354 Mr. C. J. Hargreave's Applications of the 



tion, but yft becomes an instrument in determining the form of 

 the operations to be performed on u. 



If, in the function yfrx (which we regard as a function actually 

 expressed in terms of x and constants), we give x a particular 

 value, say a, the effect on the right-hand side of the equation is 

 to produce a series arranged in powers of a, the coefficients of 

 which are functions of x determinable by means of known ope- 

 rations performed upon u. The left-hand side of the equation 

 in that case assumes a remarkable form ; it represents an opera- 

 tion performed upon u of the following nature : multiplication 

 by an expression containing x and a, explicitly <j>(x, a) ; differ- 

 ential operations with regard to x ; and lastly, the change of x 

 into a specific value a whenever x appears explicitly, u remaining 

 throughout an implicit function of x. The right-hand side of 

 the equation is the expression of this result in the form of an 

 expansion in powers of a. 



If {$D . yfr(x, a)} denote the above operation, we have 



Now the function ^(v + fl) in this expression has no other 

 effect than simply to determine the form of the differential ope- 

 rations which are to be performed upon u, the function of x. If, 

 after having thus determined the form into which <f>D is changed, 

 we consider x as changed into a in the function u, and consider 

 D as now meaning differentiation with regard to «, it is evident 

 that we thus obtain the value of </>D{T/r# . u} when x is made 

 equal to a throughout the whole expression. We have, then, 

 {<f>D.ty(x, a)}u = yfr(V + a)((j)D) .u y u remaining a function of #. 

 $D {yjr(x, a).u] (when#= a generally) = yjr( y -f a)(j>D . %«, u being 



XX, and D denoting now -r*. 



As an example, let ty(x, a) be (#— a) n , n positive. Then we 

 have <p((#-a)\ u) (when x=a) = V n <^D . M=</> (n) D . u. This 

 equation is true in two senses. It is true, first, if u continue to 

 be a function of x, the change of x into a being supposed to be 

 made only as part of the operation performed upon u. It is also 

 true, secondly, if x be made equal to a generally, the D on the 

 second side then having relation to a, of which u becomes a 

 function. 



These may be thus verified : — By the original theorem we have 



<l)lJ^{x-a) n .u) = (x-a) n .<l)D.u^n{x-a) n - 1 .(l> , D.u 

 +i/i(n-l)(a?-fl)"- 2 .</>"D.w+ ...; 



