of Surfaces of the Second Order. 439 



meeting the cone in the points r t', the rectangle m r x m 't 

 = square of the distance of m from the vertex of the cone. 

 Hence if a sphere is described through the circular base and 

 vertex of a cone, the tangent plane to the sphere at the vertex 

 of the cone is parallel to the second circular section of the 

 cone. 



Prop. V. — From the four foci of a surface of the second 

 order, let fall perpendiculars on a tangent plane ; multiplying 

 together the perpendiculars from those foci which are situated 

 on the same diameter and taking the sum, we have 



{Fit" ~\ 



b l sin 2 v H r cos 2 v V 9 v denoting the 



angle the perpendiculars make with the axis of Z. 



When (2) is a surface of revolution round the transverse 

 axe P = P', p = p', b = c and u = a; hence Pp = b°\ 



When the surface is an elliptic paraboloid, P and P' are 

 infinite, and we obtain 



{p + P 1 ) cos X = / sin 2 v + /' cos 2 v. 

 X being the angle the perpendicular makes with the axis of X. 



Prop. VI. — Through'any point of a surface of the second 

 order, let two cords be drawn, passing through the extremities 

 of the cord of the surface Q Q', which joins a pair of conju- 

 gate foci, and meeting one of the conjugate directrix planes 

 in the points m 9 m! 9 these points m 9 m' 9 subtend at the focal 

 centre a right angle. 



Prop. VII. — Let a plane quadrilateral be inscribed in a 

 surface of the second order, whose sides, a 9 (5, y, 8, are pro- 

 duced to meet one of the directrix planes in the points 

 A, B, C, D. The sum of the angles which the points A, B 

 and C, D subtend at the focal centre is equal to two right 

 angles. 



Let two of the sides a, (3 of the quadrilateral be fixed, and 

 let y, 8 be variable ; then A and B are fixed, and therefore 

 the angle A O B is constant ; hence also the angle C O D is 

 constant. 



Def. — The right line in which two tangent planes inter- 

 sect, and the line joining the points of contact, are called 

 conjugate polars relative to the given surface. 



Prop. VIII. — Two conjugate polars to a surface (£), meet 

 one of the directrix planes in two points, which subtend a 

 right angle at the focal centre. 



When the conjugate polars become conjugate tangents, the 

 proposition still holds, and when the conjugate tangents are 

 tangents to the lines of curvature they are at right angles. 

 Hence if a tangent plane be drawn to any point of a surface, 



