520 
ON THE CONIC TORUS. 
[615 
A surface of the last-mentioned form is 
mz + *J(xy) + \/(w 2 — z-) = 0, 
viz. this has the nodal conic z=0, xy — tv 2 = 0, the nodes [x = 0, y = 0, (to 2 + l)z 2 — w 2 = 0}, 
and (z = 0, tv = 0, x = 0), (z — 0, tv = 0, y = 0), and the tropes x = 0, y = 0, z + tv = 0, 
z — tv = 0; but the planes z + w — 0 and z — w = 0 are ordinary tropal planes each 
touching the surface in a proper conic; the planes x = 0, y= 0 special planes each 
touching along a line-pair. 
Fig. l. 
B 
A 
C 
B 
The equation in question, writing therein w = 1 and x + iy, x — iy in place of (x, y) 
respectively, is 
by the change of x into V(A 2 + y 2 ) 1 and the surface is consequently the torus generated 
by the rotation of the conic (x + mz) 2 = 1 — z 2 about its diameter. Or, what is the 
same thing, the surface 
mz + \/(xy) + V(w 2 — z 2 ) = 0, 
regarding therein (x, y) as circular coordinates and tv as being = 1, is a torus. The 
rational equation is TJ = 0, where we have 
U = {(mi 1 -|-1) z 2 — tv 2 + xy} 2 — 4im 2 z 2 xy 
= [xy + (1 — to 2 ) z 2 — tv 2 ] 2 + 4m 2 z 2 (z 2 — w 2 ) 
= a?y* + (m 2 + 1 ) 2 sA + w 4 + (2 — 2 to 2 ) z 2 xy — (2 + 2m 2 ) zhv' 1 — 2 xyw 2 . 
I find that the Hessian H of this function U contains the factor xy + (1 — to 2 ) z 2 — w 2 , 
viz. that we have 
H = [xy + (1 — m 2 ) z 2 — tv 2 ] H’,