68 
Substitute these values in (60), (61), then 
Y*-zY + x = 0 (64) 
Y 2 + zY + ?/ = 0 (65) 
And, t = x + y - z 2 
R = z{x- y) 
14. The values of Y in (64), (65), which satisfy (50) also, 
are the roots of the quartic 
V 4 + (x + y - z 2 ) V 2 + z(x - y)Y + xy = 0 (66) 
Equation (66) will coincide with the quartic 
Y 4 + aY 2 ±6Y + c = 0 (67) 
If x , y, 0 are determined from 
x + y-z z = a (68) 
z(x - y) = +6 (69) 
(70) 
These values depend upon the well known cubic 
z 6 + 2az 4 + (a 2 - 4c)z 2 - 6 2 - 0 (80) 
See Todhunter’s theory of Eqs., p. 112, 2nd ed. 
Havant , Sept., 1881. 
Postscript.- 
which gives 
-Equation (6) is made linear by the relation 
4a 3 4- 27m 2 dz 
Gand-Sma 1 zdx 
y= 
(81) 
dh d , /(4a 3 + 27m 2 ) 6 Wz /2am 1 - SmedY' 4 A 
' M W-3 ^* j^ + 6g ( + 27m* > = Q -( 82 ) 
c^ 2 + dx 
