INFINITESIMAL CALCULUS 71 The series for powers of these functions arc then investigated ; and it is found, e.g., that the square may be written ( - | sin a am V if J if 4K(K-E 1 ) - i 4g 2 cos 4x
cos 6.r 1-g 8 253. This enables the second kind of elliptic integral to be evolved in a series. The form introduced being called Z is related to Legeudre s E by the equations 2Kx
u, < = am u , Z(tt)
TT and the expansion is found 2Ka;/2K_2Ei_/2iE IT TT TT J IT ,j sin 2,e ( q" sin
sin 6x sin 2 am dx. . (B) 254. Before proceeding to the serial development of the third kind of integrals, the theorems concerning their reduction to depend on functions containing only two variables are given. It is shown first assuming two angles <r, 5, such that F(<) + F(a) = F(0-) and / <t> Jo 2 sin a cos a Act . sin 2 < d<f> so that the third kind of elliptic integrals, which involves three elements, the modulus k, the amplitude 0, and the parameter a (the quantity /L- 2 sin 2 a is what Legcndro called n the parameter, 206) is reduced to integrals of the first and second kind, and to the new transcendent / _ir^r , each of these depending only on two -"o A(0^ elements. This new transcendent we see from the above equation, by letting F(a 2 ) = 2F(a), and so <r= 2 , 5 = 0, for <p = a, satisfies the equation /" F sin a cos a Aa . siu 2 fofo T*" 2 = >~ that is to say, for the new transcendent we may substitute the definite integral of the third kind in which the amplitude is equal to the parameter ; another statement of the redueibility to functions depending on two elements only. The above equation (C) may be transformed by the identity derived from the formula of 207, s ; nV - gin 2 ? - 4 sin a cos a Act s *" ^ cos .f . A ^ which gives, on introducing am u for <, am re for a, and consequently am ( + ) for <r and am (zi-a) for 8, and integrating, /"" / rf?t|sin 2 - sin 2 am(w - a.)} = ?_ s l n i m _ a cos am a A am a sin 2 am u 1 - Psin 2 am a . sin ? am u 255. Jacobi accordingly accepts as canonical for an integral of the third kind the form written above. He defines it by ""k sin am a . cos am a . A am a . sin 2 am u . du 1 - fc 2 sin-am a . sin-am u Again, denoting by Q(u) the expression Q(u) the integi ation of the series for Z(u) ( 253) gives f Z(ti)du 2K cfa - whence 9(*)
v 1
25C. This is the first introduction in the Fundaments, of these functions, which have been called tMa functions from the original notation adopted for them by Jacobi, and by many writers have been named after him Jacobian functions. The connexion of the integral of the third kind with these func tions follows at once from 254. In fact, calling du (). it is (a) 0(w - a) But, since is an even function in u, Q(u whence n(a, ) = aZ(u) + 4 log f Hence, subtracting, U(u, a) - n(a, u) = uZ(a) - aZ(u), which is in this notation the theorem that an integral of the third kind can always be reduced to another in which its parameter ana amplitude are interchanged, as was noticed by Legendre ( 216). The development of U(u) in a series is found by aid of the series for sin 2 am u and of the last equation in 254. It is n (2Kx 2KA 2Iw; z /2KA
" */ IT If J
( q sin 2 A sin 2x q" 2 sin 4 A sin 4x g 3 sin_6A sin Qx &g ; 257. Returning from the integrals to the elliptic functions, the expressions in infinite products (A, 252) are resamed. The occur rence of the function is apparent in each of the denominators. Introducing the definition of a function H, cos 2x + g 8 ) . . . H /2Ka and replacing by u, it is easily seen that the expressions are sn am u = V k () 0(w) Again, it is easilyseen that (u + 2K) = () and H(w + 2K) = - H(w). Also as by substituting in for u ( 235) we get L _tan am(, /fc ) whence, integrating, it follows that as also that Q(u -}- z K ) = i whence 7r(K iu) "() ; 4K ( M ) . 7r(K -2iu) 4K sin am u . and by successively replacing u by u + iK? it is hence Keen that e(u) and H(ii) have one real period 4K common with the elliptic functions, and that e*& e(u} and e4KK 7 H(w) have another imagin ary period 4iK in common with them. 25S. The expansion of the and H functions in series of cosines and sines of multiple arcs by the method of indeterminate coeffici ents determines ) = 1 - 2q cos 2x + 2g 4 cos ix - 2<? 9 cos 6x + 2q 16 cos 8a; - , 2<? sin a; - sin 5x - 1^ sin 7 and hence a new development of elliptic functions as well as of the integrals arises. The developments of the numerators of the cos am and A am may be written down from the above in the notation subsequently used by Jacobi 0x = l -2qcos2x+2q 4 cos 4x - cos 4# + 2y !l c as, for instance, in his lectures, in which, without any pre supposition from the theory of elliptic transcendents, he established the relations which these series fulfil, and from them a theorem of addition for the quotients of the series, and from this the differ ential formulae which lead immediately to the elliptic integrals. All these formulae consist of series of exponential quantities, extending in both directions to infinity, in which the ordering element in the exponent rises to the second degree. Their general form may therefore be written 2e" 2 +2&i<+c ) where v takes all positive and negative integer values. 259. The Fundamcnta Nova appeared almost at the date of the death of Abel. Of Abel s works, besides those which we have men-
