On the Motion of the Apse-Line in the Pendulum Oval, 303 



have 



y^{av^-\'bu^}+2^{aw^-\-cu^}-\-2i/s.aviv=zl. 



Now the ellipses D and D' coincide as often as the normal to 

 the plane of the waves coincides with a principal axis. When 

 the plane of the waves runs parallel with a principal axis^ both 

 these sections in the direction of the said axis possess axes of 

 equal length. It seems, in fine, that for homogeneous media 

 these two sections differ from each other only in an inappreciable 

 degree ; that is to say, for these media the following relations 

 appear to be very nearly correct : 



A + B-2/=0, A4-C-2e=0, A + i-e~/=0. 



In order to obtain the directions of vibration and the velocities 

 for a given plane P, let the diametral section D' of the ellipsoid 

 E' parallel with P be constructed. The directions of its axes are 

 the directions of vibration, the planes of which are parallel to P, 

 and the reciprocal values of its semiaxes furnish the correspond- 

 ing velocities. 



A surface whose rays are equal to the reciprocal values of the 

 rays of the ellipsoid E' will be represented by the equation 



This is the surface of construction of Fresnel, the axes of 

 which coincide with the principal axes of the medium, and are 

 equal to double the principal velocities. 



and 



A diametral section thereof, the plane of which is parallel with 

 P, furnishes in the directions of its semiaxes and the direct 

 values of their lengths, the directions and velocities of the vibra- 

 tions which belong to P ; and this is the principle which forms 

 the basis of the construction of Fresnel. 



XL VI. On the Motion of the Apse-Line in the Pendulum C^val. 

 By the Rev. J. A. Coombe, M.A.j late Fellow of St. John's 

 College, Cambridge. 



To the Editors of the Philosophical Magazine and Journal, 

 Gentlemen, 



THE following method of investigating the motion qf the 

 apse-line in the pendulum experiment presents some pe- 

 culiarities which may be interesting to some of your readers. It 



Y2 



