gives A\a{a-\-G—V)]u''-^-\- [B (a + 1) {a + c) — AA] i^« - ^ + [C (a + 2) (a + c + 1) - BA] u^ + [])(« + 3) (a + c + 2) - CA] t^« +^ + &c. =: 0, which must clearly be an identical equation, or be satisfied independently of u. The first term of the equation is evi- dently reduced to naught by putting « = 0, or a H- c — 1 == 0, which gives a = 1 — 2W + (48AC)y« - which must clearly be an identical equation. 2i 0, 554 DIFFERENTIAL EQUATIONS OF THE Hence, we must have 2aB - 12aB + 2A- - SA^ = 0, or B 2aC - SOaC + 4AB - 32AB = 0, or C 2aA — 2aA =0, or A is arbitrary ; ba ' 3A« 5a^' 2aD-56aD + (2BH 4 AC) - (32B2+48AC) = 0, 31A^ ^=-45^' and so on ; consequently, from the substitution of these values of A, B, C, &c., we shall have for X expressed in a series of ascending powers of y^ in which A is the. arbitrary constant. To find a? in a series of descending powers of y^ we have x = A'y + W + Q'y-^ + J)'y-^ +, &c., whose differential coejQ&cients are dy d?x CV-^-2DV-3-,&c., and ^ = 20^-' + 6DV-'' +,&c; consequently, from the substitution of these values in the proposed equation, we have 2a A V + 2aB' + 2aCV-' + &c. * ~2aQ'y-^ — ho.. 2AfY + 4A'BV + 2B^2 ^ 4,wC'y~^ + &c. - 2A'y + 4A'C' + 4A'DV-' + &c. + 4A'C^ + 8A'DV-^ + &c. = a SECOND AND HIGHER ORDERS. 555 whicli must be an identical equation. Hence 2A'2 — 2A'^ = 0, or A' is arbitrary ; 2(^A' + 4A'B' = 0, or B'= "^ 2' 2 2aB' + 2B'^ 4- 8A^C' =0, or . C' = ^^ ; 4B'C' + 12A'D' = 0, or D^ - g^^, and so on ; consequently, from the substitution of tbese values of A', B', C\ &c., we shall have for the value of x, when expressed in a series of descending powers of y, in which A' is the arbitrary constant. Because the proposed equation is of the second order of differentials, its complete integral must involve two arbi- trary constants ; consequently, from the addition of the two particular values of a?, we get the complete value of , ., 3A= , 3A^ g . ., a a" 0, = Ky- ^t + g-^2/« - &o. + A'y-- + ^,- + ,&c., as required. Eemarks. — 1. If, with Mr. Young, at p. 260 of his "In- tegral Calculus," we integrate the equation ll + -~jy = l by the preceding methods, we shall get y = l+^2{x-a)^-l{x-a)-\-^{x-af -. &c., O ib in which h is the value of y that corresponds to a? = a, which is clearly equivalent to the determination of the arbitrary constant 656 DIFFERENTIAL EQUATIONS OF THE 2. This question can clearly be integrated without using series, bj regarding x as being a function of y. For the equation can be reduced to dy ^ dx y ^1 . — 1 ^ dx ^' dy 1—y 1— y 1— y which gives dx = — dy -^ - — — ; and thence x = -y-log{l-2/) = -y + log ^— — , which needs no correction, supposing y and x to commence together. (14.) To what has been done, it may be added, that differ- ential equations, of the first order in particular, may often be elegantly integrated in a continued fraction. Thus, by taking the differential equation (See Lacroix, vol. ii., p. 427), and putting A^ and Ax^=::X, P'= P+ QX4-RX^ + S ^, ^ l-\- y dx Q' = 2P + QX+S^, E' = P, S^=-SX, we shall have the transformed equation p' + qy + ^Y- + s'% = 0. 3x^ If in this equation we put y' = - — '- — 77 , and in the pre- ceding results change P', Q', K', S', and ~ , SECOND AND HIGHER ORDERS. 657 dij" into Y\ Q", E'', S'', -j~ , we shall, in like manner, get for the transformation of the preceding transformed equation, and so on, to any required extent. To make what has been said more evident, take the par- ticular example ^y + (1 + ^) ^ - 0. Then y = - — '- — ,, supposing A,7j^ and y' very small, may approximately be reduced to y = Aa?'', which gives and thence the proposed equation is approximately reduced to (mA + a A) x"" + Aaa?«-^ = 0, which is approximately satisfied by putting a = 0, and omitting mA on account of its supposed minuteness ; con- A sequently, we may put y = j , and shall thence get dy _ dx dx~ ~ (1 + yj ' Hence, from the substitutions of these values of y and dy ~- , the proposed equation becomes which is easily reduced to dy' — m — my' + (1 + a?) -^ = 0. 658 DIFFERENTIAL EQUATIONS OF THE By changing, as before, 'i/ into Ba?*, and -~- into JBa^~\ this equation becomes — m — mBaj* + ^-Ba;* + JBoj*-^ = ; "which is clearly satisfied, as required, by putting 5 = 1, and making B = m, when terms of the order m- are omitted ; consequently, 77 is reduced to ^ ; noticing, that > if ^ if we have hence reduced y to y 1 + / Baj« ma? If for 1/ in the equation - m -my' ■\-{l + x)^= 0, we put its equal, after a slight reduction, we shall get the equation {rn-l)x-\-\l + {m-l) a^] y" + y'"-^ {l+x)x ^ = 0. Putting Caf and cCaf - ^ for y'' and -|— in this, we get (m— l)aj + [1 + (m — l)a?] Ca?<^ + C'^ar' + cC{l + x)x' = 0; which, by putting c = 1, omitting the common factor x, and retaining only the principal terms, reduces to m - 1 + 2C = 0, and gives = - '^- ; consequently, =——77, becomes ~^^^ — • Proceeding SECOKI in this A ) AND HIGHER OB way, we shall get LDEKS. ^"1 4- rrix 1- m — 1 1 X '2 1 + m -{- I X 3 '2 ^ 771—2 ^ 3 X 2 1 + "^ + 2 a? J- -1 5 '2 1 - m — 3 5 •2 559 1 +, &c., for the sought continued fraction, the same as found by Lacroix, at p. 429, vol. ii. Because the equation my + (l + a^)^ = • , ., . , dy mdx IS reducible to ^j^ — — _ y 1 + a?' . . , . Its integral gives y = ^y—y^i', which gives y =^ C when a? = 0, and the continued fraction when x — gives y = A, and thence C = A ; consequent- ly, by putting A — 1, we shall have 1^_ _ _J (1 + aj)"* ~" 1 + 7rix 771 — 1 X 1 ^ +, &c., 560 DIFFERENTIAL EQUATIONS OF THE or, taking the reciprocals of these equals, we have (1 + aj)"* = 1 4- ^ m — 1 X 1 '2 m -f 1 OS 3 '2 1 m — 2 X ~ 3 -2 1+-+? a? •2 i_» -3 5 X '2 l+,&c. ; consequently, the binomial theorem may be considered as being reduced to the form of a continued fraction. Since the exponential, theorem (5), at page 51, reduces (1 + xY to -I ■ 1/1 N m^ rioff (l+a?)P 1 + m log (1 + ») + 12 "^' ' we hence get 1 + m log (1 + a^) + &c. = 1 + ^__ ■^ r~'2 1 +,&c. ; or, from an obvious reduction, we have log (1 + £C) + &C. = - ^ m — 1 X 1~*2 1 H-, &c., SECOND AND HIGHER ORDEES. 5G1 wliicli,»by putting m — 0, reduces to log (1+ aj) r:r . 1 + i^ ^3 2 ^3 2 ^5 2 ^5 2 1 +,&c. ; consequently, the hyperbolic logarithm of 1 + a? is reduced to a continued fraction. 