1880.] Mr Glaisher, On theorems in elementary trigonometry. 327 



These equations give, by equating the terms involving k"^, 



- 1 + n (sin a) =M^-1 + Ii (sin a), 



- 1 — n (cos a) = il/g — 1 — n (cos a), 



- 1 - 12 sin' a = 2il/, - 1 - 12 sin' a, 

 n (sin a) — n (cos a) = — 1 + |-2 sin' a' , 



n (sin a) — ^2 sin* a = M^— 1 + 11 (cos a), 



n (sin a) + 1 = - J/„ + n (cos a) + 1 2 sin' a, 



from which we obtain the four independent formulae 



n (sin a) — n (sin a) = M^, 

 n (cos a) — li (cos a) =■ — itf„, 

 2 sin' a — 2 sin' a' = — 4J/p, 

 n (cos a) — n (sin a) = 1 — J 2 sin' a. 



The coefficient of A;' in the expansion of M, which has been de- 

 noted by i¥p, 



= sin' h{a + h) sin' ^{a — b) + sin' |(c + c^) sin' ^(c—d) 



- sin' 1 {a + h') sin' i {a - V) - sin' i (c' + c^') sin' i (c' - f^') 



= sin' l{a-h) {sin' |- (a + 6) - sin' l{c->rd)] 



, + sin' I (c - c?) {sin' -| (c + (i) - sin' -| (a + h)] 



= {sin' !(« + &)- sin' | (c + c?)} {sin' ^(a-h) - sin' i (c - c?)} 



=sin|(a+& + c+cZ)sin|(a + &-c-cZ)sin|(a-6+c-c^)sin|(a-6— c+J) 



= sin a" sin 6" sin c" sin c?" 



= n (sin a"), 



and substituting this value of M^ the formulse last written become 

 identical with {B), [C), (D), (E). 



§ 11. The six elliptic function theorems quoted at the be- 

 ginning of the last section may be directly established without 

 difficulty, by the same method as that employed in § 8 for the 

 trigonometrical theorems, by means of the equations 



sn (p + q) sn (p — q) 



_ sn p — sn g- 



1 — Fsn'psn'g- ' 



^v, / , \ / \ 1 - sn'p — sn-g + Fsn'wsn'o' 



en {p + q) en (p - q) = i^ ,, \ — ^--^^ ^ , 



1 — k &n p sn' q 



1 / , \ 1 / \ 1 — /:;'sn'» — Fsn'(7 + ^^sn'»sn'o' 



dn (p + q) dn {p - q) = f ,, ,^ , ^ ^ . 



1 — k sn p sn q 



