340 Mr MURPHY'S THIRD MEMOIR ON THE 



Hence, we get the general identity, 



1 1 



K^-«)(/3-^)l"^' (/3-ar' 



i 1 ,^ + 11 1 (n + l){n + 2) 1 _JL_&cl 



(^-ar' "^ 1 -fi-a-it-ay^ 1.2 •(/?- a)^ ' (#-«)'- '[ 



^_^ ^^±1 ^^ 1 (« + l)(>^ + 2) 1 _J__£,c 



■^(/S-O'*""^ 1 •/3-a-(/3-0" 1.2 •(/3-af-(/3-#)'-' •] 



Put now a = 0, /3 = 1, and therefore (i - t = f, hence, 

 >+' "^ 1 •/" "^ 1.2 ■^"-' 



[+F^"*" 1 V""^ 172 •^-^ + *'c.j 



in which identity n must be one of the natural numbers 0, 1, 2, 3, &c. 

 and the number of terms in each series must be limited to w-f 1. 



Suppose the (ra + 1)* successive integral of each term of this expansion 



is taken after multiplying, for convenience, by 1.2.3 n, the result 



will consist, 



1st, of a logarithmic part, viz. . 



(-,)-.h.i.w{i-f.^.^.^^. <''";'.'r' '''-M 



where the part between brackets in the upper line is equivalent to 

 the function P,„ and in the lower to (-1)".P„, and therefore the whole 



to (-l)".P„.h.l. ^,. 



* This method is applicable in every case to the decomposition of fractions, the denomi- 

 nators of which contain equal factors. 



