ON THE QUARTIC SURFACES (*$Z7, V, W) 2 = 0. 
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about the axis 00' generates the torus. It thus appears that the same torus may be 
generated in three different ways by the rotation of a conic about the axis 00'. 
In the particular case where the plane of the conic passes through the axis, the 
meridian section consists it is clear of two symmetrically situate conics, intersecting the 
axis in the points T, T', which are nodes of the surface, the surface having as before 
a nodal circle generated by the rotation of the two symmetrically situate intersections 
S, S' of the two conics. The equation is included under the foregoing form, but it is 
at once obtained from that of the conic, 
(ax 2 + 2gx -t- c + by 2 ) 2 = 4y 2 (hx + f) 2 , 
by writing therein 2 for x and \J(x 2 + y 2 ) for y; viz. the equation of the torus here is 
{<az 2 + 2gz + c + b (% 2 + y 2 )} 2 = 4 (x 2 + y 2 ) Qiz + f) 2 , 
and the two nodes thus are x = 0, y = 0, az 2 + 2gz + c = 0.