Differential Equations of the Third and Hie/her Orders. 65 

 we shall get 



/a 00 * 



or tt a 



which is a linear equation between the partial differential 

 coefficients of the first order of the function f a (xys/jb). 



The auxiliary equations for the solution, by Lagrange's 

 method, of this last equation are 



0=dfju ^ ds, 0—dx, 0=ay, 



whence we get 



f=fa(xy s ft) =/ 4 J xy (fi g^ sj I =/, { ay ft } suppose, 



where 



U-S« Sa-U 



^=* — bt w+ -it' + ^ 



and/ 6 is an arbitrary function. 

 Observing that the equation 



/=/a(#W*i) 

 gives 



the substitution in (5) of the above value off gives us 

 P/wJ. i%. S. 5. Vol. 21. No. 128. Jan. 1886. F 



