INFINITESIMAL CALCULUS 63 ding of a mighty branch of analysis, and by the toil of half a life to have erected on these bases an independent theory which embraces nil integrals containing no other irrationality but a square root under which the variable rises only to the fourth degree. Eulerhad already noticed with what modifications his theorem can be extended to such integrals ; Legendre, starting from the happy thought of reducing all these integrals to fixed canonical forms, attained the knowledge, so important for the development of the theory, that they group into three essentially different kinds. Submitting then each kind to a careful investigation, he discovered many of their most important properties, of which chiefly those which belong to the third kind were very obscure and inaccessible. Only for the most persistent tenacity, which ever anew led the great-mathema tician to his subject, did the victory at last declare itself over diffi culties apparently insurmountable by the weapons at his disposal." 206. Having shown that the integral / ? where P is a J It rational function of y, and K = ( + fix + yx" + Sx 3 + ex 4 )* , can bo reduced to the fixed fundamental forms f~ , f-^ , f*Wx J K J K, J II I - , Legendre removes, by aid of the linear transforma- tion 3J = -t i? } the odd powers of the variable from the poly nomial R 2 , and shows, by enumeration of cases, that can always be reduced to the form md<f> m __i. , where c is a quantity less Vl - c 2 sin 2 than unity. Thus he reduces the general elliptic integral /, with abstraction from an algebraic part, to the* VI -c 2 sin"0 three normal forms of " elliptic functions or transcendents " = n, A being an abbreviation for the radical Vl - c 2 sin 2 0. With this reduction to fixed normal forms the foundation of the theory of elliptic integrals is laid, and the essentially irreducible integrals found which belong to a square root of a biquadratic func tion. The same reduction subsequently led to the division of the general Abelian integrals into those of the first, second, and third kinds, in accordance with the properties of these three classes of integrals, either of remaining always finite, or of becoming infinite, algebraically only at infinity, or logarithmically at two different points. It will be perceived that the epithet "elliptic" applied to these integrals is purely conventional, arising from the connexion of one of them with the arc of an ellipse ; but even at this stage it is apparent that we are concerned with matters of much greater gene rality than the name indicates. It may also be noticed that, though Legendre calls by the name elliptic functions what are now called elliptic integrals, this is a change introduced by Jacobi, which Legendre long resisted. The change consists in regarding the superior limit of the integral of the first kind as a function of the integral, the latter being now considered as the independent vari able. Expressed in symbols the change is that, in Legendre s equa tion F(K, 0) = Jacobi calls 6, K), and sin cos 0, or A0, &c. (or, ia this notation, sin am, cos am u, Aamw, &c.), are his elliptic functions. 207. Legendre proceeds, after classifying the integrals, to the com parison of his elliptic functions of the first kind. All geometers, ho says, are acquainted with the complete algebraic integral given by Euler of the differential equation equation can, without loss of generality, bo put under the form Vl-c 2 sin a Vl-c 2 sin 2 ^ and then its integral is fj. bt-ing an arbitrary constant. But the integral found by Euler s method is thus written cos cos ty - sin sin <J/Vl - c- siii 2 /* = cos /* , which he then verifies a posteriori. The expressions sin /i = lu v c 1 - c 2 sin-0 sin 2 ^ cos cos >|/ - sin sin |/ A0 A|/ 1 - c 2 sin 2 sin 2 i|/ A0A>// - c 2 sin sin i/ cos cos i|/ 1 - c 2 sin a sin-vj/ . _ are at once derived from this form of the integral ; and the corre sponding formula 1 for the amplitude of the difference between two functions follow by replacing $ by - ^. Legendrc next proceeds to the formulae for finding algebraically a multiple function of a given one, connecting the angles <f> n -i, <t>n , <?Wi , by the relations equivalent to F(0 )!+1 ) = F(0 n ) + F(0) , F(0,,-i) = F(0,,) - F(0) , which he writes 2A cos sin 0,, - -~- These can be applied in succession. Investigating the division of a function into n equal parts the equation is found to rise in general to the degree ?i 2 ; but for the complete function the equation is only of the degree $(?t 2 - 1) when n is odd. 208. Proceeding to generalization of Eulcr s addition theorem, Legendre admits that, denoting the radical by E(x), &c., the equa- , mdx ndy pdz tion = R , ,+ jT^: + yrrT+ &c., can lor integer values of in, n, ^ always be expressed in the form F(ju) = ??iF(0)-HiF(^) + &c., and so will always have a complete algebraic integral, for nothing pre vents the supposition that z and the following variables are given algebraic functions of x and y. Perhaps, he says, this is the only way of generalizing Euler s result concerning the equation ~~ ~ = 0- For, though Lagrange tried to find cases of integra- dx dij bility ot 7= + ^ , without supposing the two polynomials X and Y entirely similar,- it does not seem that he arrived at any any other result; the equation he gives (Mem. de Turin, iv. 119) is immediately reducible to Euler s. Thus, as has been remarked, Legendre was at this time very far from anticipating the very general transformations, since discovered, or the celebrated theorem of Abel which so marvellously extended this subject. 2U9. Having illustrated the functions F by the lemniscate and other curves, algebraic and transcendent, whose arcs are expressed by functions of the first kind, as well as by the expression for the time in the motion of a simple pendulum, Legendre enters, in chapter ix., on the comparison of elliptic functions of the second kind. Corresponding to the relation F(0) + F(^) - F(^) = , these functions are related by c 2 sin ^ sin sin ^. This includes Fagnani s as a particular case, and of course there is a similar relation for comparison of the arcs of hyperbolae In chap. xii. the well-known relation of Legendre is established between complete integrals of the first two kinds with complementary moduli ; b and c being moduli are said to be complementary when 2 = l. Denoting by F, E the values of F( ,A E(
2 J
and by F , E those of F( ,$), (--,&), this relation, which
* / * /
has been already demonstrated, 183, is These complete functions satisfy differential equations of the second order, viz., F satisfies dc* c rfc and the complete E with corresponding equations when & is taken as the independent variable. The complete integrals of these differential equations are assigned in terms of both sets of complete functions, and the differential equations are utilized to show the law of the develop ment of these functions in series of powers of the complement of the modulus, since when the modulus is near unity the ordinary series in powers of the modulus do not sufficiently converge. 210. In treating integrals of the third kind, the presence of a third determining magnitude, the parameter n, besides the ampli tude and modulus c, which are common to the first two kinds, is
