190 



the primary semi-axes of the three surfaces confocal to (oo K €„), 

 which intersect at its vertex. 



This theorem may be proved as follows : The equation of 

 the tangent plane to the cone is 



?'> »)'> r> being the co-ordinates of the point on the surface 

 (ao ^0 Co), touched by the plane ; and the square of the per- 

 pendicular P let fall from the centre on it is given by the 

 formula, 



P2 





But, since the point (?' r]' K) lies in the plane of contact, 

 the numerator in this expression is equal to unity. There- 

 fore, if we put 



S' = Z/ cos a, ri = L cos j3, Z' = L cos y, 



we shall obtain the result stated above. 



The quantity PL is evidently the same for the four sides 

 of the cone Z/, L', L", L'", whose directive angles are re- 

 spectively (a, j3, 7), (a, 7r-/3,7),(a, 7r-/3, tt - y),(a, (i, tt -7); 

 and, if we denote by D thesemidiameterof the surface parallel to 

 L, the quantity PD will likewise be the same for them all ; since 

 the sides of the cone are proportional to the parallel semidiarae- 

 ters. Again, the planes of L and L", L' and L'", pass through 

 the internal axis of the cone; whilst those of Z- and L', Z-"and 

 L'", L' and L", L'" and L, pass through its external axis. 



Let us now suppose the vertex of the cone to approach in- 

 definitely near to a point V on the surface : its internal axis 

 becomes the normal; and the external axes ultimately coincide 

 in direction with the tangents to the two lines of curvature 

 passing through the point F. L and L may now be regarded 

 as two successive elements of a geodetic line, since their plane 



