114 MR. A. CAYLEY ON A DIFFERENTIAL EQUATION. 



Or, observing that the right-hand side may be written 



n [0+i)\_eY 



the equation becomes 



= [/i^ + 7i— 1]^+™-% 

 or^ what is the same thing, 



«o + «J(^-I)^+I]' + a,[(/^-I)6'+I]^. . . 



4-a„_I[(?^-I)6'+I]^-'=[?^^ + ?^-I]«-'j 



so that Uq,o(,^, . . . Un-i are the coefficients of the expan- 

 sion of \nO -\- n— 1]'^~'^ (which is a rational and integral 

 function of 6, of the degree ^— i) in a factorial series, as 

 shown by the left-hand side of the equation. 

 To determine the actual values, write 



this gives 



nd-\-n—i-=—^ : 



n—i 



and we have therefore 



SO that the general expression is 



where A denotes the difference in regard to <f> (AU(p = U(p+, 

 — U(p), and, after the operation A'' is performed, (f> is to 

 be put equal to zero. 



