204 ON THE DETERMINATION OF THE ATTRACTIONS 



have only two cases to consider : First, that in which A=0, 

 and consequently also all the other coefficients of <M together 

 with the remaining ones in </>, as will be evident from the formula 

 (21). Hence, in this case 



< = 0: 



Secondly, that in which Jc Q is one of the N roots of = K, as for 

 instance, Jc ' in this case all the coefficients of (f> will become 

 multiples of A, and we shall have 



<^ being a determinate function of f t , f 2 , ...... f g . 



We thus see that when we consider functions of the form (20) 

 only, the most general solution that the equation 



= V >-^ ...................... (19') 



admits is .................. 



or, < = ; or, <f> = </> ; 

 a being a quantity independent of , f 2 , ...... ( and < any 



function which satisfies for < to the equation (19'). But by 

 affecting both sides of the equation 



with the symbol v> we get 



o = v.v'<-A'-v<A; 



and we shall afterwards prove the operations indicated by y and 

 V' to be such, that whatever <f> may be, 



Hence, the last equation becomes 



and as y</> like </> is of the form (20), it follows from what has 

 just been shewn, that either 



= v<, or, v = a <k 

 a being a quantity independent of f x , f 2 , ...... ( , . 



The first is inadmissible, since it would give (/> = 0; there- 

 fore when </> satisfies (19'), we have 



<' = a<>, i.e. = 7<> a<>. 



