INVERSE METHOD OF DEFINITE INTEGRALS. 117 



We have thus 



1 Ai A2 An c,x.{x-\) ....(ar-w + l) 



« + l "^ x + ^ "^ x + ^6 "^ x + n + 1 ~ {x ■\-\) .{x + ^)....{x -{-n + 1)' 



Multiply by x + 1, and then put x= —\\ hence c = ( — 1)", 



by ar + a, a:= — 2; -(4i= — - . — — , 



by . + 3, .= -3; 4=-"-^-^.^^^±ii^>; 



&c &c. 



1- j^/js -. « w + 1 ^ «.(«-l) (?i + l).(w + 2) .„ J 

 hence /(0 = 1- j •-]-• ^+ ^T^-^-^ H^ ^./^-&c. 



dt" 1.2.3....W 



3. Denoting by P„ the value of f{t) which has been investigated 

 in the preceding article, it possesses the remarkable property ; that 

 ftP„P„ = 0, except when m — n, and then 



r p jj __ ^ . 



•'' '" "~2w + l' 



the limits being always and 1. 



For when m and n are unequal, one of them as n is the greater, 

 P„ contains then only powers of t inferior to n, the integral of each 

 of which vanishes by the natvire of P„. 



When m = n, the last term of P„, namely 



(w + l)(w + 2)....2w , 

 1 . 2 ....n ^ ^'' 



is the only term of which, when multiplied by P„, the integral does 



• This value of _/(<) lias been shewn in the " Treatise on Electricity " to be the coefficient 

 of /j' in {1-2//. (1-2/)+/*^}-^. 



Vol. V. Part II. Q 



