Differential Equations of the Third and Higher Orders. 67 



where <f> is an arbitrary, and F, F x are definite functions ; and 

 where F(xy) is to be substituted for c after the integration 

 indicated in the last equation has been effected. 



If it is proposed to integrate the equation 



dx 

 putting 



dx 4 " dx 3 dy daP.daf dxdy 3 dy* ' 



d z z _ d 3 z d d z _ d z z _ 



dx d ~~ ' dx 2 dy~ ' dx dy 2 ~~ ' dy 6 ~ ' 



and assuming (8) to be satisfied by fix yrstu) = 0, proceeding 



in the same manner we shall find the conditions to be satisfied 



in order that (8) may have a first integral will be the three 



equations 



^ -,-» dot. -1-7- doc 



= Ra^- ^Y-y-9 



dx dy 



A ^ d Sa-V , T7 d Sa-V 

 ° = ^ U Tx-W-+ N dy-W 



°= n "dx-T- +Y dj^r-'> 



where a, — J „). / is one of the roots of the equation 



/ 0) 



= R 3 . a 4 -R 2 U . a 3 + RTV . a 2 - SV 2 . a + V 3 ; 



and the first integral, when these conditions are satisfied, 

 will be 



r+ Ra .8+ —y—.ut^au = \ j F 1 (y,c)dy + (l>{F(x,y)} ) 



where F(xy) = c is the integral of the equation 



= ~Rctdy—Ydx; 

 and FiCy, c) denotes the result of the elimination of x from 

 the quantity -^- by means of the equation F(xy) = c ; c being- 

 treated as constant, or as F(.r?/), in the manner indicated in 

 the preceding case. 



If the equation to be integrated is 



P d z q d°z ,p a z jj d z d z wfL£_7 /q\ 



a dx 5 + D <ta 4 ^y + rf^dy 8 + U <ta 2 cfy 3 + ^ <fy* dy 5 ~ n '^> 



d^z d 4 z 

 Putting r, s, &c v for the differential coefficients -r-^ , 3 , , 



F2 