1-1 ) , when w, is infinite, equals [see (J^) at p. 51], it clearly follows that if in (1 + a^)- = 1 4- inx ^ in — \x 1 2 1 +, &c., qn we put — for ,r, and suppose w. infinite, we shall get 24* 662 DIFFERENTIAL EQUATIONS OF THE X e'=:l + 1 X 1*2 ^ ^ 3 2 3 2 1 + i.^ ^5 2 1 » 5*2 1 +, &c., for the conversion of e' into a continued fraction. For another example, we will find the integral of 1 - (1 + (B") ^ , or its equivalent dy = j-^ . Bj taking the integral of % = ^ — — ;; = ( 1 -— :; I d^!, , r dx r x^dx we have V = I z, = x — I , ^ J 1 + x^ J l + af"' which needs no correction, supposing x and y to commence together; and to find y in a continued fraction, we may clearly put y = ~- — ;-, which gives aj r x^'dx 1 ^ 1 r x^dx 1 -\- y' •/l-raj"' I + y' x J 1 -\- x' whose reciprocal gives ^ \ xJl + x"") = 14-1/* ^^^ ^ A _ 1 r^^x\ xJl + x^'\ xJ 1 + xY SECOND AND HIGHER ORDERS. 563 ^^ ^ ~xJ r+ic^ ' \ ~ xJ r+~^/ "" U + 1 X J iT^/ ~ I ^ xJ i + ^j ' A consequently, by putting :^ -, = y\ we thence get aft and thence A = — — r is the numerator of the second of n + 1 the continued fractions, and or, taking its reciprocal, we have . , „ /, \ r x'^dx \ r n + 1 r x-^'dx \ ~ '^[x''-^J'l + X'' xJ 1 + W • I ~" S^^^y i"+a^7' or, after a slight reduction, „ ^ l {n-^l)x^ ^_ _ n+_l r x^Hx 1 r^dx\ ^ ~ \ 2/1 + 1 n 4- 1 a" + ^ y 1 + a" "^ a^y 1 -f ic"j "" [{n + l)(2n + 1) c^y 1 + ^" " "a^^^y 1"+^/ / n -\-l r x-'^dx \ Putting y'' = — — JJ-, , we shall easily get B ^ (/I + 1) (2n + 1) 564 DIFFERENTIAL EQUATIONS OF THE for the numerator of the third of the continued fractions ; and then taking the reciprocal, we shall have ^^y \ aj" + ^ J l + x-l and so on, to any required extent. Hence, we shall have dx X ^=A + aj" 1 . ^" 71+ 1 1 1 tvx'' ^ ' {n^ - 1) (2?i + 1) 1 + {n + 1)-V (2/1 + 1) (3^1 + 1) (2^)V ' (371 + 1) (471 + 1) 1+,&C., for the sought continued fraction. (See Lacroix, vol. ii., p. 431.) If 71 = 1, this formula gives the same expression for log (1 + x\ as at p. 660 ; and if ti = 2, we shall have dx f 3 \, — wux it/ — r— "^1.3 •+s 16^ ^ 7.9 1 +, &a SECOJTD AN'D HIGHER ORDERS. 565 (15.) We will now proceed to show liow to find the inte- grals of what are called Simultaneous Equations^ such as and Wy + N'a^ + P' ^ + Q' ^ = T', in which M, N, P, &c., are supposed to be functions of ^, considered as being the independent variable ; the equa- tions being coexistent and of analogous forms. 1. To integrate this kind of equations, after multiplying by the differential of the independent variable dt^ they may be written in the forms QLy + Na?) dt + Ydy + (^Ix =-- "Yldt, and (M^ + Wx) dt + Y'dy + Q!dx = Tdt ; and multiplying the second of these by 0, regarded as being a function of t, and adding the product to the first, we get the single equation [(M + M'^) 2/ + (N + ^'0) x] dt + (P + P'^) dy -f (Q + Q'^) c/^ = (T + TB) dt. Putting M + M^0 = Ml, N + N^6? == Ni, P + P^0 = Pi, Q + Q'0 ^ Qi, T + T'0 = Ti, we thence have (Miy + Ni,^) dt ■\- V^y + q,dx = T,dt, a form analogous to the proposed equations, and it is clear that this equation is equivalent to Ml {y + ^ x) dt + Pi (^dy -\-^dx^ = T,dt, 666 DIFFERENTIAL EQUATIONS OF THE N Putting y + ^x=z and assuming diy + ^i— x\ = dy + ~ dx^ the preceding equation is reduced to tlie form 'M.^zdt + PiC?2 = T A or dz + ^ zdt = ^ dt, which (see p. 455) is a linear equation. From by taking the indicated differentials, we have dy + ^^dx-{- xd^^ = dy+ ^^dx, or =-^ dx + xjI ^ = ~ dx, Ml Ml Pi which must be an identical equation ; consequently, ^ = 1' and cZ^' = 0, Ml Pi Ml N + N'0 Q + Q'0 , ^^ + We . ^^ MTM^ = PTT^ "^^ ^MTM-^ = ^' and by performing the indicated differentiation of the second of these equations, and eliminating 6 from the result and the preceding equation, we shall get the relation between the coef- ficients of the proposed equations that must exist, in order that their integration may be reduced to the integration of the preceding linear differential equation of the first order. It may be noticed in this place that, if we integrate the equation "^ MTM^"" ' we shall have ^^^p^ = C = a constant, and tlience SECOND AND HIGHER ORDERS. 667 reduces to F+"P'^ ~ ^ ' consequently, eliminating from these equations, we Lave ,^N-CM ^^^ , _Q-CP /> CM'-N' CP' + Q , . , . N - CM Q - CP whicligive ^^^^-^-^ = ^^^-^^^ or, reducing this to a common denominator, &c., we have , NP^ - PN + MQ- - M^Q _ KQ^-QN^ "^ PM'-MP' PM'-MP^' Hence, having found C from the solution of this quadratic, and taken the differential of its value on the supposition of the constancy of C, we shall clearly get the same result as from the preceding method. Solving the linear equation will clearly give s in terms of W t ; and thence from y + ^, x = b, we can find y in terms of X and t, which, being substituted in either of the proposed equations, will give a differential equation in x and i, whose integral gives x in t, and thence having found x and y in terms of t, by eliminating t we shall get y in terms of x, as required. 2. If the coefficients M, N, P, &c., in the first members of the proposed equations, are all constant, it is clear that M + M' is satisfied by supposing that 6 is constant ; and thence, from the solution of the equation 568 DIFFERENTIAL EQUATIONS OF THE -we shall get, by the solution of a quadratic equation, two constant values, O'and^'', of ; consequently, if m and n are the coefficients of the linear equation that correspond to 0\ and m' and n' are those that correspond to d'\ the linear equation gives the two linear equations dz + mzdt = ndt and dz + m'zdt = n'dt. Integrating these equations by the formula at p. 455, we shall have z = e-/""^'(fne/'''^'dt\ and z = e-f'^'^'(fn'ef'^'^'dt\ for the sought integrals ; noticing, that the arbitrary constants are supposed to be comprehended by the integral signs / 77, &c., / n\ kQ. By substituting the values of y + ^, a? that correspond to those different values of 2, for z in the preceding integrals, we shall have two equations in a?, ?/, and ^, which will clearly give x and y in terms of t ; consequently, from the elimination of t^ we shall get y in terms of a?. 3. For another example, we will integrate the simultaneous equations ' dy + (My + Naj + P^) dt = Tdt, dx + {Wy 4- N'a? + F'z) dt = Tdi, . dz + (M'V + N^'aj + F^'z) dt = T'dt; which may be supposed to be obtained from three equations of forms analogous to those of the preceding example, by eliminating -j- and j- from the first by means of the second and third equations, and so on for the remaining equations. SECOND AND HIGHER ORDERS. 