67 
and satisfies also the conditional equations (53), (54), is a 
root of the quartic (49), and a root of the quartic (49) is a 
solution of each of the equations 50 to 54. 
13. With a view, therefore, to obtain a solution of (53), 
(54), it will he necessary to try a few simple assumptions of 
the forms of t, It, s, which are suggested by the equations 
themselves — 
Put ^ = 1; ^ = 1 ; 
dy ax 
dR dR _ q 
dy dx 
.(55) 
(56) 
(57) 
By integrating these equations, then 
t = x + y + z x 
R = (y - x)z 2 
where z 1} z 2 are functions of 0 only. 
The above assumption, viz. (55), necessarily implies that 
S is a function of x, y only, or, = 0. 
Substitute the above values in (53), (54), then 
dS\aY 
dx) dy 
.(58) 
( T,+ » v+ <§)s-( T ‘-' v * 
If, then, the roots of the quartic (49) be such as to satisfy 
(60) 
( 61 ) 
V* + z 2 V + g = ° 
V2 -^ + l = ° 
These equations will satisfy (58), (59), providing 
-- 1 — 2z 
dz~ Z 2 
dS dS dz 2 
dy dx dz 
\y-x) 
,(62) 
,( 63 ) 
Since S is a function of x, y only, then 
dz 
dz 
dz 2 
-t-= -1 or, 
Integrating (62) (63), we have 
Z X - -z 2 
S = xy 
