INFINITESIMAL CALCULUS 61 essentially as follows, has much analogy with Fagnani s, given in 190. Let the equation (ax? + 2a x + a")y- + 2(bx- + 2b x + b")y + ex- + 2c x + c" = Ly- + 2My + N = (ay- + 2by + c)x- + 2(a if + 2b y + subsist between x and y. Differentiating either, we get (P Further, by (1, 2) taking the roots positive, (3) becomes dx dy PR (1), (2), (3). (4); (5), in which the radicals are respectively functions of x and of y ex pressible from (1) and (2). If the functions under the radicals are to be severally the same functions of x and y, the following conditions result : b- - ac = a" 2 - aa", 2bb - ac - a c = 2a b - ab" - a"b, bb" = a c , 2b b" - a c" - a"c = 2b c - be" - b"c, b" 2 - a"c" = c 2 - cc". The values of c and c being substituted from the first and third equations, in the second we get a = b. Whence first and third give a" = c, b" = c ; and the others are identical. (1) thus becomes ax-if + 2bxy(x + y) + c(x 2 + y" 2 ) + 4b xy + 2c (x + y) + c = . ( 6 ). Hence (5) takes the form dx dy V A + 2 Rx + Cz 2 + 2Dx :t + Ez 4 V A + 2 By + Cy 2 + 2Vy 3 + Ey 4 or more briefly = 0, of which the obvious transcendental integral is J VI 7 V~~ where A = c 2 - cc", B = 2b c f - be" - cc , C = 46 2 - ac" - c 2 - 2bc , D = 2bb - ac -be, E = i 2 - ac. Further, from (4) VY = (ay 2 + 2by + c)x + bif + 2b y + c , VX = (ax* + 2bx + c)y + bx- + 2b x + c . Vx - VY Hencc - = axy + b(x + y) + 2b -c. x y Squaring, we get, by (6), &c., the algebraic integral ( ~~^r~) = E( * + 2/) 2 + 2D(x + y) + (2i/ - c ) 2 - ac " The constant on the right, involving an arbitrary quantity, if A, B, C, D, E are known, may be taken as the constant of integra tion. 198. In Euler s first paper in vol. vii. of the St Petersburg Commen taries, he uses the "aequatio canonica" = a + y(x^ + y^) + 2$xy + ex 2 if as a starting point for establishing his theorem of addition, and, introducing the notation U(x) = I -7
establishes the equation dx , EC where y VA(A + Ca 2 + Ex*) ] Now, when B = 0, C = 0, these equations represent the theorem of addition for Legendre s first kind of elliptic integrals, and when A= l> fi = -(1 + K 2 ), E = 2 , A = l, B = - 2 , C = 0, they become the theorem of addition for his seoond kind. Thus it appears that already in 1761 Euler was acquainted with this fundamental theorem, of which he gave many applications to the comparison of elliptic arcs. 199. But it seems to have been Euler s paper " Do reductioneformu- larum integralium ad rectificationem ellipsis ac hyperbola " (Nor i Comment., x. p. 3-50, St Petersburg, 1766), which impelled Legendre to his investigations. With special cases of the general relation, g 197 (1), between x and y, Euler transforms integrals contained in the form / / / (W fa ^ am j distinguishes whether the integral has one ^ V fC -f- fix of the five significations, arc of ellipse, arc of hyperbola, or either or both of these along with an algebraic part, collecting in one general investigation the results of Mackurin and D Alembert on the recti fication of conies. Here we find the words, regarding the desir ability of a suitable notation by which elliptic arcs may be as conveniently expressed in calculation as logarithms and circular arcs are at present, "such signs," he says, "will afford a new sort of calculus, of which I have here attempted the exposition of the first elements," which Legendre cites in the preface to his great work in 1825 as having remained unfulfilled but for his own labours continued till that date from his first publications on the subicct in 1786. Euler himself admitted when he stated that he had not obtained this result by a regular method, but " potius tentando, vel divin- ando," and recommended mathematicians to seek a direct method. Lagrange here lays down the principle that, when the integral of a differential equation of the first degree cannot be found, the equation should be differentiated ; and, combining the result with the given equation, an integral equation of the first degree different from the proposed may be found. Then by means of these two the first differentials may be eliminated, and the result is the required inte gral. If this fail we may differentiate once more, and try to get a new equation of the second order, and so on. This enabled him to give a deduction of Euler s equation, which Euler received with the greatest admiration, and gives nearly as follows in his Institut. Calc. Int., iv. p. 466. Writing for brevity (2). suppose the differential equation between x and y to be dx_ { dy ^ Q Regard x and y as functions of a variable t, and replace (2) by the following dt -VY Assuming we get = Vx - which last is, by (1), ^ = B + C(a; + 2 /) + -|D( a; 2 + 2 / 2 We also get, from (5), -^- -^- = X-^ dt dt (3). (4), i = > (5)> ^ + y 3 ). - (6). = (x - y) which, combined with (6), gives, by (4), ie ( d ^ q dtq dt Multiplying by c -, this gives on integration, with F a constant, (L ( t* q dt whence, replacing values, /Vx - VY which is the same result as Euler s ( 197). A principal advantage of this method consists in its admitting of generalization, which Euler s method, depending on the solution of a quadratic equation, excludes. But Lagrange fails to apply it to the case of X and Y being arbitrary polynomials respectively in x and y; all assumptions lead back to the forms of X and Y in (1). 201. In a paper of Euler s in the St Petersburg Transactions for 1771, an angle is introduced as the variable into the integral for the arc of an ellipse. In another paper, in the Koii Commcntarii, 1767, he had also remarked that, in a differential such as dx VA + Ex + Ca; a + Da; 3 + Ez 4 by a substitution of the form x = = - . the odd powers of z under nz + b the radical can be abolished ; and by a like substitution, removing the odd powers of y, Euler treats, without any loss of generality, the differential equation in the form dx dy " VA + fy- -i-TV
