270 ELIMINATION OF FUNCTIONS. 



eliminated between the six equations (1), (2), (3). This 

 cannot generally be effected. Proceeding to the equations 



d 3 F d*F 



-7-3- = 0, -3-5-3- = 0, , , a = 0, -J-J- =0... (4), 



do? dx 2 dy dxdy* dy* 



we shall introduce only two new unknown functions, namely 

 $"' ( a i) an( ^ &" (<**) Hence we can obtain by elimination an 

 equation between z and its partial differential coefficients with 

 respect to y and x of the third order inclusive, which will 

 be free from the functions $,(2,) and <f> 2 (a 2 ) and their derived 

 functions. Since we have ten equations and eight quantities 

 to be eliminated, two resulting equations can generally be 

 obtained. 



249. We say that generally, in the case supposed in the 

 preceding Article, we cannot eliminate the arbitrary functions 

 by proceeding as far as the second derived equations. Cases 

 however occur, in which, owing to the forms of a, and o 2 , this 

 elimination can be effected ; for example, suppose 



fa(x- ay). 



Here ^ = </ (x + ay) + & (x - ay] , 



dz 



72 



= aW (x + ay) + <&' (x - ay) ; 



-j = a -j-s . 

 dy 1 dx 



250. Suppose we have an equation between three vari- 

 ables of the form 



involving n arbitrary functions ^>,, </> 2 , ...... <f> n of the n known 



functions a 1? c' 2 , ...... respectively. 



