we have on integration with respect to t } [B (p) u" (p) dp = [' R (t).dt, the constant of integration being zero from the initial condition. If then m is even and B(p) negative, the two members of this equation have opposite signs; hence u' = 0, so that the solution is unique. The same result follows if B(p) is positive and m odd. The integral of (34) satisfying the homogeneous boundary conditions (28) and the specified initial condition may now be obtained by the method of § 15 of the preceding paper. Assuming a solution of the form u (p) = (p) eTM, we obtain for (p) the differential equation (▲2 — k2)TM 4 (p) = − xB (p) 4 (p), which by §14 is equivalent to the homogeneous integral equation (36) (p) =λ [Gm(pq) B (q) 4 (q) dq, admitting a non-zero solution only when A is equal to one of the characteristic numbers. If B(p) is negative for an even m, and positive for an odd m, then by § 14 these singular values of λ are all positive and form an infinite series of real numbers. If a (p) satisfies the conditions specified in § 11 it may be expanded in an absolutely and uniformly convergent series of the characteristic functions (p) of the equation (36), so that and the required solution of the problem is then given by The integral of (34) satisfying the boundary conditions (29) is found as in § 16 of the paper referred to. § 17. In conclusion, the integration of the equation (37) ď u (A2 - k')" u(p) = B (p) (p) dť may be made by the method of §§ 17-18 of that paper. Assuming a solution of the form بل where (p) is a function independent of t, we have for its determination the differential equation which is of the form (30). Its solution satisfying the conditions (28) is given by the integral equation ¥ (p) = {λ3 ƒ Gm (pq) B (q) 4 (q) dq. If B(p) is negative for an even m, or positive for an odd m, the characteristic numbers of the kernel are real, positive and infinite in number; and it is only when λ' is equal to one of these that the last equation admits a non-zero solution. The investigation then proceeds exactly as in the case of m = 1 just referred to, the required integral satisfying (28) and the initial conditions u (p) = Σ 4„(p) { 4, cos√√λt + B1 sin √λt}. n=1 n ย A and B being the coefficients of (p) in the expansions of a (p) and B (p) respectively in terms of the characteristic functions. The solution satisfying (29) is found as in § 18 of the same paper. AN INVARIANT MODULAR EQUATION OF THE FIFTH ORDER. By W. E. H. BERWICK. Fn is a positive integer, the modular functions. 1. IF 'J=j (w) = and = 28 g', - 27g's l* (1 — k3)2 — qa +744 +1 96884q+214 93760g+8642 999709 +202458 56256q+ 33 32026 40600910 + 425 20233 00096q"+4465 69940 71935q1+... are known to be connected by a symmetric algebraic equation F (J, I) = 0, called the invariant equation for transformations of order n. When n is a prime number p, the degree of this equation is p+1 in J or I and 2p in J and I jointly. Its most general form is therefore where 4, 4, 4, = ra =A p+1 p = = 0. It is stated by Weber that the coefficients of the invariant equation have been evaluated only in the case when n=2. The invariant equation for cubic transformations, however, has been given by H. J. S. Smith in a paper entitled "Note on a Modular Equation for the Transformation of the third order" in ser. 1, vol. x. (1879), pp. 87-91 of the Proceedings of the London Mathematical Society and reprinted in vol. ii. (pp. 274-278) of the same writer's Collected Mathematical Papers. In this paper the invariant equation for quintic transformation is determined. The method adopted for calculating the coefficients is substantially the same as that used by Smith. * Algebra, Bd. iii. (1908), S. 245. 2. When p5 the invariant equation is F (J, I) = A (J® I* + J* 1o) + A 63 (J° 13 + J3 1o) 64 + A „, (J° 12 + J3 1o) + A。, (Jo 1 + JIo) + A ̧ (J° + Io) 69 61 60 + A „¿ (J3l2 + J'I3) + A „, (J3I+ J13) + A ̧ (J3 + 13) 52 51 50 + A„J*I*+ A „(J*13+J31*) + A„, (J*I*+ J21*) 43 42 + A ̧ (J*I + JI“) + A 10 (J* + I1) + A„J3Ï3 41 40 + A „, (J3I3 + J313) + A„, (J3I + J13) + A30 (J3 + 13) 32 31 + A „J'l3+ A„ (J'I + JI2) + A „。 (J2 + I3) A few coefficients can conveniently be found by adopting Sohncke's method of substituting for J, I their expansions as series in ascending powers of q and equating the coefficients of the different powers of q to zero. Putting J=q' + a‚ + a‚q2 + a‚qʻ + a‚q® + a‚q® +a ̧q1o +a‚q13 +a ̧q1*+..., <-10 10 I = q ̄1o+a,+a,q1o +....., in (2) and equating to zero the coefficients of we find 40, 40, 40, 40, 46+4=0. A similar process shows that if shows that if p is any odd prime the coefficients App-1) App-2 Ap+1,29 Ap+1,19 the remaining coefficients will all be rational integers, for the class-invariant (5) is a complex integer in the unlimited number of cases when j (w) is a rational or complex integer. The only remaining terms in F(J, I) which contain 9-50, 9-50, .... 2 are * 66 Aequationes modulares pro transformatione functionum ellipticarum, Crelle, xvi. 97–130. an dropping the factor 1, the coefficients of 7. 1 must all vanish. We thus find in anccession A=59, 37202.3.5.31, from 1 above. =20285 51200 2.5.13.1 95053. A1 = 50, -10% a, +159,'a, -5a,'-5a,* 51 24 66834 10950=-2.3.52.16445 56073, A-59, -5a a, — 59,0, + 59,a,' + 5a 3⁄41⁄2, — 5a,3a, +a‚3 —ba, 196 32114 89280 = 2.3′.5.31.1193, Au-5-25a,' 166 59993 61600 = 2′,5′.83299 96823, A„-50,- 20a,α, +5o ̧aà ̧ + 5a,a, — 25a,a, +50a ̧3 P 107 87892 81853 36800 = 2′.3.5.4494 95534 10557, A-5a,-15a,a,- 15a,a, +10%,a,+5a, +30a,1a-15a‚a ̧a ̧ 7 - 15a,a,a, +5aja, + 5%,a," — 50a,* — 25a ̧a ̧ + 75a‚1a, = 3 83083 60977 98112 15375 - 3.5.1021 55629 27461 63241. All the coefficients could be determined by carrying this process far enough, but the calculation now begins to be heavy, so it is simpler to use other methods for evaluating the remaining ones, 3. It has been shown by Cayley that the modular functions k', l' are connected by the equation 4 (l', 1') = 1'" 4-(-65586/+163840k-138240k+ 43520 - 3590k) + (1638404-1331204"-207360k + 133135k+43520") + 7" (-1382404" - 207360k + 691180k - 207360k — 138240k3) +7 (435204 + 1331354-207360-133120 +163840%) +7(-359044435204"-138240k + 163840k - 65536k) + X1* = 0....... (3). Elliptic Functions, p. 198. |