569 Supposing T, T', T'^, to be functions of t^ while the other coefficients in the preceding equations are constant; then, multiplying the second and third equations by the constants C and C, and adding the products and the tirst together, we shall have a single equation of the form dy + Qdx + G'dz + Q (2/ + Raj + S,2) dt == JJdt. If in this we change C and C^ into R and S, it will become dy + 'Rdx + Sc^3 + Q (?/ + Ra? + Ss) dt = XJdt ; consequently, putting y + Hx + Sa = 'y, since R and S are constants, the equation will become dv + Qvdt = JJdt, a linear equation, whose integral gives Vj or its equal y + 'Rx + Sb, equal to a function of t. 4c. The preceding process is applicable to differential equations of the higher orders, which may clearly be re- duced to those of the first order. Thus, the equations d'y + (My + N;7?) df + {Fdy + Qdx) dt = Tdt"" and d'x 4- (M^ + Wx) df +(:?'dy + Q'dx) dt = Tdt', by putting dy — pdt and dx = qdt^ are reduced to the equations dy ^ jpdt^ dx = qdt^ dp + Q^y + Naj + P^ + Qq) dt = Tdt, and dq + Q^'y + Wx + P> + Q'^) dt = Tdt, to which the preceding method can evidently be applied. (See Young, p. 264, &c. ; and Lacroix, p. 337, &;c.) 670 EQUATIONS OF THE HIGHER ORDERa By reducing the first two of the preceding equations to dy — pdt = 0, dx — qdt = 0, and multiplying the second, third, and fourth by the con- stants C, C, C, and adding the products, we have the single equation dy-[-Odx-{- C'djp +C'W^ +Q(y+ Eaj + S/? +Yq)dt =Vdt. Putting dy-\-Qx + Q'p + 0"q = d{y +Ea; H- S/? + Yq\ and C = E, C = S, C'^ = V ; then, since these values are constant, our equation is reduced to the linear form dv + Qpjdt = JJdt^ in which v is put for 2/ + Eaj + Si? + Yq, and thence, oy taking the integral, this becomes a function of t. For a simpler method of integrating simultaneous equations of the second order, under certain restrictions, we shall refer to p. 130 of Whewell's " Dynamics," or to any other work that treats of the very small vibrations of what are called Complex Pendulums, SECTION IX. INTEGRATION OF DIFFERENTIAL EQUATIONS CONTAINING THREE VARIABLES. (1.) If we have Fdx + Qdi/ -f ^ds = 0, such, that X and y are considered as independent variables, P Q and z a function of them, then, if i? = — p and ^ == — ^j^- , the equation will be reduced to the form dz ^ pdx + qdy. If this is the total differential of z, regarded as being a function of x and y, it is evident that we shall have dz , dz -J- and a = -J-. dx -^ dy and because dz is supposed to be an exact differential, its equivalent pdx + qdy must also be an exact differential ; consequently, Euler's condition of integrability (see pp. 489 and 440) must exist, which gives the differential coefl&cient of p taken relatively to y equal to the differential coefficient of q taken relatively to a?, and thence, since p> and q may contain s, we shall get dp_^dpd2^^dq^dj^dz ^^ ^^^^^^_p^^ dy dz dy dx dz dx dy " dz dx -^ dz ' or, by transposition, we have 2? = -7- and dy dx ^ dz -^ dz for the condition of integrability of dz = pdx + qdy. 572 DIFFERENTIAL EQUATIONS Because we liave supposed that P=- P , ^ = Q B' we thence get dy _ dy dy dq_ dx~ dx R^ c J 't- Q ■R dz dz ^i- ■R consequently, from the substitution of -^ , -^ , &c., in the ciy ctx preceding equation, it becomes dy dy dx dx dz dz which expresses the condition of integrabilitj of the equa- tion Vdx + Q,dy + ^dz = 0, on the supposition that when multiplied by the factor ip , it P Q is reduced to dz ■}- ^ dx -\- -dy =z 0, or its equivalent p Q dz=^ — ^dx — ^dy=z j)dx -\- qdy^ which, by supposition, is an exact differential equation. Hence, to find the integral of the differential equation Vdx + Q^y + Rf/s = 0, we examine it to see if the preceding condition of integra- bility is satisfied ; then, if it is satisfied, we multiply it by some factor m, which reduces it to the form mVdx + mQfly + inRdz = 0. CONTAINING THREE VARIABLEa 573 To determine tlie proper form of m, we may omit any one of tlie terms of the equation, as the last, then we find m such that onPdx + mQdf/ =^ shall be an exact differential, on the supposition of the con- stancy of 2 ; and putting du = mTdx + rnQdi/, by taking the integral, we have u = J {mVdx + mQ,di/) + (s) = V + (2) ; in which (s) = a function of 2, is used for the arbitrary constant, since, in the integration, z has been considered as a constant. To find (p {z\ we differentiate the members of the equa- tion u ~Y -\- (s), relatively to z only, and thence get die _ dV d4> (z) ^ dz ~~ dz dz '' consequently, since u is here supposed to be the integral of mVdx + onQ^dy + rn^dz = 0, -y- = wE, O/Z and thence ^ dV , d{z) d4>{z) ^ dV mR = — - 4- -^-^ , or -^-^ = mR , dz dz dz dz which gives iz) = I ( wE — -j~j dz, and thence the integral becomes known. Because (s) is independent of either x or y, it is clear that wdien the factor m. is correctly found, it must be inde- pendent of either a? or 7/. 574 DIFFERENTIAL EQUATIONS Thus, to integrate ysdx — xzdy + yxdz = 0. "We have P = ys, Q = — a?^, and K = ya?, and thence the equation of condition becomes dy dy dx ax az dz yzx — yxz — yxz -\- xzy — xzy + yzx = 0, and the condition being satisfied, the proposed equation must be integrable. To find the integral, we omit the last term^ and thence get 2 {ydx — xdy) = 0, which becomes an exact integral, by multiplying it by the 1 zx factor m = -"2 , the integral being w = — + <^ (^) J if J ^, du z d(f) (z) consequently, _ = _ + -^J ; or, smce du _yx _x dz - -2 - .,» X. M J. XX doiz) we shall get - = - 4- —^ , y y dz ' which gives d {z) = and (2) = for one of the integrals ; and taking the differential coefficient of this relative to s, we get -7 — | ^— - , which, being put equal to Em, the coefficient of dz, in the equation Tmdx + Qmdy + E//?c?s, dY dcp{z) ^ gives -7- H ~ = Em ° dz dz 676 DIFFERENTIAL EQUATIONS for the other equation ; consequently Y+H^) = and '^J. + ^i^) = -Rm, ^ ^ a 2 dz in whicTi ^ iz) is an arbitrary function of z^ which satisfy the equation Vmdx + Qmdy + "Rmdz = 0. Thus, of ydr/ + zdx — dz = 0^ regarding z as invariable, the mnltiplier m- is 2, and thus the equation to be integrated becomes 2ydi/ + 2zdx — 2dz = 0; the integral of its first two terms, regarding z as const., is y' + 2zx + 0(2) = 0, and thence -^ H — ~-^ = B>m dz dz P^-% or 2. + ''#) «0 dz becomes 2« + ^P - - 2, or 2« + "^-^ + 2 = 0, which, by putting ^{z) = 2^, is immediately reduced to a? + 3 + 1 = 0, the equation of a right line. In much the same way 2xzdx 4- 2yzdy + x^dz = can be satisfied by {0^ + f)z + 0(3) == and a^ + f + "^^ z= cc^ For another example, we will take the equation xdx + ydy dz x{x — a) -\- y {y ^^17) z' — 'g~' CONTAINING THREE VAEIABLES. 