[490 
491] 25 
V = bx + b'y 4- b"z, then 
V = 0 lie in a line 
i, that is the point 
the condition reduces 
= 0, 
i number of pairs of 
7 = 0. 
- <yz, Q = a'x + 0y + 7'z, 
- 
^—^1) (271—1) 
> 
491. 
y, ...) may lie in lined 
writing (X, Y, Z) for 
ON THE QUARTIC SURFACES V, Wf = 0. 
■Ml—1) _ 0 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xi. (1871), 
pp. 111—113.] 
lined with one of the 
[•sections of the curves 
The general Torus, or surface generated by the rotation of a conic about a fixed 
axis anywise situate, has been investigated by M. De La Gournerie, Jour, de VÊcole 
Polyt., t. xxill. (1863), pp. 1—74. The surface is one of the fourth order, having a 
nodal circle ; and with its equation of the form 7 2 - UW = 0, consequently of the form 
in question. The leading points of the theory are as follows: 
lmnpq _ () 
Consider (fig. 1) the plane of the conic in any particular position thereof ; let this 
ith the curves £7=0, 
Fig. 1. 
O 
/N 
it have been expected. 
Q/ 
/ 
/ A 
M 
O' 
meet the axis of rotation 00' in the point M, and let the projection of 00' on the plane 
of the conic be MU. Take P any point of the conic ; draw PQ in the plane of the 
C. VIII. 4