1880-] Mr Glaisher, On theorems in elementary trigonometry. 325 



cos (a + /3) cos (a — /9) cos (7 + S) cos (7 — B) 



— sin (a + /3) sin (a — /3) sin (7 + S) sin (7 — B) 

 = l-isin^(7+;8)-isin^(7-;S)- Jsin^(a + S)-isin^(a-S) 



m- 



Putting, for brevity, 



X — sin a, 



y = sin /3, 

 z = sin 7, 

 w = sin 8, 

 and using the formulae, 



sin (p + q) sin (p — q) = sin^ j9 — sin^ g-, 

 cos (p + q) cos (jo — 2') = 1 — sin^^ — sin^ q, 

 the equations (A'), ... (^') become 



(^'^ - if) {z' - w') + {l-x''- f) {l-z'- w') 



= (z' - y') {x' - w') + {l-z'- if) (1-x'- w'), 



(x' - f) {z' - w') - {z' - f) {x' - w') + {x' - z') [f - w'), 



{l-x'-f){l-z''-w'') = {l-z'-y'){l-x'-w')-{x''-z-'){f-w'), 



2 ix^ ^f- ^^xhf -^z'^w'- 2^V - 2/' - ^' + ^y'^z'- x^- w'+ Ix^uf) 



= _ 4 (^^ - z^) {f - w\ 



(1 _ ^2 _ f) (1 _ z^-y,^) - (x'-lf) (/ - W') 



= 1 _ (2/2 _,. ^2 _ 2_j/2^2 ^ ^2 _j_ ^2 _ 2a;2ty2-)^ 



which identities are readily verified. 



§ 9. The theorems (.4),... (E) are, so far as I know, new; 

 except that (C) is in effect involved in two formulse of Lexell's, 

 given by him in his memoir De proprietatihus circuloruin in super- 

 Jicie sphwrica descriptorum {Acta Acad. Petropol. for 1782 (1786), 

 p. 88). 



These formulse are : — if ABDG be a spherical quadrilateral 

 inscribed in a small circle and if AB = a, BD = h, DC= c, CA = d, 

 then 



sini(^+Z)) 



'cos l(a + b+c+d)cosl{a-b-c+d)cosl(a—h+c-d)cosl(a+b-c—d) 

 cos ^ a cos I b cos ^ c cos ^ d ' 



