the formulce of Cauchy for the Motion of Light. 301 



we tlius obtain for the equation of the ellipsoid of polarization 

 the following : .„. ^„ 



+ y\v''{.f + q^) + u\^ + 7^) + w\t,^ + o?) } 



■\- ^.yz ,a? .vw -\- ^.xz .^^ Aiw -YA.xy .^'^ .uv— (77— ) . 



\27r/ 



Ifj in the first place^ the normal to the wave coincides with 

 the X axis^ that is_, if v — ^, iv=.^jU — \, the equation of the ellip- 

 soid of polarization will he 



The velocities of the vibrations which proceed parallel to the y 

 axis and the z axis respectively will therefore be 



\/^2^7^^ and V^^^pK 



If, secondly, the plane of the waves stand perpendicular to the 

 y axis_, we obtain in a perfectly similar manner for the velocity 

 of the vibrations which are parallel to the x axis and to the z axis 

 respectively, 



V7;^ + 7'^ and V tf" -\- c^^ . 



If, finally, the plane of the waves stand perpendicular to the 

 z axis, the velocities of the oscillations which are parallel to the 

 axes X and y respectively are 



\/?M^ and 'V/^^T^ 



Experiment teaches, however, that a ray whose plane of pola- 

 rization coincides with a principal section possesses one and the 

 same velocity, whatever its direction may be in other respects ; 

 or, if we assume that the plane of oscillation is peyyendicular to 

 the plane of polarization, that oscillations which are parallel to a 

 principal axis are propagated with equal velocities, whatever the 

 direction of the plane of the waves in other respects may be. 

 According to this, we are justified in assuming that we have 



9;2 + ry2=^ + ^2^ p _^ ^2 ^ ^ _l_ ^2^ f H /32=7;2 + «2 nearly. 



These relations, to the assumption of which we are equally 

 entitled as to the assumptions regarding the connexion between 

 the planes of polarization and vibration required by the opposite 

 notion, reduce the equation of the ellipsoid of polarization to the 

 following : 



Phil, Mag, S. 4. Vol. 2. No. 11. Oct, 1851. Y ^ 



