EOKM OF TANGENTIAL EQUATION. 
371 
This is the analytical statement of the theorem we get by projecting that of the last 
article ; and since in projection the line at infinity becomes a finite line, it may be 
expressed as follows: — Being given a curve of the third degree, U=0, and two fixed 
lines L and N ; then if O, O' be two variable points on L and N respectively, such that 
the polar line of O with respect to U passes through O', the envelope of the line 00' 
is a curve of the third class. 
10. Since the highest power of v contained in equation (14) is the first, the tangen- 
tial cubic A, which it represents has one double tangent, namely the line joining the 
points X and p, which we may call the line (Xf). Similarly the line (pv) is a single 
tangent. The same thing can be shown geome- 
trically, as follows: — Let the lines L and N in- 
tersect in C, then C is the point whose equation 
is gj=0. Now since, the polar line of O passes 
through O', then the polar conic of O' passes 
through O ; but this conic intersects the line L 
in two points, and the line joining O' to each of 
them is a tangent to A x . Hence from any point 
of the line CN can in general two tangents be 
drawn to A x ; and we shall see immediately that CN itself is a tangent. This agrees 
with the fact of the curve being of the third class. Let the polar conic of C intersect 
CL in the points H, H', then the lines CO, CO' are tangents to A : ; in other words, CL is 
a double tangent, and it is plain that O, O' are its points of contact. Again, let the 
polar line of C intersect CN in H, then H is a point of contact, so that CN is a tangent. 
11. Since the point O' moves on CN, its polar conic will pass through four fixed 
points, namely, the four poles of CN with respect to U. Hence any line will be cut in 
involution by the polar conics of the points O' ; and we have the following theorem : — 
If from any three points in CN three pairs of tangents be drawn to A 1? these will meet 
its double tangent in six points in involution, and the two points of contact of the double 
tangent belong to the involution. 
12. We find the limiting points of the involution as follows: — Let the pole conic of 
the line CL with respect to U intersect CN in the points S, S'; then since the pole- 
conic is the locus of points whose polar conics touch CL, the polar conics of the points 
S, S' will touch CL. Let the points where they touch it be denoted by A, A', then 
A, A' will be the double points of the involution. Or thus, the double points will be 
the points of contact of the two conics, which can be drawn throuah the four poles of 
CN to touch CL. 
13. From the last article, it is plain that each of the lines SA, S' A' is a pair of 
coincident tangents to the curve A : ; and since CN is itself a tangent we see that from 
each of the points S, S' can be drawn only two tangents to ; but the curve is of 
the third class, therefore it must pass through S and S'i Hence we have the folio wirg 
theorem : — 
