9. The tangents to a curve at its infinitely distant points are called its _asymptotes_ if they pass through a finite part of the plane. Given the asymptotes and a finite point of a conic, to construct the conic. 10. Given an asymptote and three finite points on the conic, to determine the conic. 11. Given four points, one of which is at infinity, and given also that the line at infinity is a tangent line, to construct the conic. CHAPTER V - PENCILS OF RAYS OF THE SECOND ORDER *79. Pencil of rays of the second order defined.* If the corresponding points of two projective point-rows be joined by straight lines, a system of lines is obtained which is called a pencil of rays of the second order. This name arises from the fact, easily shown (§ 57), that at most two lines of the system may pass through any arbitrary point in the plane. For if through any point there should pass three lines of the system, then this point might be taken as the center of two projective pencils, one projecting one point-row and the other projecting the other. Since, now, these pencils have three rays of one coincident with the corresponding rays of the other, the two are identical and the two point-rows are in perspective position, which was not supposed. [Figure 19] FIG. 19 *80. Tangents to a circle.* To get a clear notion of this system of lines, we may first show that the tangents to a circle form a system of this kind. For take any two tangents, _u_ and _u’_, to a circle, and let _A_ and _B_ be the points of contact (Fig. 19). Let now _t_ be any third tangent with point of contact at _C_ and meeting _u_ and _u’_ in _P_ and _P’_ respectively. Join _A_, _B_, _P_, _P’_, and _C_ to _O_, the center of the circle. Tangents from any point to a circle are equal, and therefore the triangles _POA_ and _POC_ are equal, as also are the triangles _P’OB_ and _P’OC_. Therefore the angle _POP’_ is constant, being equal to half the constant angle _AOC + COB_. This being true, if we take any four harmonic points, _P__1_, _P__2_, _P__3_, _P__4_, on the line _u_, they will project to _O_ in four harmonic lines, and the tangents to the circle from these four points will meet _u’_ in four harmonic points, _P’__1_, _P’__2_, _P’__3_, _P’__4_, because the lines from these points to _O_ inclose the same angles as the lines from the points _P__1_, _P__2_, _P__3_, _P__4_ on _u_. The point-row on _u_ is therefore projective to the point-row on _u’_. Thus the tangents to a circle are seen to join corresponding points on two projective point-rows, and so, according to the definition, form a pencil of rays of the second order.