(3.) 



SO The Rev. R. Murphy on a new Theorem in Analysis. 



= ^--- dra'^d^[^-'-''^d-a)' 



Applying this formula to differentiate equation (2,) relative 

 to j:, we get 



d'*+^u _ p, du d_ / jy, du\ 

 d^' ~ '""da ■*■ 6/« V ''" dJ 



But if we write w + 1 for n in the formula (2.), we also have 



d^'u p du d /p du\ cP / du\ 



d^^ = ^hn+i . ^+ J^l^2.«+i^ j + 5^4^^. »+^ da) 



+ &C., 



which compared with the preceding expression shows the 

 following law for the formation of the functions F„ „, viz. 



Pm,n-|-1 = P'»«,n + 4^'- P»n-l,n .... (4.) 



And since P;„.i is known, (for Pj ^ = <$>', P2 j = 0, Ps^i 

 = 0, &c.,) we can thus form successively the quantities P^ 'gj 

 P^ 3, &c., and the general law of these quantities may be 

 thus found. 



Put 



1.2.3 ... m P.,„ = (rP -A^.^ (r-^r^'^^ + B^^^ (r-r-^ 



-C^^^(<^--f'"^+&c. 

 Hence 



1.2.3 ...,» F„..= (W"^"-A„Mr-r'^" 



_A„4.'(r-')"' 



_A.*'(r-r"' + 2B„<f$'(r-^r"' 



Also 



1.2.3 ... m 4^' P^_,, „ = m (^' (<^-if ^"^ - m A^_, ^ <^' (<^'»-^)"^"^ 



+ 7«B^_,<^2<j>'(r-'r''^ + &c. 



