526 PROFESSOR CHALLIS, ON THE TRANSMISSION OF LIGHT 



o-iven particle of the sether, according to the principle of the coexistence of small vibrations, may be 

 a vibration in a certain resulting direction, of the same period as that of the component vibrations. 

 Consequently waves which would produce the same vibration of the aetherial particle may be pro- 

 nao-ated in that direction. But the displacement of the atotns of the medium does not necessarily 

 take place in the same direction. If this displacement be resolved in two directions, one coinciding 

 with the direction of vibration of the aetherial particles, and the other perpendicular to this, the 

 resolved part of the displacement in the latter direction, will give rise to aetherial vibrations which 

 will be propau-ated laterally and produce no sensation of light. With reference to phsenoraena 

 of licht the other part alone requires to be taken account of. The above considerations will 

 enable us to determine the effective elasticity in any direction in the medium, in terms of the 

 elasticities in the directions of the axes. 



6. Let V be the velocity of a particle of the a^ther, the vibrations of which are due to waves 

 propagated in a direction making angles a, /3, 7, with the axes of elasticity ; and let v' be the 

 resolved part in that direction of the velocity of an atom of the medium situated where the 

 velocity of the aether is v. Then by Art. 2, the accelerative force of the retardation is equal to 



. (dv dv'\ „j, , ^ dv 



If now the velocity v be resolved in the directions of the axes, the accelerative forces of retardation 

 corresponding to the resolved parts of the velocity will be, 



- .fiT^ (1 - <?,) cos a ^ , -A'^(l -«?2)cos^— , - KSO -<Iz)cosy~. 



And by the considerations in Art. 5, the accelerative force of the retardation in the given direction 

 of propagation, is equal to the resultant of these forces. Hence 



-KH^-q)'jZ = -^^ir- {(l -9.)':os'a + (l - q,) cos' (i + (l -q,)coi'y}. 

 at at 



Let now ?-2 be the constant of elasticity in the direction of propagation. Then by the equations in 



Art. 4, we have, 



--l=A'^(l-9), ^-1=^-^1-9,), ^-i = A'^(l-<7,), f, -1=^^(1-9,), 

 Hence, by substitution in the foregoing equation, 



^ _ 1 = (^ - 1) cos'^a + (^, - 1) cos'^/3 + ( J - 1) cos^7. 



Consequently, 



I cos^ a cos' i3 cos' y 



— = ■\ — + . 



r' a/ 6; c/ 



The surface of which this is the equation in polar co-ordinates, may be called the surface of 

 elasticity. It is evidently that of an ellipsoid. The radius vector r, represents the velocity of 

 propagation of plane waves in any direction coinciding with that of r 



7. We have now to find the velocity of propagation in Sl filament of the asther corresponding 

 to a ray of light. In considering the motion in a filament of a medium the elasticity of which 

 varies with the direction, I shall proceed in a method analogous to that employed in my Paper 

 on Luminous Rays. {Camh. Phil. Trans. Vol. viii. Part 111. p. 365). It will be supposed that 

 in the filament there is an axis of no transverse velocity. This is taken for the axis of z. The 



