connected with the Theory of Probabilities. 



171 



Proposition II. with connected problem. 

 IfV be a rational and integral function of the n variables x,y, . . z, 

 consisting wholly of positive terms, but involving no powers of the 

 variables higher than the first, and if V* represent the aggre- 

 gate of those terms in V which contain x as a factor, Y v the aggre- 

 gate of those which contain y as a factor, and so on ; then if we 

 form the system of n equations 



V V V 



=P, ^=?...£W,&c, . . . (14) 



V 



V 



V 



that system cannot have more than one solution in positive values 

 of x, y, . . z. And in order that it may have one such solution, 

 certain conditions must in general be satisfied by the quantities 

 p, q, . . r ; which conditions it is required to investigate. 



Of the general proposition I shall not attempt to give here a 

 fully developed proof. I purpose only to give such an account 

 of the pi'oof, together with special illustrations, as may place the 

 reader in a position to supply deficiencies. 



I shall begin with the investigation of the conditions between 

 p, q, &c. 



When x, y, z are positive, the first member of each equation 

 of the system (14) is in general the sum of a series of proper 

 fractions. Thus if 



V = axyz + byz + cxz + dxy + e, 



in which case the system (14) becomes 



axyz + cxz + dxy 



axyz + byz + cxz -+■ dxy -\-e 



axyz + byz + dxy 

 axyz + byz + cxz + dxy + e 



axyz + byz + cxz 

 axyz + byz + cxz + dxy + e J 



(15) 



and if we make 

 axyz 

 V 



. byz 



dxy 



V 



it is evident that \, p, v, p, and a will be proper fractions, and 

 that the system will assume the form 



X + v + p =p 



\ + p, + p~q 



X + /A + v = r 



\-t-/*-f v + p + a=l 

 llic last equation resulting from definition. And to these (qua- 



(16) 



