3 1 6 On the Surfaces of the Second Order. 



where a, /3, 7 are the angles which the axis of x makes with the 

 axes of , ij, respectively. In the first case, the equation (22) 

 shows that the directive planes of B are perpendicular to the 

 right lines expressed by the equation (13) ; in the second case, 

 the equation (23) shows that the directive planes of B are per- 

 pendicular to the right lines expressed by the equation (20). 



When the surface A is a paraboloid, and the distance of the 

 point R from its vertex is indefinitely increased, the plane 

 touching the surface at E. approaches indefinitely to parallel- 

 ism with its axis, and the right line SK, perpendicular to that 

 plane, increases without limit. Therefore the surface B passes 

 through the point S, and is touched in that point by a plane 

 perpendicular to the axis of A. 



When the point S lies upon the surface A, the co-efficient 

 of the square of one of the variables, in the equation (22) or 

 (23), is reduced to zero, and the surface B is a paraboloid 

 having its axis parallel to the normal applied at S to the 

 surface A. This also appears from considering that when S is 

 a point of the surface A, the normal at that point is the only 

 right line passing through S, which meets the surface B at an 

 infinite distance. 



If a series of surfaces be confocal, their reciprocal surfaces, 

 taken with respect to any given sphere, will be condirective. 



When the equations of any two condirective surfaces are ex- 

 pressed by co-ordinates perpendicular to their principal planes, 

 the constants in the equations may be always so taken that the 

 differences of the co-efficients of the squares of the variables in 

 one equation shall be equal to the corresponding differences in 

 the other. Then, by subtracting the one equation from the 

 other, we get the equation of a sphere. Therefore when two 

 condirective surfaces intersect each other, their intersection is, 

 in general, a spherical curve. But when the surfaces are two 

 paraboloids of the same species, their intersection is a plane 

 curve. 



18. Through any point S of a given surface four bifocal 

 right lines may in general be drawn. Supposing the surface 



