174 MR. GRAVES ON A RECTIFICATION OF THE 



consequently it is impossible to develop f~^ 6 according to the ascending 

 integral powers of 6. Let us then proceed to develop according to the 

 ascending powers of 1 — ^fc; (c being a constant, and introduced — be it 

 remarked in advance — on account of the power it possesses, if properly chosen, 

 of rendering the intended development of f~^ ^convergent.) 

 To effect this purpose, let 



l-6fc = » [14] 



Hence 



fl = (1 - w) {fc)~^ ; or since, by [8], (f c) "^ = f — c ; fl = (l - ») f — c : 



Accordingly, after substituting in [13] {I — &i) f— c for 6, and therefore 

 — <y f — c d <y for d ^, we find 



df~'{(i -«)f-c} . 



dw ^ ' 



Hence, continuing to derive the successive differential coefficients, we obtain 



^^ i= //-1. 1.2. ..«-!. (I -w)-" 



n 



dco 



Hence, evidently. 



rd"f~'{(l -«))f-c}\ ^ r,rT 



^^ ^ '-)= V -1.1.2. ..n-1 [16] 



dn 

 CO 



jn»-l 



(by the notation ( — — ^ ^) being designated the value which 



\ dw" ' 



d"f-^{(l -„){-c] . - - , 



^^^ i acquires, when a = 0.) 



dco" 



Also, by [9], (f"^{(i-co)f-c})orf~*f-c = 2jV-c L^^] 



But, by Maclaurin's theorem, 



/ d"f-'{(l-co)f-c} > 

 J n ) 

 f-l{(l_«,)f_c] = (f-»{(i_„)f_c})...+ i.2"'...n «" ••• 



Substituting for the successive terms of this equation their values derived 

 from [16] and [15], we obtain 



f~M(l-«)f-c} = 2in-c+ »/'^r\ („... + f^...) [17] 



