58 



of which equation of finite differences the integral is 

 y = aa- + a" 



The equation (9) is easily reduced to 



((a' + a) +{j: + i)) (a' -^ d) = o 

 Hence a — a = o 



and a' -f- a + J' + 1= C^ 



the former equation gives a= a' and therefore 



a' = a iSfc. 

 Hence one integral of the equation (10) is 



y — ax ■\- a", where a is any constant arbitrary. 

 To find the value of a from the latter equation 



a'-\-a-\-x-\-i=:o 

 Suppose a — u-\- mx + n 

 m and n being constant, and u a new variable. 



Then a = u' -\- m (.r ^ i) + n 

 & equation (1 1) becomes u -\-u+ {2m+ I) x + 2n + (m ^\)i= o 



If we make 2m + I = o and 2 n + (m + l)i = o 



m = — ^ and n — — i. 



and u + u' = o (12) 



with these values of m and n 



a-= u — - — - in which value of a, u 

 may represent any quantity satisfying equation (12) 



let M r: i r^ (13) 



ib and r being constant quantities. 

 Then m' = hr^+' and from equation (12) 



hr^+i + b"- = 

 divide by h'^ 

 and r' + I = 



