NUMBER OF UNKNOWN QUANTITIES. 71, 



This equation is evidently of w dimensions with respect to x, and its 



roots by the first principle are 1, 2, 3 n; the left-hand member 



must therefore be identical with the product 



c{x-l\{x -2){x-S) (x-ti), 



whatever value may be assigned to x. 



i — lY 



Put therefore x = 0. Hence c = - — ^ ' , 



1.2.3 « 



X ^^ X.»..>........V[ ^^ -^ fif 



^-~" 2a— ,g ^ , 



n . (w — 1) 



1\¥ 



_ _ «(w-l)(w-2) 



* ^ a:,-- 1222.32 



&c &c 



and generally, «,„ = ^^ ^ . 3 ^)^~ •(-!)• 



We may verify this result by observing, that if we substitute this 

 quantity for 25,„ in the general or x^^ equation, then its left-hand member 

 becomes 



n . ( w-1) X . ( x + l) w.(w-l)(w-2) ^(£+_l)_(a; + 2) , ' 

 ^"■"'^'*" 1.2 •~T:2 1.2 . 3 • 1.2 7 3 ^*'''- 



This quantity is evidently the part which does not contain h in the 

 product, 



f, , x{x + l) ,„ x{x + l)ix + 2) ,3,o„l f, n , n{n- l) 1 1 



or in (l-//)-'.(l-|)". 



it is therefore the coefficient of //" in the expansion of 



But this coefficient is manifestly when x is any positive integer, which 

 evidently agrees with the proposed conditions. 



