ON A SEXTIC TORSE. 
69 
[652 652] 
two half-bianches of the curve coincide together with the portions which lie outside 
the cylinder x 2 + y-=l ) in fact, the portions referred to above, of the nodal curve in 
the plane in question; the portions which lie inside the cylinder are acnodal or isolated 
cm ves without any real sheet through them. It may be added, in the way of 
general description, that the section of the surface by any cylinder a? + y 2 = c 2 (c > 1) 
is a curve of the form z=Gcos(26 + B), 6 the angle along the base of the cylinder 
from the intersection with the axis of ¡s; G, B are functions of c; viz. we have for 
the two half sheets respectively 
z=Gcos(26 + B) and z = C cos (26 - B), 
each curve having thus the two maxima + G, and the two minima —G; and the two 
curves intersect each other at the four points in the two principal planes respectively; 
viz. the points for which 6 = 0, 90°, 180°, 270°, and z=GcosB, -GcosB, GcosB, —CcosB 
accordingly. 
Proceeding to discuss the surface analytically, we have for the equations of a 
. (1877), 
generating line 
x — cos <6 y — sin <6 z — COS 2d) 
-sin* = cos* =-2sin2*' = i> sup P° se ' 
or say 
defined by 
X = COS (f) — P sin (f), 
3nt one for 
y = sin (f) + P COS (f), 
it of view. 
z = cos 2(f) — 2p sin 2(f), 
-2/2 = 1 with 
which equations, considering therein p, cf> as arbitrary parameters, determine the surface. 
g its vertex 
1 
r the curve 
on the line 
Writing x = 0, we find y — > and then £ = ~ 3 + 2 sin 2 0, viz. we have 
2 
is a sextic 
x = 0, z = — 3 + —, for section in plane yz; 
regarding (f) 
and, similarly, writing y = 0, we find x — ^ , and then z— 3 — 2 cos- <f), viz. 
section with 
2 
here are in 
and in the 
y — 0, z = 3 —- for section by plane xz. 
7 he sections 
By what precedes, these are nodal curves, crunodal for the portions 
irface; they 
(y = ±l to + oo, z = -l to - 3) and (x = ± 1 to ± oo, z- 1 to 3) 
; 1 tO ± OC , 
respectively, acnodal for the remaining portions y < ±\, x < +1 respectively. 
y increases 
Writing x = rcos6, y — r sin 6, so that the coordinates of a point on the surface 
are r, 6, z, where r = \J(x 2 + y 2 ) is the projected distance, 6 is the azimuth from the axis 
5. The two 
of x, and z is the altitude, w T e have 
these nodal 
r COS 6 = COS (j) — P sin (f), 
me through 
1 such that 
r sin 6 = sin <f) + P cos (f>, 
, y = 0, the 
z = cos 2(f) — 2p sin 2(f>.