386 PROCEEDINGS OF- THE AMERICAN ACADEMY 



t' is the pole, and Tj and u on that of which T is the pole ; but the cir- 

 cles of which t' and T are the poles are tangents to the supplementary 

 conic. Hence the theorem is established. 



22. The directrix, or director arc, is the conic polar of the focus. 

 We have, as a particular case of Art. 21 : The arc joining the focus to 

 the conic pole of any arc passing through the locus is perpendicular to 

 the latter arc. 



Also every tangent to a conic and the arc joining its point of con- 

 tact to the conic pole of a cyclic arc meet that cyclic arc in two points 

 a quadrant apart. 



By the theory of projections, explained in Art. 21, we have: The 

 arcs drawn from the focus of a conic to the point of intersection of 

 two tangents, and to the point where the arc through the points of 

 contact meets the director arc, are at right angles to each other ; or, in 

 other words, if any chord rp' cut the directrix in d, then fd is the 

 external bisector of Z!pfp'. 



Then, by the same reasoning as in Art. 21, the converse of this can 

 be proved. The arc passing through the conic pole of a cyclic arc 

 and through the point of intersection of two tangent arcs meets the 

 cyclic arc at a point a quadrant distant from that at which the cyclic 

 arc is met by the arc joining the points of contact of these tangents. 



23. The angle subtended at the focus by the part cut off on a vari; 

 able tangent by two fixed tangents is constant. This is a right angle, 

 if the two fixed tangents intersect on the directrix. If through two 

 fixed points on a conic two arcs be drawn intersecting in a third point 

 of the conic, they intercept a constant segment on the cyclic arc. This 

 is a quadrant, if the arc joining the two fixed points passes through 

 the conic pole of the cyclic arc. 



Also, from the corresponding property of plane conies, the sum of 

 the cotangents of the segments of a focal chord is constant. Recipro- 

 cally : — If, from a point upon the cyclic arc, tangents be drawn to a 

 conic, the sum of the trigonometric cotangents of the angles which 

 they make with the cyclic arc is constant; for these angles are meas- 

 ured by the segments of the focal chord of the supplementary conic. 



The rectangle under the tangents of the segments of a focal chord 

 is proportional to the sum of the tangents of the segments. 



If, from a point upon the cyclic arc, tangents be drawn to a conic, 

 the product of the trigonometric tangents of the angles which they 

 make with the cyclic arc is proportional to the sum of the tangents of 

 the angles. 



