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



and so on. 



One remarkable development of the character above allnded 

 to is the expansion of ifrj£ _ l x in powers of x, yfrx and %x being 

 given forms. If a be a root of yx> then 



£((^)>*)- s - 



8 + 



where a? is made equal to a in the coefficients of the powers of x 

 after the differentiations have been performed. It is now 

 required to express these coefficients explicitly as functions of a ; 

 for which purpose we have the general term 



the D now denoting differentiation with regard to «. 

 For example, let 



%x=ax + bx 2 -{-cx 3 + . . . , 

 so that a root of %# is zero ; then 



X x 

 whence 



\yxj 



= i/rO + Ap/r'0 .#+(A 2 ^'0 + B 2 ^0).|- 



+ (A 4 t (lv) + 3B 4 ^''0 + 3.2^ 



If, however, %x be a function which does not vanish with x, 

 but has a root a, then we have 



( ? y= ( s v 



