320 Mr Glaisher, On theorems in elementary trigonometry. [Feb. 9, 



this is the theorem implicitly involved in the well-known formulae 



. o T „ sin -i- s sin i (s — a) sin ^(s — b) sin ^(5 — c) 



sm" iL = z f-j -. , 



* cos ^ a cos ^ b cos ^ c 



2 1 Tr_ cos|-5Cos|(s — a) cos \{s — b) cos-^(s- c) 



cos 4 £^ — ^ 7^ -■. , 



cos fa cos ^6 cos I c 



which by division give Lhuilier's expression for the spherical 

 excess. 



It may be observed that (a) may also be written in the form 



cos {h + c) cos (c + a) cos (a + ^) = sin a sin h sin c sin (a + & 4- c) 



+ cos a cos h cos c cos (a + 5 + c). 



§ 3. By expanding in series the sines and cosines in {A) and 

 equating terms of the different orders we obtain algebraical ex- 

 pressions of the form : symmetrical function of a, h, c, d = a, similar 

 symmetrical function of a, b', c, d', viz. we thus obtain algebraical 

 expressions of different orders involving a, b, c, d symmetrically, 

 which remain unaltered when a, b, c, d arc replaced by a- — a, 

 a — b, a — c, a— d. 



Thus equating the second powers, we have 



■ a'+b' + c'+d' = a" + b" + c"+d" (1), 



which is a well-known identity ; and equating the fourth powers, 

 we have 



a^ +¥ +c* +d'' +6 (6' c^ +c'a'' + a' 5' ) + 24a bed 

 = a'' + h'i + c'* + d" + Q {b"c" + c"a" + a%") + 24^ab'cd', 

 or, as we may write it for brevity, 



ta* + Qtccb^ + 24>abcd = ta" + Q^a'b" + 24<a'b'cd'. . . (2). 

 By squaring (1), we have 



la' + 2Xa'b' = la"+2ta%" (3), 



whence multiplying (3) by 3 and subtracting (2) from it, 



Xa'-12abcd = l.a"-12ab'cd' (4). 



Also, from (2) and (3), 



Xa,%'^ + Qabcd = %a'%'^ + Qa'b'c'd'. 

 Equating the terms of the sixth order in {A), we have 

 2a' + lota'W -f 902a'6V + 12Qabcd 2a' 

 = a similar expression involving a, b', c, d! (5). 



