68 Mr. R. Moon on the Integration of Partial 



&c., and assuming (9) to be satisfied by f(xyrstuv) = 0, we 

 shall find the conditions of (9) having a first integral to be the 

 four following equations, viz. : — 



= Ba- T -+W^-' 



dx ay 



"~ dx R dy R 



° = Ra ^ IP +W Ty W ' 



n j, d V-W , w d V-R« 



where a= .,; ^ is one of the roots of the equation 



/ M 



= RV - R 3 Y. « 4 + R 2 U W. a 3 - RT W 2 . cc 2 + W 3 S . * - W 4 ; 



and the solution, when such exists, will be 



Sa-W a RT.« 2 -SW.a + W 2 V-Ra 



^ + -R^"' S + RV .^_^ r -. aM +^ 



=^{yc)dy + <t>{F(x,y)}; 



where F(<^?/) = c is the integral of 



= Ra,dy-Wdx; 



and Fi(y, c) is the result of eliminating x from the quantity 



™- by means of the equation ¥(ay) = c. 



Pursuing the same method, I have already (in effect) shown 

 in the pages of the Philosophical Magazine that the condition 

 of the equation 



R £ +s ^| +T |= v • • • ( io > 



admitting of a first integral,, in other words of its being inte- 

 grable by Monge's method, is that we have 



ax ay 

 where « satisfies the equation 



= R* 2 -Sa + T; 

 and when this holds, the first integral of (10) will be 

 p + cq = j ¥ 1 <j/c)dy + <f> { F(xy) } , 



