50 MR. ROBERT RAWSON ON DIFFERENTIAL 



The values of ft, ol, C are determined from (34), and are as 

 follows, a = — a ; /3= — 1, and C = a\ 



The constant 3C I =a 3 is found by substituting V, U, S 

 in the cubic (33) . Hence, 



\=x + y-a (43) 



is a root of the general cubic. 



V 3 + 3«V Z + 3(« Z — xy)V-x 3 — y 3 — $axy + a l = o. . (44) 



The remaining two roots must be obtained from 



V z + (x+y + 2a)Y + x z + y z — xy + ax -{- ay + a z = 0. .(4.5) 



11. The values of a?, y in (44) can be found by a qua- 

 dratic so as to make (44) coincide with the classical cubic 



V 3 + 3aV* + 36V4-c=o (46) 



For this purpose the equations 



a z -xy = b, (47) 



x z +y z -\-2axy = d i — c . . . (48) 



will be necessary. 



From these two equations there results 



_7,ab — 2a}—c 1 



x ' = ~ ^ + 2 s/(3ab~2a'-c) 2 -^(a z -by, 



?ab—2a 3 — c 1 



y* = 5 - -4/( 3 ab-2a*-cy-4(a z -by. 



The solution of a quartic by a partial differential resol- 

 vent with three independent variables : — 

 12. Let 



V* + fV* + RV + S = o .... (49) 



be a quartic in which t, R, S are functions of the three 

 variables x, y, z. 



