65 
From these two equations it follows that 
/rf R 
vdV _ 
fd R 
, v+ ^' 
vdY 
dyj 
’ dx 
\dx 
dx, 
Idy' 
Equation (34), which is a partial differential equation, is the 
first partial differential resolvent of the cubic (33) together 
with the conditional equation (36). 
Hence it follows that the value of Y, 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 + a (37) 
where a, is a constant quantity. 
Substitute Y as given in (37) in (36), then 
/dR dRY x dS dS A /OON 
K^-^) iX + y + a) + dy-dr = (> < 38 > 
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. 
f < 39 > 
Substitute this value in (38), and it becomes 
dS dS „ 9 
j3x 2 + afy - a/3% (40) 
The integrals of (39) and (40), are 
R = fixy + C (41) 
3S = fiy s + (3x 3 - 3a (3xy + 3Ci (42) 
where C, Ci are independent of x and y. 
The values of /3, a, C are determined from (34), and are 
as follows, a= - a; /3 = - 1, and C = a 2 . 
The constant 3Ci = a 3 is found by substituting Y, R, S in 
the cubic (33). Hence, 
Y = x + y -a (43) 
is a root of the general cubic. 
