56 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
constant angle sliding around a circle generates a circle, while the 
chord of contact envelopes a circle which is concentric with the other 
two, may be thus stated: Any point in the plane being taken as the 
point which is to become the pole of the line which was the line of in¬ 
finity, in ■piano , if parallels be drawn to the sides of the constant angle 
they will intersect the sides on the line at infinity, and contain be¬ 
tween them a constant angle, while the sect intercepted on the line at 
infinity will be constant. If the plane be now supposed to assume 
a uniform curvature, the chosen center need not coincide with the ver¬ 
tex of the constant angle, because this property, on account of its 
graphical nature, must hold independently of the curvature. The 
line at infinity will now become the cyclic line, and the circles traced 
by vertex and enveloping line will become conics having the same 
cyclic line, which is the reciprocal of 17. Other results may be de¬ 
duced in the same way. But the close analogy between these pro¬ 
cesses and the methods of projection is best shown in the following: 
Two spherical conics have cyclic lines intersecting in A and B. Let 
P, the pole of AB, be taken as a center, and the hemisphere be sup¬ 
posed to grow indefinitely flat, the two conics will become similar. 
PROJECTION. 
10. (2°) shows that a spherical conic is the projection of two cir¬ 
cles, or that a cyclic cone has two circular sections, which are parallel 
to the cyclic lines on the sphere. 
7 shows that if sections of the cone are taken perpedicular to OF 
(the focal line), 0 being the center of the sphere; 0 F will be the locus 
of their foci. A spherical conic is then a curve, the intersection of a 
sphere and a cyclic cone, whose vertex is at the center of the sphere. 
The cyclic arcs are determined by planes drawn through the vertex 
parallel to the cyclic sections, and the foci are the intersection of the 
sphere and the focal lines of the cone. Many of the foregoing results 
are easily deduced by projection from the properties of the circle and 
plane conic. Thus the reciprocal of 17 is deduced by projection from 
equal chords in a circle envelope a concentric circle. Of course all the 
graphical relations are true for their projections. 
