On the Surfaces of the Second Order. 295 



Two cones so related may be called reciprocal cones. When 

 one is of the second order, it will be found that the other is also 

 of the second order, and that, in their equations relative to their 

 axes, which are obviously parallel or coincident, the coefficients 

 of the squares of the corresponding variables are reciprocally 

 proportional, so that the equations 



0, + + =(), (1) 



express two such cones which have a common vertex. These 

 cones have the same internal axis, but the directive axis of the one 

 coincides with the mean axis of the other, and it may be shown 

 from the equations that the directive planes of the one are per- 

 pendicular to the focal lines of the other. The two curves in 

 which these cones are intersected by a sphere, having its centre 

 at their common vertex, are reciprocal spherical conies. In 

 general, two curves traced on the surface of a sphere may be said 

 to be reciprocal to each other, when the cones passing through 

 them, and having a common vertex at the centre of the sphere, 

 are reciprocal cones. Any given point of the one curve corre- 

 sponds to a certain point of the other, and the great circle which 

 touches either curve at any point is distant by a quadrant from 

 the corresponding point of the other curve. 



By means of these relations any property of a cone of the 

 second order, or of a spherical conic, may be made to produce a 

 reciprocal property. Thus, we have seen that the tangent plane 

 of a cone makes equal angles with two planes passing through 

 the side of contact and through each of the focal lines ; there- 

 fore, drawing right lines perpendicular to the planes, and planes 

 perpendicular to the right lines here mentioned, we have, in the 

 reciprocal cone, a side making equal angles with the right lines 

 in which the directive planes of this cone are intersected by a 

 plane touching it along that side. It is therefore a property of 

 the cone, that the intersections of a tangent plane with the two 

 directive planes make equal angles with the side of contact ; a 



