214 



it is easily seen, will destroy the term spoken of. If, then, we cancel 

 this term, and substitute for P, A x - B 2} their values, we shall have, 



A 



observing that such a quantity as — may be replaced very nearly by 



unity, and rejecting any terms multiplied by products of B - A and 

 C- A, &c, for this part of JSf 



3 15 3 3 B-Af 2 1 



8 8 2 ABC \2n - n l 2n - 3n 



r*(n-n>) j Wl D ° 



The terms found are the only ones depending upon the same com- 

 bination of x{z x , &c, and having the same divisor D xt and not multi- 

 plied by sin H„ or higher powers of B - A. 



There is one other term, however, it will be advisable to take into 

 account. It arises thus : the term P sin 2n - 2n l , when introduced into 

 cos 6 - 6 l , &c, in the equations for a x and a. 2 , produces not only terms 

 having n - n l , but also terms having 3n - 3n l for their arguments ; so 

 that &c, will contain terms of the form IT cos 3n- 3n l , &c. Now, 

 these terms will obviously contain the same combination of x^, &c, 

 in their coefficients, that A ly &c, do; and for this reason they had 

 better be retained. Their divisors, however, will be deficient ; but if 

 we neglect quantities depending upon products of 



C-B C-A 



it is plain that the divisor of the latter may be pat equal to \ of the 

 divisor of the former ; in fact, iTwill become \ A X P. 



In like manner if # 2 contains the term K sin 2>n- Sn 1 , we shall have 

 K= B 2 P. The effect of these terms will be to introduce into t the term 



X A — B 



— - 1 - \ P sin 2n-2>n\ 

 12 2n - 3n l ' 



with similar terms in 0 and y{r ; and these, when introduced into the 

 term - sin i sin 20 - 36 will give 



15 1 1 



2 12 2?i-3n 



- X PA X 



that is, they will reduce the term multiplied by 



1 



2n - 3n A 



in equation C to -|rds the value it has there. 



