203 



This coefficient, it is easily seen, is equal to -J [M x + M y ), so that we 

 may replace the above expression by 



n - m 



±M X + M v x x y x z x 



A 



in like manner it will be seen that the part multiplied by n becomes 



1 M x + M, B-C 



2—J)— X ^-AB n 



thus, the two terms in question become 



1M X + M vl xm x % x (n-n x B-C 

 + — n 



2 D 2n-n x \ A AB 

 and in like manner the latter pair become 



l(M x +-M y \ x x y x % x fn-n l A-C 



2 V B J 2n-n i \ B AB 

 therefore the sum of the four is 



1 M x +M y x x y x % x I (I 1\ B-B 



D 2n-n t \ \ A B AB 



thus the constant terms of the right-hand side of the equation for # 3 

 become 



P 15 (, 5 • , i q > \ 1 B-A[M x + 3f yl 



— 1 + cos 1 — sin 2 i 1 + 3 cos t 4 



\ 1 B 



) 2 ~A 



r* 8 V 4 / 2 ABC 



also on putting for a and their values, it is easily seen that the value 

 of i^is 



15 fi f 5 . 



. 1 + cos 1 — sin 2 i 1 + 3 cos 1 x,y x z x 

 8 r 4 V 4 / 



so that the expression just found for the constant term in the deve- 

 lopment of N reduces itself to 



- 



2 ABC V D 



And this is identical with that part on the other side of the equation for 

 which arises from the multiplication together of u x and a 2 , so that 



at 



these terms identically destroy each other. 



