483 



lar to the intersection of the two corresponding tangent 

 planes of the second cone ; and as these two sides ap- 

 proach indefinitely to each other, their plane approaches to 

 a tangent plane, while the intersection of the two correspond- 

 ing tangent planes of the second cone approaches indefi- 

 nitely to a side of the cone. Thus any given side of the one 

 cone corresponds to a certain side of the other ; and any 

 side of either cone is perpendicular to the plane which 

 touches the other along the corresponding side. This rea- 

 soning applies to cones of any kind. 



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 coinci- 

 dent, the coefficients of the squares of the corresponding va- 

 riables are reciprocally proportional, so that the equations 



Px^ + Qy^-f-R;.^ = 0, ^-f--il + ll = o, (I) 



V Q R 



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 perpendicular 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 sur- 

 face 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 corresponds 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 



