M. B. PORTER—SPHERICS. 
51 
16. If a right angle turn about a fixed 
point P, and cut two fixed straight lines 
EM and EM', successively assuming 
the positions QPQ, STS and RPR', the 
points QRQ/R'M'M, where M and M 7 , 
Q and Q 7 are the points of intersection 
of the legs of the right angle in its first 
position, will be on one conic, because 
Q'. R'M'MR/\P. R'SQ'R7\P.R'QS'R,a 
Q.R'M'MR, 1 ° . This result shows that 
if a right angle turn about any point on 
a sphere, it intersects a spherical conic 
in four points, whose joins touch another spherical conic (II. 3°). 
The vertex of the right angle will be the focus of the second conic by 
7, while the directrix, which passes through E and the intersection of 
QQ' and MM 7 , is the polar of P with respect to the conic QQ'MM’RR'. 
2°. If P is at the center of the conic, the envelope is a circle 3 ° if 
P is a point on the curve, the envelope becomes a sect which connects 
P and a certain point through which all the subtended chords pass. 
If the right angle turn so that one leg is tangent, the other is the nor¬ 
mal, so that this property affords an elegant method of drawing a nor¬ 
mal at any point of the curve. The reciprocal theorems are 1° if a 
quadrant be displaced along any line, and tangent drawn from its ex¬ 
tremities to a conic, they intersect on a conic having this line for its 
cyclic line, which has the same pole with reference to both conics. 
2 ° . When the cyclic line of the second bisects the angle between the 
cyclic lines of the first conic the locus is a circle; 3 ° when it touches, 
the first conic the curve becomes a geodesic. 
1 7 . By 13 Cor. The reciprocal of a conic with respect to another 
conic is a conic, and it is evident that there will be a point which will 
have the same polar with respect to all three, hence the reciprocal of 
one conic with respect to another having the same focus and directrix 
is a confocal conic with the same directrix. It is not difficult to prove 
that the chord of contact subtends a constant angle at the focus. 
This reciprocated yields: If a constant sect be displaced along the 
cyclic line, tangents from its extremities intersect on a conic, and the 
chord of contact envelopes another conic, all of which have the cyclic 
line in common. 
18. Many other interesting properties of the focus and directrix fol¬ 
low from the foregoing. 
