48 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
The part of any third tangent intercepted between these two fixed 
tangents subtends a constant angle at the focus equal to angle TFP. 
8. If F'T is produced so that TK=FT, triangles FPS and F'PK are 
equal, PF=PK, PS=PF'and FS=FT'+T'S=FT / +T'F'=F'K^AA' 
Thus angles FPS and F'PK are equal, and since FPF'is common, 
SPF=KPF, or TPF=T'PF'. 
9. From 8: If a system of conics touch two lines and have one 
focus on a third passing through the intersection of the tangents; the 
locus of the other focus is determined by 2 ° . 
10. For the reciprocal curve , 1 ° above becomes: the intercept of 
any two tangents on the cyclic lines is bisected by their chord of con¬ 
tact. 2 ° becomes: if, from any two fixed points on the curve, lines 
be drawn to a variable third also on the curve; they cutoff a constant 
sect on the cyclic arcs. 
Hence five points determine the reciprocal curve; for two determine 
the reciprocal curve. 3° becomes: in any secant, the two short (long) 
sects cut off between the curve and the cyclic lines are equal. Figure 
14. an=n'r. 
II. 
If any four points on the polar reciprocal are joined with a variable 
fifth point also on the curve, these lines determine on the cyclic lines 
two ranges of four points each, which are said to be perspective. Dis¬ 
place one of these ranges along its line, and the range on the other line 
may be so displaced that it will be perspective with the other (by 10. 
2°). Further, if the range on the first cyclic line be arbitrarily 
chosen, and three elements of the range on the other be similarly 
taken, the fourth element is determined by the perspective relation, 
and will remain invariable, no matter how either range be displaced 
along its cyclic line. 2 °. Let the polar of the perspective-center 
be drawn. It will make with the cyclic lines angles measured by the 
sect between the perspective-center and the poles of the cyclic lines. 
If lines, symmetrical to the cyclic lines with respect to the polar of 
the center of perspective, be drawn, these ranges will be projected into 
equal ranges on these symmetrical lines. If a reciprocal curve be 
traced through the center of perspective having for cyclic lines the 
lines symmetrical to the former ones, it may be shown, as before, that 
having assumed seven sects, the eighth is determined by the perspec¬ 
tive relation. By 2° it is possible to vary the angle between the per¬ 
spective ranges through all values, so that two projective rows will be 
placed in perspective, when three elements of one are perspective to 
these elements of the other, no matter what the angle is between them. 
The functional relation which subsists thus evidently between these 
sects is called by Clifford their cross-ratio , by Chasles their anhar- 
