52 MR. ROBERT RAWSON ON DIFFERENTIAL 



The above assumption, viz. (55), necessarily implies that 

 S is a function of x, y only, or 7- = 0. 



Substitute the above values in (53), (54); then 



( v * + ^+f)£=(!- v ' + t<»-^)|>> 



If, then, the roots of the quartic (49) be such as to satisfy 

 V*+* z V+ig=o, (60) 



these equations will satisfy (58), (59), provided that 



dt= 2g » (62) 



Since S is a function of x, y only, then 



dz 

 dz 



Integrating (62), (63) we have 



~r=— 1, or z z =—z. 



Z,= —Z Z 1 



Substitute these values in (60), (61), then 



Y z -zV + x=o, ....... (64) 



V* + zY + y=o (65) 



And 



t-=oc + y— z z y 



R=z(x-y). 



