46 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
Besides the axioms laid down in Dr. Halsted’s Geometry ( Spherics ), 
the following principles are made use of. 
Defining the polar reciprocal of a curve to be the curve enveloped by 
the lines quadrantally polar to every point on the given curve; the 
area of any curve is equal to the area of the hemisphere, less the scalar 
perimeter of the polar reciprocal multiplied by a steradian. 
This follows from the theorem in elementary geometry concerning 
polar polygons, by supposing the vertices (indefinitely near together) 
to be situated on the curve, whose area it is desired to measure. As 
the points approach coincidence they would form a polygon more and 
more nearly equal to the required area, and their polars would en¬ 
velope a curve more and more nearly equal to the polar reciprocal. 
Besides the point to line correspondence, there is a point to point 
correspondence also, for if two tangents, indefinitely near to each other, 
intersect, they determine a point corresponding to the polar of this 
tangent. That the polar of a circle is a circle is easily proven from the 
most elementary consideration, such as symmetry or congruent tri¬ 
angles. The remarks on projection have been added appendically to 
explain the terms cyclic line and conic. 
SPHERICAL CONICS. 
I. 
1. In this paper, the stim of any number of sects means, the sect 
between the extreme ends of the first and last when they are all placed 
end to end along any line so that none overlap. 
2. A spherical conic is the locus of a point which moves in such a 
manner that the sum of the two sects connecting it with two fixed 
points (foci) shall be equal to a constant sect. If the symmetrical 
conic be constructed by producing the focal sects one-half line, its foci 
will be <P and <P\ and these points will be situated on the line through 
F and F', the foci of the first, and one-half line from them. Both curves 
may be regarded as constituting one locus. 
In figure 1, FP-j-FT^a con¬ 
stant and FPx^P are equal 
to one-half line. Therefore 
the difference between FT* 
and <PP is constant. 
With respect to the foci F'and 0, the locus may be regarded as a 
spherical hyperbola. While from the above considerations it is clear 
that the locus is symmetrical with respect to four centers midway be¬ 
tween F, F 7 , ( I>, and with respect to three axes the line through 
FF 7 , $$ 7 and the perpendiculars to it at the midpoints of FF 7 and F$ 7 . 
