490 



Then, considering r, r' as radii of the section xz of the sur- 

 face, we have obviously 



1 cosHk , sinHfc , n 1\ . , n 1\ 



-H = = 2 1- + -j — 2 I cosk'; 



r'"* R p \p R/ \p r/ ' 



observing that when these formulae give a negative value for 

 r^ or r'^, in which case the surface expressed by the equation 

 (2) must be a hyperboloid, the direction of r or r' meets, not 

 that surface, but the surface of the conjugate hyperboloid 

 expressed by the equation 



^ + ^ + i^=-l. (4) 



P Q R ^ ^ 



Now calling and Q' the angles made by the tangent plane 

 of the cones with the directive planes of the given surface, 

 which are also the directive planes of each cone, the angles 

 K, k' depend on the sum or diflference of and 0'. If the 

 latter angles be taken so that their sum may be equal to the 

 supplement of k, their difference will be equal to k', and the 

 formulae (3) will become 





-L^ifi^lN -n IN .- - (■') 



by which the semiaxes of any central section are expressed 

 in terms of the non-directive semiaxes of the surface, and of 

 the angles which the plane of section makes with the direc- 

 tive planes.* 



* See the Transactions of the Royal Irish Academy, vol. xxi., as before 

 cited. The formulae (5) were first given, for the case of the ellipsoid, by 

 Fresnel, in his Theory of Double Refraction, Memoires de I'lnstitut, tom. vii., 

 p. 153. 



