DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 



37 



differentiated with regard to the independent variables, the results are identities. By 

 hypothesis, no new equation is derivable when first derivatives are formed : we 

 therefore form the derivatives of the second order, being six in all ; viz., they are 



dot? 



= 0, 



- 

 dydz~ ' 



dz da; 



= 0, 



dx dy 



= 0, 

 = 0, 



equations which contain derivatives of v of order 4. Let these fifteen derivatives be 

 denoted by s 1} s 2 , . . . , s l5 , their definitions being given by the scheme 



dx + dy 



do. i 



r 



dy, 



dz 



r s 



o 



? 'io 



*u 



'14- 



'13 



12 



Further, let (XX) denote the part of *y^ which is free from derivatives of the fourth 



order, (XY) the corresponding part of j^, (XZ) that of ^-^, and so on. Then the 

 six equations are 



(XX) + Ar x + Hr 2 + Gr, + Br, + Fr s + Cr ( , 

 (XY) + Ar 2 + Hr, + G- B + Br 7 + Fr 8 + Gr, 

 (XZ) + AJ-S + Hr 5 + Gr G +Br 8 + Fr + Cr ]0 



(YY) + Ar, 

 (YZ) + A- 8 

 (ZZ) + Ar 6 



Hr + Gr 



Hr 8 



Hr 



Gr 



10 



Br n 

 Br 12 



Br 13 



Fr is + 



18 



Fr 



Cr = 



ls 



^ 





 

 

 

 





As before, let the variables be changed from x, y, z, to x, y, , where is a function of 

 x, y, z, as yet undetermined, whence also z is a function of x, y, u. For the 



