AND PARTIAL DIFFERENTIAL RESOLVENTS. 49 



From these two equations it follows that 



SdU dS\dV_/dU dS\dY 



\dy dy J dx~~\dx dx ) dy' ^ ' 



Equation (34), which is a partial differential equation, is 

 the first partial differential resolvent of the cubic (33) to- 

 gether with the conditional equation (36) . 



Hence it follows that the value of V which satisfies (34), 

 and satisfies also the conditional equation (36), is a root 

 of the cubic (33) ; and each of the roots of the cubic (33) 

 is a solution of the partial differential equations (34), (35), 

 and (36). 



10. Since R, S are arbitrary functions of x, y, it remains 

 to determine them so as to satisfy the equations (34), (35), 

 and (36) when 



V=x + y+u, (37) 



where a is a constant quantity. 



Substitute V as given in (37) in (36), then 



Now, if the first term of (38) is a quadratic in terms of 

 x> y, then the second term must be a quadratic also. This 

 readily suggests the following equation, viz. 



G?R ttR r\t \ 1 \ 



■*"&=*(—'> (39) 



Substitute this value in (38), and it becomes 



ty~~dx =l3yZ ~ /3 * Z + 0ll3y ~~" 13 ^ ' ' (40) 

 The integrals of (39) and (40) are 



B,=fay + C, (41) 



3S=/3^ + ^ ? -3«/3a?2/ + 3C I ;. - . (42) 

 where C, C, are independent of x and y. 



SER. III. VOL. VIII. E 



