412 MISCELLANEOUS PROPOSITIONS. 



tion becomes <f>(a + h, b + k) = 0. Now expand < (a + h, b + k} 

 by Chapter xiv. Suppose that every, differential coefficient 



d^&tx, y) 

 , 7 r \ , vanishes when x = a and y = b, so long as r + s is 



less than n. Then we may denote the expansion symbolically 

 thus : 



where u stands for <f> (x, y) and v for < (x + 6h, y + 6k], 

 6 being some proper fraction ; and after the differentiations 

 have been performed we are to put x = a and y = b. 



Now if we suppose h and k indefinitely small we have ulti- 

 mately for determining the ratio of k to h an equation which 

 may be expressed symbolically thus : 



/. d ' d\ n 



U T- + &-T- w = 0, 

 V dx dy] 



or more explicitly thus : 



where after the differentiations have been performed we are 

 to put x = a and y = b. 



It is obvious, as in Art. 195, that when h and k are indefi- 

 nitely small y coincides in meaning with -~- for the case in 

 which x = a and y = b. 



381. As an example of the preceding Article suppose 

 we have the equation cc 4 / c 2 (c a?) 2 (c*4-a 2 ) = 0. Here 



when x = c and y = we have -j- = and -^- = ; also then 



d?u d 2 u , d?u m , i , 



- =9 - 4c 4 , -4-V- = 0, and jr- 2 = 2c*. Thus we obtain 

 ax 2 dxdy dy 



Jc 



therefore 7 = V 2 - 



h 