577 Tliis equation can be immediately satisfied by putting X (x — a) -\- y {y — h) = (p(2) := a. function of 0, wbich reduces the proposed equation to 2xdx + 2ydy = — ^^ c?s, whose integral is x^ + y^ = 2 I — — - dz. It is easy to perceive that, by putting (2) = — (s — c) 0, the integral becomes x^ -\- y^ = - s^ -{. W, or a;2 + j/f^ + s' = B the equation of a spheric surface, in which W is used for the constant. It may be added, that, if we put y = x, the differential is immediately reduced to = ;r = , whose -^ z — c 2x — a — b' integral is clearly 3 — = C {2x — a — h). (3.) It may be observed that the differentials and their integrals here considered being of algebraic forms, their in- tegrals are sometimes called algebraic integrals. Algebraic integrals of differential equations can some- times be obtained from the simplest principles. Thus, to find the algebraic integral of dxVl+i(^ -{- dy Vl -{-f = 0, or of its equivalent we may proceed as follows. 24 578 DIFFERENTIAL EQUATIONS By assuming X +1 V or a^ — {l + v)x-i-l — v=0, and using x and y to represent its roots, we shall, from the well-known theory of equations, get X -{- y = 1 + V and xy = 1 — v, whose sum gives x -{- y + xy = 2 for an algebraic integral of the equation. Because x and y are roots of ar^ — (1 + -y) a? + 1 — -y = 0, it is clear that we shall have / a^ _ a, + 1 _ /y> -y + 1 _ ^ x + 1 - ^ y + 1 -"■' consequently, by erasing this common factor from the second form of the proposed equation, we shall get the differential equation {1 -{- x) dx + {I -\- y) dy = Oj whose integral is X^ 4- y'^ a? -f y H -^— = C = the arbitrary constant From the algebraic equation we have x + y == 2 — xy^ which, substituted in - + y + ""-^ = c, 2-^ + ^^' = C, or (^^ = C-2, reduces it to 2 — xy or, more simply, {x — yY = Q' = constant^ an integral that is evidently of an algebraic form. CONTAINING THREE VARIABLES. 579 Remark. — If we proceed in like manner to integrate dx M= = o, we shall get a;^ — (1 + -y) a? + 1 — "^ = 0, and thence a? + .y + a?y = 2 for the algebraic integral, the same as before. Hence, from i/ ^ + 1 ^ a/ y + ^ ^ X' -X +1 ^ f -y ^V the proposed differential equation reduces to dx ^df^_^^^ ^^^^ {\ + y)dx + {l-\-x)dy = 0, 1 + a? 1 + y whose integral is a; + 2/ + a?y=:C = the arbitrary constant, which, by putting 2 for C, becomes x -\- y -\- xy =i 2^ which is the same as the preceding algebraic integral. For another example, we will show how to find the algebraic integrals of dx Vl -V ix^ -{■ dy Vl + f + dz \n^ + z^ z= 0. Because cfo vr+w= dx {X- + X) /^^~ ty putting ^lt+i =^' we have a^ -{- {I — v) ay^ -\- vx — v = ] and supposing x, y, and 2, to be its roots, we shall have x + y-\-z = v — l, xy i- xz + yz = V, and xyz = v ; 680 DIFFERENTIAL EQUATIONS consequently, eliminating v from these equations, we shall have x}/ + X3 + y2 — {x + y + z) = l and xi/ -\- xz -i- yz =z xySj which clearly correspond to two of the algebraic equations. To get the other algebraic equation, we reject the factor 3 + 1 1 ar» -f aj2 y" + y^ 2' -h s^ )/v' which is common to all the terms of the proposed equation, and thence get the differential equation (ar" + a;) c?a? + (jr* + y) dy + {z^ + z) dz = 0; and by taking the integral of this, we have sc^ x^ 1/ 'i? z^ z^ , , or 2 (ar^ + f -\- z') + 3 (a.- + 2/' + ^\= C, which is clearly an algebraic equation, as required. For the last example of this method of finding algebraic integrals, we will take dx Vi -\- x^ ^- dy ^1"+ y^ — 0. By putting a?' = x' and y' = y\ we shall change the equation to 1 ^ / 4 A+'^ 1 ^ , ./I + y"" Putting 1 4- a;'^ 1 + t/'^ y— = ^ — ^ ^e get a?'2 — 'ya?' + 1 = 0, X y whose roots being x' and y\ we have x'+y' = V, or ar^+x^^ = v, and ai'i/' = 1, or xY = 1. Rejecting the factor - from the proposed equation, we get I CONTAIN-INa THREE VARIABLES. 681 dx' + dy' — 0, whose integral gives a?^ + 3/' = a;^ + ?/^ == C ; and thence y?]!^ =1, a?^ + 2/^ = C, are algebraic integrals of the proposed equation. For fuller information on algebraic integrals, see pages 383-404 of Professor Gill's "Mathematical Miscellany," published at Flushing, L. I., during 1836, 1837, &c. ; and for other methods of finding algebraic integrals, together with their applications to elliptic functions, see the " Exer- cices de Calcul Integral," of Legendre, and p. 471, &c., of Lacroix. (4.) Eesuming dz = -^dx -^ -^dy = ^pdx + qdy, in which 2 is a function of x and y, considered as being in- dependent variables, so that dx and dy (see p. 34) must be constant in differentiating the equation ; consequently, by taking the differential of the equation, we shall have d^z =z —-- dx^ + 2 -^ — J- dxdij + -^-^ dip- : dx^ dxdy ^ dy^ -^ ' d^z d?z d''z GlZ or, representing ^ , -^ = ^-- , ;^„ by r, ., and t, it becomes d^z = rdx^ + zsdxdy -\- tdy'^] and from — = p and -^ = q , , , dz d^z -J d}z -, we also have a -y- = -7-5 aa? + -j—r ctV dx dx^ dxdy . , dz d^z -, d'^z y and d-^ = , , dx -\- -r-^ dy, dy dydx dy^ ^' or their equivalents dp = rdx + sdy and dq — sdx + tdy. 582 EQUATIONS OF THREE VARIABLES. If we differentiate the equation {x- a)' + (y - hy + (2- cf = R^ successively, according to the preceding principles, we shall get (x — a) dx -\- iy — b) dy -\- {z — c) dz = 0^ day" + dy^ -{- dz^ + (s — c) d'z = 0, and Mzd^z -\- {z — c) d^z = 0, for its first, second, and third differentials; and so on, to any required extent * It is evident that the preceding forms will be very useful in finding, by reverse processes, the second, third, &;c., inte- grals of differential equations between ai, y, and 2, when z is a function of x and y regarded as being independent variables. SECTION X. PARTIAL DIFFERENTIAL EQUATION'S. (1.) Integration of partial differential equations of tlie first order between a?, y, and 2, z being considered as being a function of x and y, regarded as being independent variables. A partial differential equation between