646] 
47 
646. 
ON THE GENERAL EQUATION OF DIFFERENCES OF THE SECOND 
ORDER. 
[Iiom the Quarterly Journal of Pure and Applied Mathematics, vol. xiv, (1877), 
pp. 23—25.] 
Consider the equation of differences 
U X = a x-l U X-1 + G—2 U X—2 > 
viz. we have 
u 2 = a x u x + b 0 u 0 , 
u 3 = a 2 u 2 + bill!, 
u 4 = a 3 u 3 4 b 2 u 2 , 
u 3 a 4 u 4 -I - b 3 u 3 , 
u 6 = a s u 5 + b 4 u 4 , 
&c., 
u 3 = a 2 «1 
■u x 4- a 2 
b( } ii,j 
1 + b x 
' 
u 4 = a 3 a 2 a x 
Mi + a 3 a 2 
bo u 0 , 
+ ci 3 b x 
+ aj) 2 
+ b 2 
5 = a^a.^ 
u x 4- a 4 a 3 a 2 
b 0 Ug , 
4- a 4 a 3 b x 
+ a 4 b 2 
+ a 4 a x b 2 
+ a 2 a x b 3 
4* b x b 3 
+ a 2 b 3 
and thence