152 Mr. Murphy on the 



In many instances we have had occasion to take the coefficient 



of - in f{x), or which is the same, the term independent of x in 



x 



xf{x), there may easily be obtained several theorems, for facili- 

 tating the research of this quantity, the following may perhaps, 

 from the elegance of its form, be not deemed unworthy of notice, 

 viz. 



The term independent of x in <p {x - - j 



,, », \x-d>"(x)\" \a?tb'"(x)\'" . 

 = <p(x)-{x<p'(x)\' + ' y ^ M - £^£ +&c, 



x being put = l in the series. 



For <f>(x) may be represented by 2a„.a", the symbol 2, denoting 

 the sum with respect to n, 



Hence (a - -) - 2«, ■(*--)• 



But(.-i)"=(i-i)".(i + -)»; 



the part of which, independent of x, is 



, (n.n — lY- » 



■ 1 -' + 1 (tt)"^ 



or = x "-\ x -dx-) + r.tf >&c - 



supposing * to be put = l, after the differentiations are performed. 

 Multiply by «„, and take the 2 with respect to n ■ observing, 

 generally, that 



2a " dx m ~ dx m 



and we obtain the proposed theorem. 