a?, y, and 2, is said uz nz to be of the first order, wlien it involves -7- or -^ , or both of these differential coefficients, together with constants and one or more of the variables, according to the nature of the case. It is hence clear that a partial differential coefficient can not exist between only two variables, as x and y ; since if one of them, as y, is a function of the other, the coefficient -~ must evidently be complete or total^ and not partial or incomplete. (2.) The simplest partial differential coefficient that can exist between ^, and a?, y, must evidently be of the form — =z (2, obtained by regarding y as constant in the differen- CLX tiation; consequently, reversing the process, in the integra- tion, we multiply by c?a?, and thence get dz = adx, whose integral gives z = ax -\- y. fiXA.MPLES. dz 1. The integi*al of -^ = a^ + yx + a is required. Multiplying by dx and integrating, since y and a; are in- dependent variables, clearly gives s = -g + y ^ 4- «aj + «;^y. 2. To integrate dz _ 2x J ^^ _ y The answers are s = log (y" + aP) -\- (fyy and s = '/a?^ + y^ + «^a?. 3. To integrate dz 1 , ^s 1 and -^ c2aj |/(y3 - aj2) dy ^{x^ + y^)* The answers are X z = sin-^ - + y and 2 = log [|/(ar' + y^) + y] + ». 4. The integrals of ^ =/(^j 2/) ^^d Z" ~ ^ ^ ~ ^ function of a? and y are required. PARTIAL DIFFERENTIAL EQUATIONS. 585 The answers are z = J f{x, y)dx -\- m the value of dz^ it will be reduced to \ dz , , , 1 dz , dz = -.TTz. -J- du. or to dz = j- do. IN dx m dx dz Hence, since -7- is arbitrary, we may clearly suppose it to be so taken that c?3 = 7^ -7- du ZN dx may be exactly integrable, and of course ^ 1 dz 7;r7 -J- = Fw = a function of w, IN dx ' and thence 2 = ^i^ ; and, in like manner, from 1 dz dz ^= — -r- dv m dx we shall get z = i})V, a function of v, which must clearly be the same as the preceding function. Thus, to integrate . '^(|)-2'(S) = ^' by compaiing it to the general formula we get M = — y and N = a?, and thence I (Ndx — Mdy) = da becomes I {xdx + yd/j/) = du^ which gives 1 = 2 and u = x^ + y^] consequently, z =

{x^ + y^) becomes the general equa- tion of surfaces of revolution, when the axis of revolution coincides with the axis of 2. For another example, we will take the equation dx ^ dy~~ ' Comparing this to the general equation, we get M = a? and N" = y, ' which reduce I (Ndx — M.dy) — du to I {^dx — xdy) = du ; and putting ^ = — ^ , the integral becomes /ydx — xdy _ ^ _ X and thence z z=: 6 ^: also 771 /'Ndx — Mdy\ , /ydx — xdy\ I — ■ — ^;^^ -I becomes m \^ -\ = dv. which, by putting m = - , gives - = '?^, i/ J and thence s = (/> -, the same as before. (See Young's "Dif- ferential Calculus," p. 199, &c.) (4.) We will now show how to integrate equations of the ^- _^©^<^(|)-^ = «. : on the supposition that the variables P, Q, E, are functions 588 PARTIAL DIFFERENTIAL EQUATIONS. of a?, y, z. Dividing the equation by one of the variables, as by P, and representing the quotients p and p by M and N, it becomes ^+M^ + N = 0, or its equivalent ^ -|- M/^ + N = ; and from dz = -^ dx -\- -^ dy we also have dz = pdx + qdy ; consequently, eliminating^, we shall get dz + '^dx = q {dy — Mc&»), in which $-, being clearly arbitrary, we must put dz + Nc(?aj = and dy — M.dx = 0. If M does not contain z, the equation dy — M.dx = admits of a factor m, which makes m (<^y — Mdx) = an exact differential, whose integral gives F (a?, y) = C = constant. Hence, if N does not contain s, by eliminating y from F (aj, y) = C, we shall get y in a form that may be ex- pressed by y —f {x, c), which, substituted in dz 4- 'Ndx = 0, will give an integral of the form z = — I Ydx, V being a function of x and c ; consequently, the indicated integral can be found, whose constant ought clearly, for generality, to be an arbitrary function of the constant C. Thus, to find the integral of dz dz ,/ « o^ by comparing it with the proposed form, we have U = ^ and N=-ai'-^±iQ. X X . , PARTIAL DIFFERENTIAL EQUATION'S. 589 Hence the equations ds + l^dx = and dy — Mdx = will become dz — adx — ^ ^ = and dy— - dx = or ^^ — ^— = : X X X and it is clear that the factor - reduces the second of these X equations to — ^ ^ ^ = d- = 0, X . X whose integral gives - = C or y = Gx. Consequently, from the value of y in the first of these equations, we shall get dz=:adx^il + 0"), , whose integral may clearly be expressed by z=.ax x/{l + C-) + 0C, «^C being an arbitrary function of C. Because C = -, from the substitution of this value, we thence have z = a Vx^ + y^ + - , X for the equation between a?, y, and z. Eesuming the equa- tion dz + ^dx = q {dy — Mdx), on the supposition that the first member does not contain y, and that dy — Mdx does not contain s, then, if we have the factors 7n' and m, which we may have, such that m^ {dz + l^dx) = du and m {dy — Mdx) = dV shall be exact differentials, they will give 590 PARTIAL DIFF.ERENTTAL EQUATIONS. dz + '^dx = —7 and dy — Mob = — , whicli will reduce the preceding equation to du =^ Q — dV\ ^ m consequently, since the first member of this equation is an exact differential, the second member must also be an exact differential, which it may be (on account of the arbitrariness rn! of g) by putting q — equal to a function of Y, and thence, 'ffh by taking the integral, we shall have u = (pY, or u must be an arbitrary function of Y. Thus, if we take the equation dz X dz ^ _ n dx y dy a? ~~ ' we shall have M = - and N = , y » and thence the equation dz + l^dx = gi {dy — Mc?ic) will become , s , / , a? 7 \ xdz — zdx lydy — xdx\ dz--dx = q[dy--dx) or = q\^--^ ), and thence 7n' = - and m = 2y give X J -^ = - = u and f^ydy — 2xdx = 7/"- — x" ; consequently, we shall have — =z (^^ ^ or) for the sought X integral. It may be added, that if we eliminate q from the equations jp + Mq + N.= and dz =pdx -f qdy, PARTIAL DIFFERENTIAL EQUATIONS. 