The Rev. R. Murphy on a fiew Theorem in Analysis. 31 



And by equation (4.) the sum of these expressions must be 

 equal to 1.2.3 ... m P,„,„+i, that is, 



From whence we have 

 Am = m,2Bm = m A^_i or B^ = -^-^ ^ 



3 C^ = m B„,.i or C^ = — ^ -^-^^~ ' &c. ; 



and therefore 



1.2.3 ... m p„.„ = (^-r-^ - ;;2 <^ {r-'r""^ 



which series terminates at the m^^ term, since (^""-""p"^ = 0. 



If we put 1, 2, 3, &c. successively for w in this formula, and 

 substitute in the expression (2), stopping at P„^ „, we should 



c?" u 

 have -7—^ explicitly obtained ; but there is no necessity for 



this. 



Again, since A (p = ^ (jr + ^, y) — 4> (jr, y), therefore 



(A p)"^ = {^ (ar 4- /?,y)r -rn<p{x, y) {^ {x + A,y)}*»-^ 



+ !^l!!^) {^ (^^ ^) }2 ^^ (.^ + ^, ^) p-2 ^ &c. 



Each term in this series may be expanded according to powers 

 of h by Taylor's theorem ; and if we take the coefficient of 



h* . h^ 



in each term, the coefficient of -z-rr^- in (A (p)"* 



x»z,o ... n l.J.o ... n 



is exactly the same as the series (5.). 



By this comparison we find that P^^ „ is the coefficient of 



h"* . , . n iA(p)'" 



rTTiy in the expansion of —-^ — . 



1.2.3,,. n ^ 1.2.3 „.m 



Recur now to the series (2.), and we get 



d" n . h* 



-j—„ = the coefficient of ^tttt: in the series following, viz. 



^ da^ da\ 1.2 rfflj ^ rfa* L 1.2.3 rfaj ^ 



