66 Mr. R. Moon on the Integration of Partial 



or 



= E«/ 6 '( A -) + U/»'(y) + Va/ZOh), 

 + * 1 E "^ "TUT" + U dy ~BT /* ^» 





But s and t can only appear in this last equation as parts of 



Q( TT 



the symbol fa or r-\ =p s + at : hence the coefficients of 



s and t in the last equation must separately =0 : or we must 

 have 



o =R 4* +u *', 



a# ay 



n t-. d Sa-U , TT rf Sa-U 

 = Ea^ =p hlii =p ; 



which are the conditions to be satisfied in order that the given 

 equation may admit of a first integral : and which being 

 satisfied, we shall have for the final and complete determina- 

 tion off the equation 



o=B«/ 4 'H+t[/i%) +v«//oo- 



We may take as the auxiliary equations for the solution of 



this last, 



Ya 



= ~Rotdy — XJda : Q=dpb 1 ~- -jr-dy; 



from the first of which we may always derive 

 F(#, y) = c, a constant. 

 By means of this last equation we may always eliminate x 

 from -pf • ^ e t E\(y, q) be the result of such elimination; 

 then the second auxiliary equation will stand, 



= dfi 1 — F 1 (y,c)dy; 

 whence we derive 



const. =fa-$F 1 {y,c)dy, 



c having to be regarded as constant in the integration. Hence 

 we get finally for the first integral of (1), 



IH-^ty,e)dy=4>(F(x,y)}, 



or 



Q( TT 



