530 PROFESSOR CHALLIS, ON THE TRANSMISSION OF LIGHT 



being supposed to be in the planes of symmetry. Let, therefore, in the last obtained equation, 

 /be a function of w only. Then, 



d'f k7i 



Now in the Paper on the Polarization of Light, (p. 37.S), the particular value of / for a polarized 



ray was found to be cos n\/ka:. By substituting this value in the equation above, we obtain 



the equation of condition h = \, or a' = c'K It would appear therefore that a polarized ray 



cannot be transmitted in the medium, the transverse elasticity being different from that in the 



direction of propagation, if the velocity of propagation really be c", or c y/l + k. For the 



transmission of the polarized ray it is necessary to suppose an alteration of the rate of propagation. 



This may be conceived to take place as follows: First, suppose h = \, and a polarized ray in which 



2 7r 



the breadth of the waves is X, or — , to be transmitted with the velocity c'\/l + k. Then 



suppose the elasticity in the direction of the plane of polarization to be altered from c"" to a"-, and a 

 polarized ray to be still propagated. By hypothesis the nature of the medium is such as to allow 

 of this taking place. Now as/, and consequently the transverse section of the ray, do not alter 

 by the supposed change of elasticity, the only way in which the condensation can be altered is 

 by a change of \. The time of vibration of a given aetherial particle remaining constant, the rate 

 of propagation will be altered in the same ratio. Let therefore / = cos n \/k,v, and let \' be 

 the new value of \. By substitution in the foregoing equation, we obtain the equation of 

 condition 



« n'c^ ^, M X c' 



T = —77- ■ Hence — , or - = - ; 

 fi a- n K a 



and the velocity of propagation 



X , / r a! 



= c'\/i + k X -=c'\/l+kx— = a' \/l + k. 

 \ c 



The foregoing reasoning involves the inference that the rate of propagation of a ray in a medium 



is not solely due to the effective elasticity in the direction of its axis, but is affected also by the 



circumstance that the medium is incapable of transmitting any but a polarized ray, and that 



for such a ray A; is a constant. 



9- We are now prepared to find the equation of a surface, the radius-vector of which 

 drawn in any direction from a fixed point, shall represent the velocity of propagation of a ray 

 in that direction. As we found the velocity of propagation to be that due to the elasticity in the 

 direction of a line drawn perpendicular to the axis of the ray in the plane of polarization, the 

 process will evidently be the following. Cut the surface of the ellipsoid of elasticity by a plane 

 perpendicular to the direction of propagation. The semi-axes of the section will be the radius- 

 vectors in that direction of the surface required. Let a, b, c be the semi-axes of the ellipsoid. 

 Its equation in rectangular co-ordinates referred to the axes and the centre will be 



or' y" s^ 



-i + h + -i = ^- 



a' o" cr 



Let the directions of the rectangular axes be changed by substituting for x, y, and z the 

 following values : 



■» = aw' + fiy + yz', 



y = ax + p,'y' + y'z, 



z = a" CO + /3"y' + y"z'. 



