1 98 CLASS OF ALTERNANTS. 



I sin a sin 3a sin 5a cos 3a \ _ p ^ _^, ^-^^^ ^^^^ ^ ^^,^ 



I sin a sin 3a cos a cos 3a | ^ v ~„ (2« + 2/5 + 2y), (IV) 



I sin a sin 3a cos a cos 5o I r- v, o ., o /-v7-\ 



^ — ^ , — \- =ijr :=^sin 2(1 — s/n ::ii 2a- (V) 



I sin a stn oa cos a cos 3a | 



(3) Of course, these results, when known, can be verified by 

 evaluating the five alternants concerned. Following the sure but 

 lengthy process based on the substitution of exponential values for 

 sines and cosines, we find 



I sin a S/.7I 3a cos a cos 3o | = — 16- sui (( — y) sin (c — /j) . . . 



sin Qj — a) 

 = - 16- II sin {( — y), say, (VI) 



and the others the appropriate multiples of this. 



(4) Further, since 



16 ros a cos j3 cos y cos c ■ cos ^ a, or, say, 16 II cos a ' cos 2S a, 

 = 1 + 2 cos 2o + :i: cos (2a + 2/5) + :2 COS (2o + 2/3 + 2y) + COS S 2n 



(VII) 

 we have 



16- II COS a • COS i:a = l — A+B 



= - I sin a sill 3o cos a cos 3o | + | sin 3a s/n 5« cos a cos 3« | 

 ' . . 1 . . 



+ I sin a sin 5a cos a cos 3a | -!- | -■</n a s/n 3n cos ,i cos 3a | 



But the three determinants in the numerator here have two 

 columns in common, and each takes its other pair of columns from 

 a common three; their sum can therefore be expressed as one 

 determinant, namely, 



I sm a + sin 3a sin 3a + sin 5a cos a cos 3a | , 



and tliis is equal to 



I 2 siw 2a ^'O.sa 2 .s//( 4o cos a cos a cos 3a \ 

 = 4-11 cos a • I sin 2a sin 4a 1 4 cos-a — 3 | 

 = 8-11 cos a ■ I sin 2a sin 4a 1 cos 2a \ 



so that, on dividing by 16 11 cos o, we obtain 



/ , .i , , -\ I M *■'" 2a sin 4a cos 2a I /itttt\ 



cos (a + ^ + y + r) = i I ., _^ ! ^_^ (VIII) 



I stn a sm 6a cos a cos oa \ 



a more important result than any of those from which it is 

 derived. 



(5) It is noteworthy that the set of four equations obtained 

 by substituting in (I) for a, li, y, c their complementaries has the 

 same determinant as the original four. There is thus secured 

 without effort the evaluation of four additional alternants, namely 



I cos a cos 3a cos 5a sm 3n | = j " f '" (^" + ^'^\ , ,, , j 'D (IX) 

 ' ' ( + :S stn (2a + 2;3 + 2y). ) '^ ' 



