30 ROYAL SOCIETY OF CANADA 
To find tangents at C and D, take C and D as radiant points and 
C A, D Aas base lines. Then projecting B from C and D we get Bo and 
By; and projecting # from C and D we get Ecand Ep. The intersec- 
tion K,of Be Bp and Ee Ep, gives the intersection of the tangents at 
Cand D. 
Again, taking B and C as radiant points, and A B, A Cas base lines, 
and projecting Æ from these points, we get Ze, Ez, which are the same 
as Cr, Ca. Hence the intersection of tangents at B and C lies on Cy Cz, 
on which also lies the intersection of the tangents at À and #. Similarly 
the intersection of tangents at C and Æ lies on C4 Cr, on which also lies 
the intersection of tangents at A and B. 
We thus see that if a quadrilateral be described about a conic, the 
intersection of lines joining opposite points of contact (B # and A ©) is 
also the intersection of lines joining opposite intersections of tangents 
(G Land FM). 
Next consider the five tangents (at À, B, C, D, E) as rays of a sheaf 
of the second order. Take F and G as radiant points and F# P, G P as 
base lines. Sheaves from F' and G will be in perspective. But G Z and 
F M intersect at C4, and F H and G JN intersect at D4. Hence B E is 
the line on which corresponding lines from F and G intersect, and B and 
E are “ points of contact.” 
Hence when five points are given (which uniquely determine a curve 
of the second order), and the construction for tangents at these five 
points is made, the five tangents, regarded as the basis of a sheaf of rays 
of the second order and uniquely determining the sheaf of rays, have for 
‘ points of contact ” the five original points. 
Again, À and B being radiant points, and À #, B E base lines, if 
any ray, g, be given, we construct for g,, the ray corresponding to g, and 
so get À, a sixth point on the curve of the second order. 
To construct tangent at À, take A and # as radiant points, and À B, 
R B as base lines. Then projecting Æ on these lines from A and À, we 
get Qand Hp. The intersection of Q Hr with the tangent at A gives #, 
and enables us to draw the tangent at A. 
But this tangent (at R) is a ray of the set of five tangents, viewed 
as a sheaf of rays of the second order. For it is also got by projecting Q, 
a point on A # (which is the line joining “ points of contact” on F &, 
P G), from P and F on the base lines F G, P G, since P, Q, X are in 
the same straight line, and also F, Q, Y in the same straight line. 
It has thus been proved that the “tangents ” at points of a curve of 
the second order form a sheaf of rays of the second order, and the points 
at which the tangents are drawn are the “ points of contact ” for the rays 
of this sheaf of the second order. 
