240 LAPLACE'S THEOREMS ON DEVELOPMENT OF FUNCTIONS IN SERIES. [16 



Hence by Maclaurin's Theorem 



a l b m c n ~ d l+m+n 



_ 



II mini lda l db m dc H \DJ_\ a 



rl ll-.m,,n Jl+m + 



when 77, replace x, ?/, z, in P, Q, .R, F on the right. Now 



,' 



Z> 



Collect the developments of 



we have 



!7m,,! Jl+m+n r f JD rl ( P G R\~\ 



at) C P lm < l - - -> ^ ' V> 



--- ~ ~ ~-~ --- 





 X* 7 . --- ,~ , 



II ml nl 



G R\~\~\ 



' V> H 



f. --- ^ f 



?, C) JJ 



If we collect the coefficients of a'b m c n from the different terms, the 

 result agrees with that given by Laplace, Memoires de I'Academie, 1777. 



[March, 1869. 



