114 MR. A. CAYLEY ON A DIFFERENTIAL EQUATION. 



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

 the equation becomes 



or, what is the same thing, 



+ a„_,[(?^-I)l9+I]™-'=[^^^ + ?^-I]^-'; 



so that 0^0^ ^i^ • • • ^n-i are the coefficients of the expan- 

 sion of [nO -{■ n— iY~'^ (which is a rational and integral 

 function of 6, of the degree n—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 





w 



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

 — U(p), and, after the operation A* is performed, is to 

 be put equal to zero. 



