Mr. Jarrett on Alscebraic Notation. 91 



00 7T3'"-' 



0= yJil'""' 



— .d;"-'u, by (1), 



Cor. For u write £j, . u, 



then E,.E,.u = S,„Dx\"'~'d;''-' : E,.u 



00 , 00 , 



= £ ^„r^ *""" 5^"- ' d;"-" : 4,"- ' : n, (Art. 18.); 

 and in the same manner we shall find 



E^.E,. £, .u = S,„ S"',, S% 'D^ "'" " . S^ "-^ Dil^ ' ' 4"-" : <^;'"'' : d/"' : m, 

 and so for any number of variables. 



Ex. 6. Maclaurin's Theorem : u = S^x""-' rf,,„'"-' : u. 



00 



Assume u = S„a„_^x"^-^ (1), where a„_, is either zero or some 

 finite quantity. 



■ Then d"'^u = S„a„_^ . \n-l .x"""'-' 



= S\a„_,. \n-l .3f-"'-' + a,„.|m.3;° + 5„a„,^,. |m + w .jc". 



m mm 



m2 



