ME. A. CAYLEY’S SIXTH MEMOIE UPON QUAHTICS. 
79 
points (ineunts of the conic), and the harmonic in relation to these two points of the 
given point has for its locus a line which is the polar of the given point. The polar 
passes through the points of contact of the conic with the tangents through the given 
point. 
In hke manner considering the conic and a line ; from any point of the line we may 
draw two tangents to the conic, and the harmonic of the given line with respect to the 
two tangents envelopes a point which is the pole of the given line. The pole is the 
point of intersection of the tangents of the conic at the points of intersection with the 
given line. 
The polars of the several points of a line envelope a point which is the pole of the 
line ; and the poles of the several lines through a point generate a line which is the 
polar of the point ; and this proposition shows how the theory of poles and polars gives 
rise to a theory of reciprocity. 
201. If the point-equation of a conic be 
(a, b, c,f,g, hXir,g,zf=0, 
the point-equation of the polar with respect to this conic of the point (of, y\ z') is 
{a, h, c,f, g, hjx, y, z^cd, y\ z)=0. 
But it has been seen that the line-equation of the same conic is 
and the line-equation of the pole with respect to this conic of the line (|', Z') (that is, 
the line whose point-equation is ^'x-\-yi'y-\-Z!z = 0) is 
(91, 3$, c, #, 0, mi ar, r)=o, 
in other words, the point-coordinates of the pole are 
202. If U=0, V=0 be the point-equations of any two curves of the same order, 
then X, gj being arbitrary coefficients, 
is the equation of a curve of the same order passing through the points of intersection 
(common ineunts) of the two curves ; such curve is said to be in involution with the given 
curves. The discussion of the general theory of involution is reserved for another occasion. 
203. In particular, if U=0 be the equation of a conic, and P=0, Q=0 the equations 
of two lines, then 
U+XPQ=0 
is the equation of a conic passing through the points of intersection of the conic with 
the two lines ; and if the two lines coincide, then 
U+?iP^=0 
is the equation of a conic having double contact with the conic U=0 at its points of 
intersection with the line P=0. Such conic is said to be inscribed in the conic U=0; 
M 2 
