386 Mr Glaisher, Addition to a previous paper on [May 17, 



which are (B) and (C) of § 7. The equation (vi) is in fact (D), 

 and (A) is deducible by addition from (B) and (€), so that 

 we have thus obtained the five theorems {A), ... (E) of § 7.* 



§ 17. The formula (B), viz. 11 (sin a) = IT (sin a) + H (sin a"), 

 is in fact a particular case of a more general theorem, connecting 

 products of four sines, which may be written 



sin {a —f) sin (a - g) sin (a — K) sin (6 — c) 

 + sin (6 — y) sin {h — g) sin (h — h) sin (c — a) 

 + sin (c — y) sin (c — ^) sin (c — h) sin (a — h) 

 + sin (6 — c) sin (c — a) sin (a — h) sin (a + J + c —f—g—h) = 0...(p), 



where a, b, c,f, g, h are any six quantities. The formula {p) how- 

 ever involves only five independent quantities. 



K we put g 4-h = 0, a+h + c =/, 



so that the last term vanishes, then (p) becomes 



sin {h + c) sin (a — g) sin {a + g) sin {b — c) 

 + sin (c + a) sin (b — g) sin (& + g) sin (c — a) 

 + sin (a + b) sin (c — ^g') sin (c + ^r) sin [a — b) = 0, 



which is in fact (B) in the form {B'), (§ 8). 



§ 18. The formula (p) is in effect due to Professor Cayley and 

 Mr R. F. Scott. In the Messenger of Mathematics, vol. v., p. 164, 

 Professor Cayley gave the theorem : if 



A-^B-^ C+F+ G + H=0, 



then I sin {A + F) sin {B + i^) sin ( (7 + F), cos F, sin F = 0, 

 sin {A + G) sin {B + G) sin (C+ G), cos G, sin G 

 sin {A + H) sin {B + H) sin (C + ^), cos H, sin J/ 



and in the Messenger, vol. viii., p. 155, Mr Scott evaluated this 

 determinant, and showed that, the letters being unrestricted, it 

 was equal to 



&m(G-H)sm(H-F)sm{F-G) sin {A+B+ C+F+ G + H). 



On expanding the determinant, and replacing A, B, G, F, G, H 

 by — /, —g, —h, a, b, c respectively, we obtain the equation (p). 



* Since this paper was read to the Society some additional examples, &c. in 

 connexion with §§ 12 — 16 have been published in the Messenger of Mathematics, 

 ¥ol. X. pp. 26—34 (June and July, 1880). 