591 we shall have 'Kdz -f No?y =j9 (McZa? — dy) ; or, since jp is arbitrary, as before, we sball get the equations ^idz + '^dy = and dy — Mdx = 0, and it is clear that we may proceed with these equations in much the same way as before. Where it may be noticed that we may use the first of these equations Kdz + 'Ndy = instead of dz + l^dx = (the first of those before found), since the second equations are identical. If we take the equation dz X dz xy ^ . . ,^ x . '^ xy -y - + -^z=0, It gives M = and N = -^, ax a dy az ^ a az' and these reduce the above equations to 2ydy — ^zdz = and 'iiady + 2xdx = 0, whose integrals are y^ — z^ and 2ay + x^ ; and thence from ti, =zz (pY, by putting y^ — z^ for u^ and 2ay + x^ for V, we shall have y^ — z^ = (p {2ay + x^). (See p. 50 of vol. ii. of "Wright's "Commentary on Newton's Principia.") (5.) We will here venture some remarks on the integra- tion of the partial differential equations of the second order between x, y, and z, when z is considered as a function of x and y. 1. A partial differential of the second order must involve one of the coefficients d'z d^z d^z drz dx^'' dy'^'' dxdy dydx'' at the least; and may contain other terms like those that are contained in partial differential equations of the first order. 692 PARTIAL DIFFERENTIAL EQUATIONS. 2. The method of integrating equations of this order is, in some respects, quite analogous to that of integrating partial differential equations of the first order. We will now pro- ceed to integrate some of the simpler forms of equations of the second order. 8. To integrate the forms ^- — — — d ^^ — ^^ _ cb^ ~~ ' dy- ~ ' dxdy ~~ dydx ~ The first of these equations, multiplied by c?a?, gives -— = 0, whose inteojral is ^- = y ; dx ^ ° dx ^ ^ which, multiplied by dx^ gives dz = <^ydxj whose integral is 2 = y + ypy ; in which y for the integral of -T-3 = P, and z = I I I Fdx + 0y j dx + tfyy is thence the integral of ds, and we have z z=z I i(fix -\- I Vdy\ dy + -^a? d^z for the second integral, resulting from -y-g = P ; and in like manner we have f\4>y + y P^^) % + ^-^ for the integral resulting from , , = P ; noticing, that the equations ^_ d^'z _ d^z _ -p o (P, Q, R, &c., being functions of x and y), may be treated in much the same way. 4. -^ + P -7^ =: Q, in which P and Q are functions of X and y, can also be easily integrated. dz For by putting —- =z u, the equation becomes cix -T- -}- Pw = Q, or <^?^ + Vitd^x = QdXj a linear equation, whose integral is expressed by ic = 6-/p^^ IfQef^^^'^dx + y] dy -\- \px. 5. It is manifest, from the elimination of f{ax + hy) from X =/{ax H- hy\ at p. 26, which gives the equation a^ - h—- dy dx ~ ^ PARTIAL DIFFERENTIAL EQUATIONS. 695 an equation of partial differential coefficients, tliat equations of partial differential coefficients of the first order must re- sult from the elimination of arbitrary functions from equa- tions, in a way very analogous to that in which ordinary differential equations result from the elimination of arbitrary constants from equations. Hence, it is clear that in finding the integrals of partial differential equations, we ought analogically to add arbitrary functions to correct the integrals, instead of using arbitrary constants for that purpose ; noticing, that the forms of the arbitrary functions must, in particular cases, be determined from the nature of the question. Thus, if we take the partial differential equation adz hdz _ dx dy ' to find its integral, we may proceed as follows : — Eepresenting -y^ and -j^ , as usual, by p and ^, the pro- ws? dy posed equation becomes ap -\- l)q ^= 1\ and since s is a function of x and y, we also have ^^ ^ ^ ^^ + ^ % = I>d^ + ^dy- Hence, by eliminating^ from the equations op + bq ^= 1 and pdx + qdy = dz, we get ■ q {hdx — ady) =z dx — adz ; which, on account of the arbitrariness of ^, is equivalent to hdx — ady =■ and dx — adz = ; or eliminating dx from the first of these equations by means of the second, we have dy — hdz = and dx — adz ~ 0, 596 PARTIAL DIFFERENTIAL EQUATIONS. whose integrals will clearly be of the form y — 1)2 = A and a? — a^ = B ; consequently, since these equations must clearly be coex- istent, we must have A = {x — as), for the sought integral ; indeed, if (as at p. 26) we eliminate the arbitrary function denoted by from it, we shall get the proposed equation a -z — [- h -j- = 1 from it ; noticing, that from what is done at p. 211, y — hz = (p {x — az) is plainly the general form of the equation of cylindrical surfaces, in which the nature of the directrix is undeter- mined. If, however, the equation of the directrix on the plane x, y, is of the known form y =fxj then the nature of the function can easily be found. For by putting naught for z in y — bz =^ (t>{x — az), it becomes y = (px; consequently, since y =. fx we have {x — az) will become of the known form y — hz —f{x — az). For further illustration, we will show how to find the 1 /. dz dz ^ mtegral of —x + ^ = px + q = 0. Since dz = pdx + qdy, by substituting q = — ^J>a?, from the preceding equation, for q in it, we shall get dz = p {dx — xdy) ; PARTIAL DIFFERENTIAL EQUATIOIJTS. 597 whicli, on account of tlie arbitrariness of j?, gives dz = and xdy — dx ^= 0; or, multiplying by --, , '-1-^ = 0, By taking the integrals of tbese differentials, we have z = a and - = h : X consequently, since these integrals are coexistent, we must have a = (ph'j or, substituting the values of a and 5, we shall have 3 z=z (f) -^ which, if we please, may be written in the X form 0-^2 = - ; which belongs to what are called conoidal X surfaces, whose right directrix coincides with the axis of s, without reference to the nature of its curvilinear directrix. If the curvilinear directrix is given, together with- the position of the axis of the conoid, then, putting z=: u^ we shall, from the equations of the curve of double curvature, which represent the curvilinear directrix and the axis, find », y, and 2, in terms of u ; consequently, having found x and y in terms of s, we shall get - in a function of b, and X shall thence get 0~^3 in a known form, which will give

