Refraction of Light at the Surface of a Crystal. 115 



If we take the case in which only one wave traverses the 

 crystal we find the following relations by eliminating the 

 ratios S/S, and S/S':— 



tanfl = tanfl'cos(0-0') + ^-^gjm , • ■ (20) 

 T/ cos 0' sin (0 + 0'J v y 



hn fl - tanfl 008 ^"^ I 2 sin 20 sin 2 0' tan ^ 



1 " cos (0 - 0') + sin 2 (0 -0') sin (0 + 0') cos 0'.' 



(21) 



The first of these equations w T as tested by me experimen- 

 tally, the second medium being Iceland spar (see Phil. Trans. 

 1882, Part II.). There was fair agreement between the 

 theory and experiment; but the errors of the experiment were 

 larger than they need have been, in consequence of some 

 want of annealing in one of the lenses of the telescope used 

 in the observations, which was not discovered till too late 

 (Proc. K. S. vol. xxxiv. p. 233). Equations the same as (20) 

 and (21) have been obtained by Neumann, MacCullagh, 

 Kirchhoff, and others. 



Equation (4) gives iis a value for D' — D. We find 



D- D'= sin 0(S sin 6 + S x sin 0,) 



— S' cos %' (sin & sin 0' — tan %' cos 0') — terms in S" 

 = S' cos %' sin & cosec 0' (sin 2 — sin 2 0') 



+ S' sin tf cos 0' + terms in S" &c. 

 = S' cos %' cosec 0' { sin (0 + 0') sin (0 — 0') sin & 



+ tan x ! cos 0' sin <p'\ + terms in S" &c. . (22) 



Thus the problem of reflexion and refraction at a crystal 

 on the Labile iEther theory is fully solved, and some of the 

 results exhibited in a form which can be tested by experiment, 

 though the experimental results will not discriminate between 

 it and the electric theory. 



It will be noticed that the terms in D and D' arising from 

 the surface action have no place in the electric problem. 

 Equation (26) of my paper in the C. P. S. Proceedings, 

 already referred to, which is the same as equation (16) above, 

 expresses the conditions either that the electric displacement 

 along the normal, or that the magnetic force along the axis of z 

 is the same in the two media. So long as we suppose the surface 

 to remain unelectrified, these conditions lead to the same 

 equations. On the Labile iEther theory the two conditions 



