OF. TWO INDEPENDENT VARIABLES. 221 



224. If we wish the series for ff>(a;+h, y + k) to close 

 after a finite number of terms, we can put the expansion 

 for f (a) under the form 



-' (0) . ~ 



and from this the required form for <j> (cc + h, y + k) can be 

 obtained. For example, if n = 3, 



^ + k 



^ 



s? 



where stands for <f>(x+ dh, y + 6k}. 



225. In the formula established in Art. 223, put as = 0, 

 and = 0; then 



, . 



where M O , ^-, -^-, -r-a 2 -, stand for the values of 



du du d*u . . . 



M, ^ , -7-, -r-s , when in these expressions we put 



a; = 0, and y = 0. If we change h and A? into x and y respec- 

 tively in the above formula, we have 



+ I-T: \x* -j4 + 2#v -j-$- + y 9 vt[ 

 |_2_[ dx* " dxdy y of] 



+ 



