398] 
AND SEXTIC SURFACE CONNECTED THEREWITH. 
91 
plane of the quadric surface touches the tangent cone of the cubic surface: to show 
this it is only necessary to observe that at the point («' = 0, y'= 0, z = 0) the tangent 
cone is y'z + zx + x'y' = 0, and the tangent plane is X'x' + yy' + v'z = 0, and that 
these touch in virtue of the above-mentioned relation V (V) + \/(y) + V (V) = 0. It 
follows that on the curve of intersection, or cuspidal edge of the developable, each of 
the summits is a cuspidal or stationary point, that is, the cuspidal curve has four 
stationary points; this agrees with the character of the curve as given, Salmon “ On 
the Classification of Curves of Double Curvature,” Carnb. and Dubl. Math. Jour. vol. v. 
(1850), p. 39, viz. the character is there given 
a = 6, in — 6, n — 4, r = 6, g = 3, h = 0, a = 0, ft = 4, x = 4, y = 6, 
(ft = 4, that is, there are as stated 4 stationary points). 
To find the equations of the nodal curve, instead of transforming the equations 
as given in terms of (a, b, c, d, e), it is better to deduce these from the equation 
of the surface; viz. if there is a nodal curve, we must have 
8 X J : 8 y J : 8,7 : 8*7 : 18J 
— 8 X J : hy'J : 8 Z J : 8 W J : I. 
Writing these under the form 8 X J + 8'8 X ’J — 0, «See., where 6' is regarded as an arbitrary 
parameter ( J ), we have 
\'w' + y!z + v y + 6' (y'z + y'w' + z'w') = 0, 
X'z + fiw' + vx + 6' (z'x + z'w' + x'w') = 0, 
X'y' + y!oc + vw + 6' {x'y' + x'w' + y'w') = 0, 
X'x' + g!y + vz + 8' {y'z + z'x' + x'y') = 0, 
which equations (eliminating &) must be equivalent to two equations only. 
I remark that the first three equations may be regarded as a set of linear 
equations in 1, w', 8', 8'w'; and determining from them the ratios of these quantities, 
we have, suppose, 
1 : -w : 8' : -8'w' = A : B : G : D, 
where 
A = 
X', y'z', y' + z' 
) 
B = 
y'z', y' + z', y'z' + v'y' 
y!, z'x!, z + x 
z'x', z' + x', vx' + X'z 
v, x'y', x + y' 
x'y’, x' + y, x'y' + yx 
G = 
y' + z', y'z' + v'y 
, v 
, D = 
y'z' + v'y', X', y'z' 
• 
Z + X , v x' + X'z 
> / 
v'x' + X'z', y, z’x! 
a + y'> x 'y' + y' x 
X'y'+y'x, v, x'y 
1 The value of 6’ is in fact = - 1 ^'-, that is, instead of the four equations involving an arbitrary 
parameter 0\ we have really four determinate equations. 
12—2