190 



the primary semi-axes of the three surfaces confocal to (aobg Co), 

 which intersect at its vertex. 



This theorem may be proved as follows : The equation of 

 the tangent plane to the cone is 



?> n\ Tj being the co-ordinates of the point on the surface 

 (flo ^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, 



f ^'Ko r/rjo r^o \^ 



But, since the point (!' ij' Z') lies in the plane of contact, 

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

 fore, if we put 



I' = Z cos a, Tf] ^L cos j3, Z^ = L cos y, 

 we shall olatain the result stated above. 



The quantity PL is evidently the same for the four sides 

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

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

 and, if wedenotebyDthesemidiameterof the surface parallel to 

 jL, the quantity PP> 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 Z,", U and L'\ pass through 

 the internal axis of the cone; whilst those of Zand i', Z,"and 

 jL'", U and i", LI" and Z/, 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 V. L and L" may now be regarded 

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



