588 Dr. A. 0. Rankine and Dr. L. Silberstein on 



But since \g lK \ = —1, and, therefore, 7 44 |<w ==— 1, we- 

 have at once 



3)?=— 7 44 .SM. 



Thus, the relation of 9Jt to M is exactly the same as that of 

 (i to E, or, if we write, in the usual way, (£ = KE, s 3Jt = /LtM,. 

 the permittivity operator A" is identical with the permeability 

 operator, 



K=fi—— 744.5, (4) 



both operators having the principal axes of co and 744 times- 

 its principal values. Now, such being the case, the velocity 

 of propagation \) can easily be proved to be independent of 

 the orientation of the light vector. In fact if n be the wave- 

 normal, we have, from equations (M), with p = 0, 



l; ATE = VMn; l ^ /jM=VnE, . . . . (5> 



C G 



and since the operator fju is identical with A~, 



\y 



/vE + Vn(A^ 1 A^nE)=0 



K~ l being the inverse operator of K. If K u etc. be the 

 principal values of AT*, and nj, etc. the components of n 

 along them (or the direction cosines of n), the last equation 

 becomes 



j ^KxKjKj-wx^-n^Ks— VK 8 ]E"f n(nJCE) = 0, 



whatever the direction of the vector E. But since by the 

 first of (5), nKE = 0, we have ultimately 



\f _ ■ /r i tt 1 2 + K 2 rc 2 2 -+K 3 7i 3 2 



c 2 ~ KxKjB:, • • ; • {> 



Thus, as was announced, the velocity of propagation of 

 light is, in the most general gravitational field, a function 

 of n, but entirely independent of the orientation of the light 

 vector. 



Thus, also, we see that, according to Einstein's theory, no 

 difference whatever is to be expected between our above 

 said two velocities c v and c/ t . As to other theories, none of 

 them contemplates any connexion at all between light pro- 

 pagation and gravity. But, apart from any theory, it 



* Of these three values K x , K 2 , K 3 two are, as a matter of fact, equal 

 in any gravitational field. But we can prove our statement even without 

 profiting from this axial symmetry. 



