52 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
In figure 9. A and B are fixed points, P 
■d movable, and F is the focus. 
Angle AFC=| supplement of PFA (14.) 
and angle BFD=i sup. of PFB; so that angle 
CFD=| angle AFB. 
Fig- 9 - The reciprocal is: if a variable tangent 
intersect two fixed tangents, and the points of intersection be joined 
with the pole with respect to the conic of the cyclic line, these joining 
lines determine an invariable sect on the cyclic line. 
p 
19. In figure 10. Let P be a point and Q 
its quadrantal polar. If SL is a quadrantal 
sect always passing through R, and having 
its extremity on QS, the polar of P, L will 
trace a spherical conic passing through P and 
R, and whose cyclic lines are the quadrantal 
^ g polars of P and R. Drop a perpendicular 
Fig lo. on QS from P, it will intersect at L, for SL 
would be equal to PS (a quadrant). Thus the locus of the vertex of a 
right-angled triangle whose base is fixed is a spherical conic. 
Reciprocating this last, if a quadrant sect slide between any two 
lines it envelopes a conic. 
By means of [11], it is possible to solve a number of problems re¬ 
lating to the construction of polygons inscribed in polygons and conics 
whose sides shall fulfill certain conditions. The celebrated problem 
of Pappus, generalized by Cramer and Poncelet, to inscribe in a conic 
or a polygon of w sides, a polygon of n sides, the sides of which shall 
pass through n given points taken in any assigned order. 
20. In figure 11, taking for simplicity the triangle, 
let A, B, C be the three points. Choosing any point 
at random on the curve, as (1), draw a line through 
(1) and A to cut the curve at s; draw through j and C 
to cut the curve again, and so continue with all the 
points in the order assigned, the last one cutting 
at (IQ. Determine thus three points (l 7 ) (2 7 ) (3 7 ), 
and find a fourth, such that 0.1 / 2 / 3'4/\’0. 1234, (4) being a double 
point in the series 12341 / 2 / 3 / 4, is a vertex of the polygon. There are 
generally two solutions. 
DESCRIPTION OF CONICS BY LINES AND POINTS. 
21. Besides the methods already described (2, 3, 4, 11. . ..), for the 
generation of a conic, the method of Newton is easily deduced from 
the property of projective pencils. Two constant angles turn about 
vertices A and B, if two of the sides intersecting trace a conic through 
A and B, the other two will trace another through A and B. 
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