NUMBER OF UNKNOWN QUANTITIES. 75 



From these equations taken in the inverse order, we get 



^ 1 



"' ~ 1.2 .3 nh"' 



«„-!= - na, 



n> 



„ _ n(n-\) 



1 . 2 

 &C.= &C 



Hence the required expansion is 



To apply this principle to equations, we may observe that when 

 the general or a;"' equation is cleared of fractions and its right-hand 

 member transposed, it is of the form 



-P+ XiXi +^2X2 + +i8„X„ = 0, 



where ssi, sss a, are the unknown quantities, and P, Xx, X^.... 



known functions of x. 



The left-hand member must, by the reasoning of the preceding 

 Section, be divisible by (« — l)(x— 2) {x—n). 



Let Xi, Xi, &c. when divided by this quantity leave the re- 

 mainders Q'l, Q'2, &c. and P, the remainder P', hence 



where all the functions are necessarily of less than n dimensions, the 

 application of the process above described, would then determine the 

 quantities ssi, asj, »„. 



R. MURPHY. 



Caics College, 

 March 5, 1833. 



K« 



