506 



the axes of ^, i;, Z, 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 perpendicular 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 R approaches indefinitely to 

 parallelism with its axis, and the right line SK, perpendicu- 

 lar 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 coeflficient 

 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 hav- 

 ing 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 sur- 

 faces, taken with respect to any given sphere, will be condi- 

 rective. 



When the equations of any two condirective surfaces are 

 expressed by coordinates perpendicular to their principal 

 planes, the constants in the equations may be always so taken 

 that the differences of the coefiicients of the squares of the 

 variables in one equation shall be equal to the corresponding 

 differences in the other. Then by subtracting the one equa- 

 tion from the other, we get the equation of a sphere. There- 

 fore 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. 



