218 GENERAL THEOREMS 



Effecting the integrations indicated in H, we see that 



j^cos Saw + N f sin 2)aw 

 = D ~ ' 



where D = (a 2 + (Sam) 2 } . . . {c 2 + (Sap) 2 }, 



and JV and ^V' follow the same law of formation as in the 



particular case already considered, except that for - , &c. 



Ct 



we substitute , &c. With this remark we perceive that 

 H= 



[a + V(-l)Saw}...{c + V(-l) 



(3). 



The assumption now to be made is that 



i =S J\. , 2 , 



(a + Saw) . . . (c + Sap) A /' 



where A is the product of every set of s factors taken out of the 

 whole number of r factors 



a + Saw, . . . c + Sap, 

 and F is independent of a t . . . a 8 . 



r t r 1. ... r 5+1 



There will thus be - - '-^ - disposable quantities 



F, which will be found to be the number required to make (2') 

 identically true. Consequently we shall have 



,v cos Saw + v sin Saw 



where S is the product of s factors of the form a 2 + (Sam) 2 ; and v 

 and v are formed just as in the case of 5 = 2: that is to say, we 

 shall have 



* = a...c(l-<7 a +(7 4 -&c.), v f = a...c((7 1 - 

 where C t is the sum of the products of every combination 

 that can be made of the s quantities - , &c., taken t and 

 t together. 



