222 EXPANSION OP A FUNCTION. 



x and y being each put equal to zero in u and its differential 

 .coefficients after the differentiations have been performed. 



In this manner the formula of Maclaurin is extended to 

 the expansion of functions of two variables. 



226. The expression for the nth differential coefficient 

 of /(a), in Art. 223, is 



J2 dx n i dy t * c 



which, for abbreviation, may be written 



j, d ,, d\ n , 



h -r- + k -7- / 



ax dy) J 



provided we interpret this expression thus : [ h' -j- + k' ^~ 

 is to be expanded by the Binomial Theorem as if h 1 -j- were 

 one term and k' -7- the other term : when the expansion is 



effected, every such term as ( h' -,-} Ik' -,- ) f which occurs 



V dx) \ dy) J 



is to be replaced by h" l ~ r k' r j^~T- r If we adopt this mode 

 of abbreviation the result of Art. 223 may be written 



f -hi : \ fl ^-'* K j I u ~rr'\ n j~~r K lT v > 



|n-l V dx dy) [n_\ dx dy) 



where u = <f)(x, y], and v = < (x + 6h, y + 0k). 



By Art. 110 the last term of the expansion may, if we 

 please, be replaced "by 



dy 



The methods here given for the expansion of a function 

 of two independent variables may be readily extended to 

 the expansion of a function of more than two independent 

 variables. 



