112 MR. A. CAYLEY ON A DIFFERENTIAL EQUATION. 



which equation may be considered instead of the original 

 equation; and it is to be shown that y, regarded as a 

 function of Uj satisfies a certain linear differential equation 

 of the order n — i. In fact^ expanding y by Lagrange^s 

 theorem, we have 



^ 1.2 ^ ^ 1.2.3 



1.2 1.2.3'^^^ 



the law whereof is obvious, and using the ordinary nota- 

 tion of factorial viz. [nY=n (n—i) . . . {n—r+i), we 

 may write 



9,— S L^ -I /7« y(n- 1)6+1 



where 6 extends from o to 00 . 



It is now very easy to show that y satisfies the dif- 

 ferential equation 



r dy-' V n d 271- ly-' 



u-T- y = na u-^ 



L duj Ln—i du n—ij 



u^^~^y. 



In fact, using on the left-hand side the foregoing value 

 of yj and on the right-hand side the following value 

 o£u^~^y, obtained from that of 2^ by writing ^— i in the 

 place of 0, viz. 



d 

 and observing that in general the symbol u -j-, as re- 

 gards u'"\ is=w, the equation in question will be sa- 

 tisfied, if only 



= -h TflT ((?^- 1)^+1) 



