CLASS OF ALTF.RNANTS. 



199 



I cos (t ros 3a ros bu srn <, \ = J i) -w// 2«( -(- .<;/// i: 2« ( • D (X) 



I .0. :U ros 5a .<n, a .^in 3« | ='- ''% ^^^" + ^^)_ ^ . U) (XI) 



' ' — i. ros (-la + 2li + 2y)| ^ ' 



I ros a (■(>■•< ba sin a si// 3,, | = | v ros 2a — ros ri" 2n|I) (XTI) 



where I) stands for either member of (VT). 



By the same substitution (VIII) merely reproduces itself, and 

 (VII), on which (VIIT) was made to depend, becomes 



16-11 sin a ■ ros i! « = I — i: ros 2(( + ^' ros (2« + 2lS) 



— ':^ ros (2<, + 2/^ -f 2y) + ros v 2„ 



which for present puri)oses is not helpful. If, however, we 

 introduce 



16-11 ros a -rtts i: n = 2 x/n 2« + S .<.-/// (2« + 2^^) 



+ V sin (2a + 2/^ + 2y) + sin v 2a (XIII) 



or 16-11 sin a ■ sin ^ >i = — i: sin 2u + 2 xin i2n + 2/3) 



— :S sin (2« + 2/3 -\- 2y) + sin S 2« 



and proceed as in ^ 4, we obtain the desired companion to (VIII), 

 namely, 



.v/// („+,-! + y + r) = A ' _ ^ ^-o~- (XIV) 



I COS a ms 6a srn a sin 6a \ 



(6) Results similar to all of the preceding exist in the case of 

 every ev('?i number of angles. The fundamental equality corre- 

 sponding to (I) having been obtained, the others flow easily and 

 smoothly from it. It is important, therefore, and it is sufficient, 

 to show the form which this equality takes in one other case. 

 When the number of angles is si/\ it is 



A sin n -f B .*;//; 3rt + Csin 5n + F ros ,1 + G ros 3a + H ros 5«( =>/// 7(( (XV) 



where 



A E i: cos (2« + 2/3 + 2y) + 2 ros (2a + 2,} + 2y + 2o) E i! ros-. + v ros ^ say 

 B E — :S ros (2a + 2/3) — 2 ros (2a + 2/3 + 2y + 2r + 2f ) = - i: ros., - v rosl, 

 C E i; ros 2a + ros (2a + 2,-4 + 2y + 2o + 2f + 2/') E i; r<,Si + ros,-, 

 F E il sin (2a + 2<^ + 2y) — V sin (2a + 2^ + 2y + 2p) E i) s/n^— v su^, 

 G E i: sin (2a + 2f3) + ^ sin (2a + 2^ + 2y + 2i + 2f ) E — i) sin., + v .s/n-, 

 H = 2 sin 2a — ,s/'« (2a -I- 2i< + 2y + 2(1 + 2c 4- 2,') E i; .v///,— .s-zV/,. ; 



the related auxiliary equalities are 



2«. II ros a . ros ^u = l + i! rt/s^ + S roso + i! r o.s-.; + v ros ^ + v ^.^,.,. _|_ v ,.,,,,.^ 



= 1 + A - B + C' 

 2''.TI si/iasin "E <i='E sifl^ — 1i sin-^ + 'E '^in-,' — — '<in ^-{-'E s/ri;, — ^ >7//g 



=F + G + H (XVI) 



and the determinant of the coefficients of A, B. C. F, G, H, 

 namely, 



i sin a sin 3a sin 5a ros a COS 3a cos Oa | 

 = 2'2 II .sm (4-f). (XVII) 



