219 ) 



CHAPTER XIY. 



EXPANSION OF A FUNCTION OF TWO INDEPENDENT 

 VARIABLES. 



223. LET u=$(x, y] be a function of two independent 

 variables, and suppose <J> (x + Ti, y + &) is to be expanded in 

 ascending powers of h and k. Put 



h = ah', Tc = ah', 

 then < (x + Tt, y + k] = <f> (x 4- ah', y + ok'} ; 



the last expression may be considered a function of a, and 

 denoted by /(a). By Maclaurin's theorem, 



we shall now shew how the differential coefficients of f(a) 

 may be conveniently expressed. Suppose 



x + ah' = x, y + ok' = y ; 



then /(a) stands for <f> (x\ y'} and since both x' and y con- 

 tain a, we have by Art. 169, 



[ 



dx dot dy da. 



, g 



but 



^ 



dx dy 



Also, by Art. 63, 



x r , y} _ dcf> (x', y} dx' ^ 

 dx dx' ' dx ' 



dx' 