i/ + ip7/ for the in- tegral of -^-^ = 0, in which y and i/jy are the arbitrary functions. JI we put x = 0, the equation z = x(t>y + i/'y becomes z = t/>y, which represents the section of the curve surface by the plane of the -axes 2 and y. Since the axes of the co-ordinates are supposed to be rect- angular, it is clear that (py represents the tangent of the angle which the line of common section of the surface by a plane parallel to the axes of x and z makes with the axis of x. Hence, if a line is drawn in the plane of the section through the point where it cuts the curve 2 = Vy^ supposed to be drawn, at will, to make an angle with the axis of a?, having (py equal to its (natural) tangent, the line thus drawn will represent the common section of the plane and the sur- face whose equation is ^ = x(f)y ^- V'y, and thence we may readily perceive how the curve surface may be supposed to be described geometrically. PARTIAL DIFFERENTIAL EQUATIONS. 599 7. It may be added, in concluding this treatise, that the integral of a differential equation containing any number of variables, whether they are total or partial, may clearly be found by Maclaurin s theorem, as explained in {h') given at p. 25. 8. Sometimes the generating function of the integral thus- found can be obtained, and thence the integral will be ex- pressed in finite terms. Thus, if we have „ dz cPz a?^ d^z a? . ^ = ^ + ^* + ;pr:2 + d? rro +' '^'=- in which z is expressed in terms of x^ supposing it to be a func- tion of a? and y regarded as being independent variables, and dz d?z . A ^ x. ^\ ^ p dz dfz . z, -T- , -T-s , &c., are supposed to be the values oi z, ~r ,-r^, &c., dx^ dx^ ^^ dx dx^ when X is put equal to naught in them ; noticing, that if dz d^z x=.^ makes any of the quantities 2, — , -y-^, &c., infinite, (XX dx then, by putting x -\- a for x, we may proceed, as before, to find the expansion according to the ascending powers of x. It is hence clear that the preceding series may be regarded as an integral of a partial differential equation between x and z, in which x and y (or, indeed, any number of varia- bles) are independent variables, and ^ is a function of them, or depends on them. If the preceding series has been obtained from a partial differential coefficient of the first order, it is clear that z will represent an arbitraiy function of the variables supposed constant in the differential coefficient, and if the series has been obtained from a differential coefficient of the second dz order, it is plain that z and -^ will each be arbitrary func- cix 600 PARTIAL DIFFERENTIAL EQUATIONS. tions of the variables supposed to be constant in the differ- ential coefficient, and so on, to any extent that may be re- quired. cPz cPz 9. If we take -j-^ = ?/, and c {(p'y — V' V) for V^y, it will be reduced to = {y + ex) + i) {y — cx\ wliicb expresses the integral of the proposed partial differ- ential equation of the second order, which is the well-known formula for vibrating chords. (See Lacroix, voL iL, p. 639, and Monge, " Application de 1' Analyse a la Geometric," p. 415.) To find the total integral of z we may put s, Z, — , — • , &c., CLX OjX ^'"^' (^)' (''* P- ^'' noticing, that we may put x + a, y -\- b, kc, for x, y, &c., and that in this way the integral with reference to all the independent variables, or any number of them, can be found at will 2« APPENDIX. To complete tlie work, we add the following important articles : — I. To what are oilen called the singular points of curves, we add the following from Todhunter's " Treatise on the DitFer- ential Calculus." (See p. 325, &c., of that work.) 1. Points cT arret are those points of a curve at which a single l)ranch of it suddenly stops. Thus, y = a? log X = log x -. — X dx dx ^ 1 gives y:x:_. _^ __ = l^_::^.^; which shows that y = when x = 0, or the curve stops when a? = 0, which is hence a point d' arret/ but if a? is negative, then, since log x is impossible, it follows that y must be impossible. For the first part of what has been done, see p. 67. 2. A poiiit saillant is a point at which two branches of a curve meet and stop, without having a common tangent. Thus, let y = j , which gives 1 + ^ dl _ 1 e^ dx •' / i\-* 1 -^ e" x\l + e^j in which e denotes the base of hyperbolic logarithms. APPENDIX 603 If X is unlimitedlj small, tlien y in the proposed equation is unlimitedlj small also, for two reasons : first, on account of tlie smallness of x ; and second, on account of the unlim- 1 . . '- itedly great value of - in the denominator 1 -{- e""; and it is clear that the curve touches the axis of x at the origin of the co-ordinates, where y-= 0. Again, if x is negative, it is easy to see that x unlimitedlj small gives y unlimitedlj small at the origin of the co-ordinates, or where a? = ; and it is also clear that when a? = 0, we shall reduce — to ax |=.^_^ + &e.= l +&, = !, 1 4- € ^- I + ex so that -^ is the tangent of an arc of 45° ; and of course the second branch of the curve lies on the negative side of the axis of cc, and tnakes an angle with x negative of 45°, or half a right angle, and intersects the preceding branch of the curve at the origin of the co-ordinates, making an angle of 135° with it 3. If a curve has an infinite number of conjugate points, that series of points is called a branehe jpoiiitill&e. Thus, if y^ =z X sin^ a?, or y — sin x ^x, then, if a? = - in an j in- tegral multiple of tt, we shall, for all such points, have y — 0, as required. Bemarks. — Since these points do not have place in al- gebraic curves, jet, since thej maj sometimes occur in tran- scendental curves, we have deemed it right to give an account of them. II. We here propose to investigate the path that ought to be described bj a boat in crossing a river of given breadth. 604 APPENDIX. fix>m a given point on one side to a given point on the other, so as to make the passage in the least time possible ; sup- posing the simple velocity of the boat by the propelling power to be given, and that the velocity of the current, being in the same direction with the parallel sides of the river, is variable, and expressed by any given function of the perpen- dicular distance from that side of the river fi-ora which the boat sets out. It is manifest that the boat, by the propelling power alone, ■will describe a certain line, either straight or curved, passing from her point of departure to the other side of the river, which is such that the current will float her dbwn the river into another curve, which is formed by the composition of the velocity of the boat in the direction of the first curve and of the velocity of the current, and that the curve thus described, from the point of departure to the point of arrival, will be described in the same time that the pi'opelling power alone would cause her to describe the first curve mentioned, which time, by the question, is to be a minimum. Let then y, y', denote corresponding ordinates of the two curves (y belonging to the first curve), having x fdr their common abscissa, the origin of the co-ordinates being at the point of departure, the perpendicular width of the river being the line of the abscissas, and its side the line of the ordinates. Let V denote the given velocity of the propelling powei-, and t the time elapsed from the instant of departure ; also, let correspond to the first and last points of the curve, which being given, 6y'" and 6y'\ their multipliers, must be equal to naught ; and of course the preceding integral is reduced to - hy'd L, ^ \ ( ^'^-^ - x\ = 0, which clearly can not be satisfied so as to leave Sy^ arbitrary, except by putting its factor equal to naught, which gives whose integral gives by using ^ for the arbitraiy constant. This equation is clearly equivalent to the form cvy- cy VYy + Y'-x'^ = (V'-x'') vYy + y'- x'\ or we shall have CV^ = (Y^ + Ox' - x'^ VY-y + V^ - x'\ or CVy = (V^ -i- C^' - x'J (Vy + Y^ - x'% which gives whicli gives APPENDIX. 