630. ON AN EXPRESSION FOR 1 ±sin(2p+1)u IN TERMS OF sine. [From the Messenger of Mathematics, vol. v. (1876), pp. 7, 8.] White sin« — a>, then we have sin u = x, cos u - V(l - x*), sin 3m = 3,-c - 4?E», cos 3« = (1 - 4a'2) V(l - ."c2), sin 5» = 5« - 2(W 4- 16a5, cos 5« = (1 - 12a2 + 1G*4) VO -&c. &c. It is hence clear, that in general 1 - sin (2p + 1) u = (1 ± .t) ((1, xfl 1 + sin (2^ + I)it = (l + x) [(I, a;)?]2, where (1, a:)'' denotes a rational and integral function of x of the order p, and (1, - x~y the same function of - x for it is only in this manner that we can have cos2 (2p +1) u =(1 - x1) {(1, x*]*]1. We, in fact, find 1 -f sin ti = 1 + x, 1 - sin 3?? = (1 +x)(l - 2xf, 1 4 sin 5it = (1 + x) (1 + 2x - 4x*) 1 - sin 7m = (1 + x) (1 - 4t - 4^ + 8a.3)2, &c. and it thus appears that the form is 1 + (-)» sin (2;> + 1) u = (1 + c) 1(1, xy*. Table of Contents CONTENTS; [An Asterisk means that the paper is not printed in full]; 630 On an expression for 1 ±sm (2p+ i)it in terms of sine ; Messenger of Mathematics, t v (1876), pp 7, 8; 631 Synopsis of the theory of equations ; Messenger of Mathematics, t v (1876), pp 39-49; G32 On Aronhold's integration-formula ; Messenger of Mathematics, t v (1876), pp 8S-90; ^633 Note on Mr Martin's paper " On t/ie integrals of some differentials " ; Messenger of Mathematics, t v (1876), p 163; 634 Theorems in trigonometry and on partitions ; Messenger of Mathematics, t v (1876), p 1G4, p 188; 635 Note on the demonstration of Clairaut's theorem; Messenger of Mathematics, t v (1876), pp 166, 167; 63G On the theory of the singular solutions of differential equations of the first order ; Messenger of Mathematics, t vi, (1877), pp 23-27; 637 On 'a differential equation in the theory of elliptic functions; Messcngor of Mathematics, t, vi (1877), p 29; 638 On a q-formula leading to an expression for Et; Messenger of Mathomatics, t vi (1877), pp 63-66; C X; Tin