398] 
we find 
and thence 
that is 
AND SEXTIC SURFACE CONNECTED THEREWITH. 
t*v= A' 2 - fx' 2 — v' 2 + 2/a V, 
v\=- A' 2 + /¿' 2 - v 2 + 2v'X', 
A/x = A 2 — ¡x 1 + v + 2\'fx, 
HV + I/A + A/a = - (V 2 + /a' 2 + v ' 2 - 2fi'v' - 2i/V - 2AV), = 0, 
93 
^ + i + i=o, 
K fl v 
the relation which connects the new constants A, ¡x, v. Moreover 
yz —xw = 4 (?/'/ — ¿c V), 
zx —yw = 4 {z'x - y'w'), 
xy — zw = 4 («i/' — /w'), 
3w 2 — x 2 — y 2 — z 2 = 8 (yV + z'x' + x'y + a/w' + y'w' + zV), 
2 
and writing for greater convenience 0 = the equations are transformed into 
0\x = xw — yz, 
0/iy — yw — zx, 
0vz = zw — xy, 
20 (A + ¡x + v) w = Sw 2 — x 2 — y 2 — z 2 , 
where 
1 1 , 1 A 
- H h - — 0, 
A /x v 
viz. these equations, eliminating 0, give the equations of the nodal curve. 
From the first three equations eliminating 0, we deduce 
yzw (fx — v) = x (fxy 2 — vz 2 ), 
zxw {v — A) = y (vz 2 — \x 2 ), 
xyw (A — ¡x) = z (\x 2 - fxy 2 ), 
or, as these equations may be written, 
x 2 (fxy 2 - vz 2 ) y 2 (vz 2 - \x 2 ) z 2 (Xr 2 - fxy 2 ) 
'°"V=-JZr~ = ~^=A A-/a ’ 
which equations, from the mode in which they are obtained, are it is clear equivalent 
to two equations only. Using the fourth equation, and eliminating 0 by substituting 
therein for 0\, 0fx, 0v their values from the first three equations, we find 
2 w 2 (S w-^- -- a f) = 3w 2 -x 2 -y 2 -z 2 , 
\ X y Z J