366 Mr Murphy on the Inverse Method of 



and to determine iV,, N 2 &c. suppose, successively, x = — l, -2, -3, 

 &c. thus we obtain 



NM-iy, ;v>-(-i r .^; Ar 8= (-i)».^=i) . ^l**** + kc 



and therefore 



and by Ex. 2 we have therefore 



/(^(-i)Mi-f.^.^^^. ^ +1 jf +a > .f-& c .}, 



an elegant result, to which we shall hereafter have occasion to 

 refer. 



Ex. 5. Given <& (a;) = j — , ,, , — — r-n 7 rr . 



r ' (x + /i).(x + 2,k) (x + nh) 



Applying the same method we get in this case 



f-'.ji-ty- 1 



/(0= r 



2.S...(n-l).h"- 1 



9. Let us next consider logarithmic functions, beginning with 

 p 

 h.\.Q=<p(x), supposing that neither P nor Q vanishes from x = 



to ao , and that the highest powers in the rational functions 

 P and Q may be equal, and be multiplied by equal coefficients; 

 which conditions are evidently necessary to make <p (x) remain 

 finite for all positive values of x, and to vanish when x=x>. 



To find f{t), let P and Q be resolved into their simple frac- 

 tions, so that 



P=C(x + a,)(a; + a 2 ) (x + a„), 



and Q=C.(x + b l ).(x + h) (x + b„); 



