DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 17 



of the second order in three independent variables, possessing a general primitive 

 involving a couple of arbitrary functions of two arguments, but not possessing an 

 intermediary integral. 

 Let 



8^ < (u, v) = & L , g M gy e ( M> v ) = < 12) ^ < (, r) = ,,, 



and so on for higher derivatives and for other functions. Take two functions 



and consider the equation 



v = $ + p<l>i + (r<j>, + e + X0, + p.0,, 



as one from which < and # are to be eliminated by means of derivatives of order not 

 higher than two ; the quantities p, cr, X, p, being denned as 



We have 



Cfr X = ax 



l>\y + c'^, /A = a'a; 



For second derivatives, it is unnecessary to form expressions for l> and c, for the 

 latter gives the only equation which contains < 222 , and the former the only one which 

 contains # 22i , so that, when elimination is to be performed, these equations would be 

 ignored. We therefore take 



+ erh 13 + (1 + 2a) 8 n 

 k = 1 & + a' + & > 



y) 0n + (' 



0-^122 



+ 0' + y) 13 



VOL. CXCI. A. 