607 ^ ~ dx ~ (Y^j^QY + Ox'-x'^^fiGY-Y^-Cx'^-x'^f ' or we have y(¥_ (Y^ + C^^ - x") VY' - x'-' ^ dx ~ [-Q^ys _ (y2 ^ c^' _ aJ'2^^2■]i' whose integral will give the curve described bj the pro- pelling power and the action of the water upon the boat during its motion. di/ To make -j- in the preceding question real, the expression (XX in its denominator positive, so that the square root can be taken, and thence give a real result. If we omit the terms in the same expression that contain ^y ^.. ^y' X ' and its powers, and put -j- for -~-^ we shall, by a simple reduction, s^et -4- = , for the line described by ' ^ dx ^Q^ _ 1 ' -^ propelling power alone, from which the current may be sup posed to float the boat down into the actual curve described, has the preceding for its differential equation. Because Y and C are invariable, it is clear that the integral of the pre Yx ceding equation is y = —- — — , which needs no correction, supposing the origin of the co-ordinates to be at the place of departure of the boat 608 APPENDIX. Hence, the propelling power alone causes the boat to describe a right line passing through the given place of departure. To get C, we must obtain the integral of the preceding equation, in which, bj putting for y' its value at the given point of arrival, noticing that the correction may clearly be supposed to equal naught, we shall have an equation whose only unknown quantity will be C, which solved gives C ; and thence, by taking those values that are not less than 1, the first and last points of the right line, described by the propelling power alone become known, and thence the direction or directions, according as C has one or more values, will be found, and the problem solved, as required. Remarks. — The question here solved was proposed in No. 2 of the "Mathematical Diary," in the year 1825, by its much accomplished editor and profound mathematician, Robert Adrain, LL..D., then professor of mathematics in Columbia College, New York. I communicated a solution to the question in No. 3 of the same work, which received the prize awarded the solution by the editor. Since there were many mistakes in the published solution, I have con- cluded, at the earnest solicitation of a former pupil and a much accomplished scientific gentleman, to insert the correct solution of the question in tliis work. III. To illustrate what has been done, suppose the body A moves uniformly around the circumference of the circle LKI with a velocity represented by 1, or unity, while the body B, in pursuit of A, moves continually directly toward A in the curve BB'B'' witb the uniform velocity ?/* ; then APPENDIX. 609 it is proposed to show bow to find the nature of the curve described by B, or of the curve of pursuit. Let A A' and BB' be very small parts of the curves described by A and B in the same time, and they will clearly have the ratio 1 : m.. Let O be the center of the circle connected with the extremities of the arc A A' by 5:0 the radii AO, A'O, at whose extremities the tangents Aa and A! a" are drawn, crossing each other in C ; then it is evident we shall have the angle aC'A', made by the tangents, equal to the angle A'OA, subtended by the arc at the center of the circle. Now the angle. aAB — (p =z Aa'h + A'JA, AB and A'B' being the corresponding tangents of the curve of pursuit which intersect in ^, and thence we have A'hA = aAB — Aa'b — (p — Aa'h = cp - (CA'5 - AVa') = - d

dx, and ultimately eb = Ai, or eWB = A^B ; and tbence ~ d£ =z B'B — A' + B0^ -f &c.]^ z = AV<^,^ + 4ABvVZ/) + 3 (2 AC + B-) cpMcp +, — mB(l>\I

— , &c. APPENDIX. 611 Hence, from substitution and omitting the factor c?0, the equation tdt — r [d (sin 0, t) — vitdcp] becomes the identical equation A^*/) + 4AB^ + 3 (2 AC + B^) cf>^ + &c. = (2A — mA) r^ + (— •- A -f 4:B — 7nB\ r(p^ + (^ - B -f 60 - mC) a^7n) _ {2~m) (4-3??i) (52 - 9m ) - 80 (2-?/?y ■~ 60^(4 -3/«)- ~' which gives _ (2 - m) (4 - 3m) (52 - 9m) - 80 (2 - mf 60 (4 - 3my (6 - 5m) ^' and so on. Hence, 612 APPENDIX. (2 - m) 4 - 3m) (52 - 9m) - 80 (2 - ^nf . « , „ ] ^ 60^- Smf (6 - 5m) ^ + ^""'j "^ is the integral, Q' being the constant ; and if T = C is the value of t at the origin, when = ', we shall clearly have ,=.T4-((2-m)(^-,0-i (,^3^^(0-00^^ (2 _ ,n) (4 - 3m).(52 - 9m) - 80 (2-m.)^ _ | 60 (4 - ^6vif (6 - 5m) (^ ^ ) + &c. j / for the correct integral. To find a?, we take the equation dx =^ ; then, ^ m — cos from the value of t^ we get the form — dt = — [AcZ^ + 3B ((^ - ^Jd^ 4- 5C ((^ - (i>yd(l> H- &c.] ; consequently, since by putting m — 1 = m', we get whose integral gives (m' + 6) A - 36m'B + 120m"O ,^ , ^,-. , , 1 IJO^^^J — (* + ^ ) + ^'-l '•' which needs no correction, supposing it commences with APPENDIX. 613 (p = (p\ or to equal nauglit at tlie origin of the motion. If in this value of x we put as in tj we shall get the required value of x. By taking (f) — (p' sufficiently small, we can, from the for- mulas found, find the corresponding values of t and x ; and then, changing ^ into \ and putting

. I UNIVERSITY OF CALIFORNIA LIBRARY This book is DUE on the la' t date ^....r-ped below. ^^" Fine schedule: 25 cents on first day overdue g^v 50 cents on fourth day overdue *T*^ ■>; . ..y . A. . One dollar on seventh day overdue. OCT 25 1941 MAY 8 1946 REC'D LD JULl6 1959 FEB 7 1955 ,vT4 1954 LU R^C D LD SEP 8 1962 RkUl3 CD llAr26!359 IH SLACKS JUL ^ ^^^^ LD 2l-100m-12,>46(A20128 16)4120 mon^ 5^ W^ ' «'*^ THE UNIVERSITY OF CALIFORNIA LIBRARY