128 Mr MURPHY'S SECOND MEMOIR ON THE 



that is, it 



_ {pqY V .^' . 3' 



n{p+q) + l'l.{p + l)(q+l){2p + l){2q + l)....{{n-l).p+l\{{n-l).q + l\ 



, 2 _ y f 1.2.3....W 1^ 



COR. j,(p,p}n- 2n + l-\l.{p + l).{2p + l)....{{n-J).p-i.l}f- 



14. To find the reciprocal function to that denoted by L„ in Art. 8, 

 , d" {f (h.\. ty\ 

 ^' 1.2....W df ' 



namci 



L„ consists of the powers of h. 1. /, and possesses the property of 

 ftL„t' = when x<n; suppose now that we investigate a rational function 

 X„ which shall possess the property JtK {h.l. t)' = when x<n; then it 

 is evident that j^X„i„/ = when n and n are unequal; and therefore they 

 are reciprocal functions. 



Put K=l + AJ + A,f +....AJ'', 



Put Ar = 2" + 'B„ A, = 3'' + 'B, A„ = {n + lY^\B,r, 



hence we must have when x<n, 



1'-' + a"-'^, + 3"-"^2 + (« + i)"-^jB, = 0. 



Now the left-hand member of the equation is the same as 



putting t = after the differentiations. 



Hence the differential coefficients from the 1" to the w* inclusive 

 of the function between the brackets vanishes when ^=0; that function 

 of e' ought therefore to contain no power of t inferior to the (w + 1)"', 

 and conversely, a function of e* which does not contain such a power 

 of t, will fulfil the required conditions. 



