PEOCEEDINGS 



OF THE 



€ amhribg^ ^ Ijibsopljk al ^mtt^ . 



February 9, 1880. 



Professor Newton, President, in the Chair. 



W. J. Lewis, M.A., Trinity College, was ballotted for and 

 duly elected a Fellow of the Society. 



The following communications were made to the Society : — 



(1) Mr J. W. L. Glaisher, M.A., Theorems in elementary/ 



trigonometry. 



§ 1, The principal theorem referred to in the title is 

 sin a sin b sin c sin c? + cos a cos b cos c cos d 

 = sin a sin. 6' sin c sin d' + cos a cos b' cos c cos d',.....{A) ; 



where 



a = ^ (— a + b + c + d), 

 6'=i( a-b + c + d), 

 c' = I ( a + b — c + d), 

 d' = ^{ a + b + c-d), 



so that 



a, b', c, d' = <T — a, a — b, cr — c, a — d, 



where cr= |(a + 6 + c + c^). 



§ 2. If we put d = 0, the theorem reduces to' 



cos a cos b cos c = sin s sin (s — a) sin (s — 6) sin (s — c) 



+ cos s cos (5 — a) cos (s — J) cos {s — c)... (a), 

 where s = ^ (a + 6 + c) ; 



Vol. III. Pt. yiii. 24 



