NUMBER OF UNKNOWN QUANTITIES. 67 



SECTION II. 



Application of the First Principle. 



The first principle alone is sufficient, in a great number of instances, 

 to resolve the proposed equations ; we shall illustrate its application by 

 selecting three distinct classes of equations to be resolved. 



First, when the terms which compose the general or a;**" equation are 

 proper fractions. 



Example : 



To find the values of the n unknown quantities ssi, %i, sss, s,, sub- 

 ject to the n equations following, 



»1 «2 ^ »„ ^ _ 1 



3 4 "*■ 5 "^ » + 2 2' 



«i , ^ ^ , g« =_ 1 



4 "*" 5 6 "^ « + 3 3* 



» + l « + 2 ra + 3 2w »■ 



The general, or a;**" equation, when its right-hand member is trans- 

 posed, becomes 



- + — Y — -^ \- H — =0. 



a; x+1 x + ^ x + n 



N 

 Suppose these fractions are actually added, and let -^ represent the 



sum; where D = x{x-\-\)(x-\-9l) {x + n) and A'' is some function of x 



of n dimensions. . ,- • .• 



i2 



