any Function of Sines and Cosines. 



119 



So that, finally, for example, writing the indices of sines outside 

 the bracket, if B 7 , A 7 are neglected together with every B and A 

 which has a higher index than 7. 



mi o.. i 3 1 _ . 5 3 . . 7 5 r _ . 11 . Ill 



t»\ on 1 3 \ oi 5 3 _ 7 5 f _ . 13 13i 



(6) 2 =2» T . T . T .2.1. y . T .3.2. T . T 1 5, sin-g--^ cos y j. 



11 



13 



1 3 1 



5 3 



7 5 



(6) 2 cos^-(6) 1 co 8 ^=2"^.^.2.1. T . T .3.2. T . T . 1 *, 



,„. . 11 ... . 13 



(6) a siny- (6) 1 siny: 



2 "tw»44- 32 t4^.- 



By means of the foregoing table and Table III., the values of 

 any quantity jB and A may be found upon any hypothesis as regards 

 the terms which may safely be neglected. 



Similar expressions may be obtained when the series are sepa- 

 rated into two categories, the one containing only even and the 

 other uneven indices. 



Let 



1st Category, even indices. 

 2nd Operation. Elimination of B 4 , A 4 . 



(l) 1 + (l) 3 -2(l) 2 cos4 = (2) 1 

 (l) 2 +(l) 4 -2(l) 3 cos4 = (2) 2 

 (l) 2 + (l)5-2(l) 4 cos4 = (2) 3 . 



The second operation by which B 4 and A 4 are eliminated is as 

 regards J3 2 , A r 



2 jB 2 sin 1 {sin 3 + sin 7 — 2 sin 5 cos 4} 



— 2^ 2 sm 1 {cos 3 + cos 7 — 2 cos 5 cos 4} 



= 2 2 .B 2 sin5 sin l{cos 2 — cos 4} — 2 ^ 2 cos 5 sin J {cos 2— cos 4}. 



The following table gives the indices for each vertical column 

 under B { , A ( when the operations are conducted successively and 

 separately so as to eliminate all except B t , A { . 



