50 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
rantal transformation), the correspondence depending on the conic of 
reference. 
13. If tangents are drawn at the vertices of the hexagon, figure 5, 
the intersection of the tangents at D, F; and A, C, will be on the polar 
of M, so for L and N; so that their intersection will be the pole of the 
Pascal’s line. This is Brianchon’s theorem, and may be thus enun¬ 
ciated: if the opposite vertices of a hexagon, whose sides touch a 
conic, be joined, the joins pass through one common point, the pole of 
the corresponding Pascal’s line in the inscribed hexagram. Conversely 
Pascal’s theorem may be deduced from Brianchon’s. 
From the above, since the conic and its polar reciprocal are gene¬ 
rated in precisely the same manner, either by points or enveloping 
lines, the reciprocal curve is a spherical conic, and all the properties 
proved for this curve hold for the spherical ellipse. 
Cor.: The reciprocal of one conic with respect to another conic is a 
conic. 
14. The Conjugate Triad. The six joins of 
four points determine three points called a con¬ 
jugate triad, and it a conic is drawn through the 
four points, any one of the three points is the 
pole of the line passing through the other two. 
Thus, figure 7. A is the pole of BO, etc. 
In figure 7, aKbAAdK'cA/\cK'dA. Such a row 
is called harmonic. Thus the conic, the pole, and 
11 its polar divide any line through the pole in a 
harmonic range. 
When the pole is at the center of the conic the polar is the trigo¬ 
nometric polar of the center, and the sect aK=Kb, while KA is equal to 
a quadrant. Conversely, in a harmonic range, abed, if ab=bc, bd is 
equal to a quadrant, and in a four-ray pencil 0. abed, if angle aOb— 
angle bOc, angle bOd is measured b} a quadrant. 
15 . By—[8] : if a tangent is drawn from the polar of the focus to 
the conic, the part intercepted between the curve and the polar of the 
focus (directrix) subtends a right angle; also, if a line be drawn cut¬ 
ting the curve and the directrix, and the focus be joined with the three 
points of intersection, the join of the focus and the point on the direc¬ 
trix bisects the angle between the other two. 
In figure 7. If the angle at O be a right angle and turn about O 
while Ad and Aa remain fixed, the angular velocity of B about 0 is 
double the angular velocity of the sides of the right angle. 
