1880.] Mr Glaisher, On theorems in elementary trigonometry. 321 



Multiplying (4) by 2a^ = ta\ 



Sa® + Xa^lf — 12ahcd "Za^ = a similar expression (6), 



and, cubing (1), 



Xa^ + 3Sa*6^ + 62a^6^c^= a similar expression (7). 



By combining (5), (6), (7) we have 



ta' + 5ta'¥ = Xa" + bta%'\ 

 Xa' + Uta'bV = la" + 15Sa"5'V^ 

 Xa^-15ahcdXa;' = ta" - loa'b'cd'Xa'^, 



and it thus appears that the expressions 



(i) ,Xa\ 



(ii) Xa' + 2Xa%'', 



(iii) Xa^ — 12ahcd, 



(iv) Xa'+5Xa'b\ 



(v) Xa' + loXa'bV, 



(vi) Xa'^ — 15ahcdXa^, 



remain unaltered when a, h, c, d are replaced by a! , b' , c , d' . 



By combining these results it follows that the same is true 

 also of the expressions 



(vii) Xa^"" + Qabcd, 



(viii) Xa'h^ - 2,Xa%%\ 



(ix) Xa'b'' + Zabcd Xa\ 



(x) XaVc^ + alcdXa^. 



§ 4. It is not difficult to form other trigonometrical equations 

 besides [A) in which a symmetrical function of a, b, c, d is equated 

 to a similar symmetrical function of a, b', d, d'. For example 

 from the equations 



sin a sin b + sin c sin d = sin a sin 6' + sin c sin d', 

 cos a cos b -\- cos c cos d = cos a cos 5' + cos c' cos c?', 

 which are easily verified, it follows that 



sin a sin b + sin a sin c + sin a sin d 

 + sin 6 sin c + sin 6 sin c^ + sin c sin d 

 = sin a' sin &' + sin a' sin c' + sin a sin cZ' 

 + sin b' sin c' + sin b' sin cZ' + sin c sin fZ', 



24—2 



