274 ELIMINATION OF FUNCTIONS. 



twelve equations, but have twelve quantities to eliminate, 

 namely 



da di d*a tfa. d*a 



dx' dy y dx*' dxdy' dy" 



But the fact is that by adopting the method of Art. 252, 

 we have fa (a) and fa (a) occurring in such a way that they 



disappear together in our elimination of -y- and -4- . Hence 



it happens that we are able to effect the required elimination 

 without proceeding beyond the derived equations of the second 

 order. 



255. In particular cases the elimination may be effected 

 without proceeding to so many differentiations as the general 

 theory indicates. Suppose, for example, that we have three 

 arbitrary functions, we should generally have to form the de- 

 rived equations of the third order by Art. 252. But if the 

 three arbitrary functions are so related, that 



the given equations take the form 



.*>,</, *,,<, fa (), fa" (a)} =0, 

 / [x, y, z, a, fa (a), fa' (a), fa" (a) } = j 



and by proceeding as far as the second derived equations, we 

 obtain twelve equations and eleven quantities to be eliminated. 

 namely 



dy. da d z a d*a d*ot. 



dx ' dy' dx* ' dy* ' dxdy' 

 fa(], fa' (a), fa" (a), fa" (a], fa"" (a). 



Thus we can deduce one resulting equation involving x, 

 y, z, and partial differential coefficients of z up to those of the 

 second order inclusive. 



