211 



Similarly, for the increment of u> 2 we have 



S fiC-A (A t + B l . A x - B 2 . , 



^2 = t -i — 5— {7 rr— : cos n - n 1 — — sin n-n ] 



that is, A x is increased by 



3 p C-B f A, - B, 



4 r 3 A \ (n - n 1 ) n 1 



and B 2 by 



3 fi C-A ( A x -B 



4 r 3 B \{n -n l )n l 

 whence A x + B 2 is increased by 



SfifC-A C-BV A, - 



4 r 3 \ B A ) {n - n 1 ) n 1 



where A x and B 2 on the right-hand side stand for the first approximate 

 values of A x and B 2 . 



Similarly A x - B 2 is increased by 



5 t ( °- A _ ^2 + ^1 

 4 r 3 \ B A ) (n- n x ) n 1 



The increment, therefore, of N introduced by these, will be 



C r 3 9 4 V J J I {2n - n 1 ) (n - n 1 ) n 1 



(2n - n x ) (n - n x ) n l \ 



which, if we put for A 1} &c, their first approximate values, and reject 

 quantities depending upon higher powers of B - A becomes 



B-A135 />V 1 1 ( Z{x?z x +yFz x )-Zj 

 ABC 32 \?) (2n-n 1 )n 1 D\ r 2 



It now remains to take account of the variations of 0 produced by 

 those of o> 3 , 



# 



(since — =3 a 3 , &c.) 

 ut 



But with regard to these, it is easily seen that however they modify 

 the preceding results, they cannot destroy them ; and for this simple 

 reason, that every term in iv 3 which will give rise to such terms will be 



