ELIMINATION OF FUNCTIONS. 271 



If we proceed in the manner of Art. 248, and deduce 

 from this equation all its derived equations up to those of the 

 7/i th order inclusive, we shall obtain 



1+2 + 3+4+ (m+1) 



. , . . (m+1) (ra+2) ! 

 equations, that is ^ equations. 



a 



The 'number of unknown functions will be (ra+l)n, and 

 therefore, that we may be able to eliminate the arbitrary 

 functions, we must have generally 



greater than (m + 1) n, 



therefore greater than n ; 



therefore m = 2n 1 at least. 



If m = In 1, the number of equations will be w (2n + 1), 

 and the number of functions to be eliminated, 2n 2 ; hence, 

 there will be generally n resulting equations. 



251. Suppose however that the known functions a 1? 2 ,...a M , 

 are all the same function ; we shall find that it will be suffi- 

 cient to proceed to the derived equations of the n ih order 

 inclusive, in order to be able to eliminate the arbitrary func- 

 tions. For let 



F[x, y, z, <j (a), < a (a), ^> n (^)}=0; 



differentiate with respect to x and y ; thus 

 dF_ dFdz^ dF (da. dadz^ 



(jL$s (/.- ClJG CtCf. \ClJ5 Ct% (tOC 



dF dF dz dF (da didz\_ 

 dy dz dy dz \dy dz dy) 



Eliminate r- ; thus 



dF dF dz di didz 

 I i 



dx dz dx dx dz dx 



_ 

 dy dz dy dy dz dy 



