62 INFINITESIMAL CALCULUS This furnishes an essential simplification of the process of calcula tion, and leads to the results of his important and remarkable " Plenior Explicatio " in the St Petersburg Transactions for 1781, This contains in fact a proposition which includes the theorem of addition for all three kinds of elliptic integrals of Legendre. 202. Putting " where Z is an even function of z, n(x) + n(y) - n(z) can be exhibited as an algebraic function of x, y, and z, provided a certain algebraic relation holds between x, y, and z. "But now," observes Eulcr, "I have noticed that the same comparisons may be instituted if for Z be assumed any rational function of z-, as suppose one of the form in this cas", however, the difference between the sum of two such formulae and a third is no longer found to be an algebraic quantity, but can always be expressed by logarithms and circular arcs, so that the investigation is much more extensive than I hitherto conceived." To establish this, assume 2xy Vl + mz 2 + HZ* - nx^ij-z- =-- . z* = or, writing x 2 + y 2 z 2 + 2.T1/A - Solving in turn for x and y, this gives x( 1 - ny 2 z 2 ) + yA = z/l + my- + ny 4 j/(l nx 2 z 2 ) + xA = z/ 1 + mx 2 + nx* But, by differentiation, we know that dx dy . (1), (2), . (3). . (4). (5), (6). y( - nx 2 z z ) + xA x(l ny 2 z-) + yA For x = 0, the relation (3) gives y = z ; whence (5) integrated gives the known theorem of addition for integrals of the first kind. Now let X,Y be the same functions of x 2 , y- as Z is of z 2 ; then Xdx Ydy writing 1 J = (iV. .... (7), V 1 + rax- + nx 4 V 1 + my 2 + ny 4 the quantity V may be found as follows. We can eliminate by (5) either dx or dy irom (7) ; but, as there is no reason to consider V as a function of x or of y specially, intro duce a new independent variable u = xy. Then we may replace (6) by dx = [y("L nx^z~) + xAsdu, dy= [x(l -ny 2 z 2 ) + yA]sdu . . (8), 1 V 1 + mx* + nx* Vl + my 2 + ny 4 which may be written thus dy where we take Thus d dy zdu Vl + my* + ny* which give dV = V X But now is a function of xy and a? 2 + y 2 , or of u, since by (3) y 2 - x* x" + y 2 = s 2 - 2 u A + nu 2 z 2 . Thus, putting z - o -~ = U, wehavedV= - Udu. Substituting this y- x- in (7), and integrating, we have since, for x = 0, y z and u ~ I As an example, -/: Udu, a b - ab that is, Thus, if we take 1 - x 2 a 2 + a b (x~ + y 2 ) + b"-x 2 y (a b - ab )z " a" 2 + a b z 2 + 2a r b Au + (b" 2 + a b nz -)u 2 we have (ab - a b)zdu = / y a 2 + a b z- + Za b Au + (b 2 + a bnz-)u- in which after integration u is to be put =xy. By assigning special values to the constants this equation can be applied to each kind of elliptic integral. 203. In the T/ieorlc dcs Functions Analytiqucs of Lagrange there is found the remark on the relation between the summation of ellip tic integrals and spherical triangles which is involved in the formula; of 200, and of which we have given the essential formula) in 34-36. But Lagrangc s paper " On a new method of the Integral Calculus for Differentials ailectcd with the square root of a poly nomial of not more than the fourth degree" (Memoircde I Acad. dcs sciences, 1784-5, 2d part, Turin, 1786), contains additions of much greater importance to our subject. Taking Further, writing for brevity, K = " ir = V(iyv 8 )(ig / y2-j > and , 7/R V " 1 q*y* we have the equations y"E" 1 q" 2 y"~ and 2p" 2 dy dy ^dy" E, 11 11" Now, assuming q<2), we have P<2> <1>" , and, as q = q. -,-, i > q", &c. , without limit. Lagrange gives another system of equations of reduction following from the above series when continued backwards, so that the series The equations p = Pl = give conversely and the terms p, Pi,P-2 decrease, while q, q lt g . . . increase. Now, putting V). The former scries are essentially the transformation known by Landen s name ; the latter, indirectly, are ( 214) the transformation which Gauss published in 1818, and on which his theory of the arithmetico-geometric mean is based. 204. We have thus glanced at the most important contributions to this branch of our subject previous to Legendre. His "Memoire sur les integrations par d arcs d ellipse" (Ilistoirc dc I Acad., 1786) appeared a few j ears after Euler s death (1783). The geometric basis is here almost abandoned. Establishing with ease and ele gance the theorems of Fagnani, Landen, and Euler, there are per ceptible traces of a coming theory of transformation in the analytical conception of these theorems. Legendre s Memoire sur les Trans- ccndantes Elliptiqucs (Paris, 1793) contains the division of elliptic integrals into their different kinds, the reduction of integrals of each kind to the simplest normal forms, and the calculation of elliptic integrals by most accurate modes of approximation. All these in vestigations are collected in Legendre s Excrciccs (Paris, 1811-19), and later in his Traite dcs Fonctions clliptiqit.es ct dcs Integrates F lleriennes (Paris, 1825-6, 2 vols., suppl. vol., 1828). 205. "It is Legendre s undying glory," said Lejeune Dirichlet of this great work, "to have recognized in the discoveries we have just mentioned (of Fagnani, Euler, Landen, and Lagrange) the bud-
