92 Mr. Jarrett on Algebraic Notation. 



But \n — \ = Oj from « = 1 to n — m; 



m 



,00 



.-. d"',u = a,„ . '^ + S„ a,„ ^ „ . [m + n x", 



and d"',,„u = a,„ .\m, or a„, = d'"„„ : u. 

 Also a» = <„ : m; .-. n„._i = <„"-' : u, \"^ ~ ] . , 



CO 



Hence, substituting in (1), m = iS',„x'"-V,,;"-' : u. 



Ex. 7. Maclaurin's Theorem applied to a function of two 

 variables : 



OCi CO 



(1-1 



CO CO 



Assume u = 'S'„.x"'~' . 5'„a,„_,,„_,^"~', 

 where a,„_,,„_, is not infinite. 



Then rf»,M = S,„x"'-' . S„a^.,,^,.\n-J. .y"''-' ; 



s 



CO 



••• d%,„u = 5„,a?"'~' .«,„_,, J. [s, as in the last example, 



00 



or <,o:m = -SmJ^'"' «„.-!,„ • 



00 



Also d°„. :m = S,„a:'"-'a„,_,,„; 



• d "-' ■ u - S x'^-' a ■ f»i = 1 ) 



.. Ci^, .a — O^X "m-l, n-l^ 1 I • 



'■W = 00-' 



Hence d'^^ . d,_ „"-' : u = S,„\m-\ . x' 



-r-i 



and d^,„. rf,,„"-' : m = .S^lr^.flr,.,-! , as in the last example. 



Also rf",, „.</,,„"-' : u = a„,„_,, 



'•a..o -a,,, : M = a„_, , j ^ and ^ \. 



