1 1 2 On the Laws of Crystalline 



eliminate the two ratios of the three transversals, together with 

 the angle at which the incident transversal is inclined to the 

 plane of incidence. In the equation produced by this elimina- 

 tion, the angle of incidence would be the polarizing angle, and 

 the other quantities would be known functions of that angle ; 

 whence the angle itself would be known. 



a quantity which ought to be equal to 



mi (2V - TV) ; 

 and consequently the equation 



TIT'I cos (0i - 0'i) = TST'S cos (63 - 0' 3 ) (v.] 



ought to be true, This equation, by help of the expressions (6) for n, ra, and the 

 like expressions for T'I, r'z, becomes 



sin (t 1 + i a ) sin (t 1 + t' 2 ) (1 + tan 5 tan 0'J 



= sin (t x - 1 2 ) sin (t x - t' 2 ) (1 + tan 3 tan 9' 3 ) ; (vi.) 



which again, by substituting the values (13) and the other similar values, is 

 changed into 



sin (i a + i' 2 ) {cos (t 2 - i' a ) + cotan 2 cotan 0' a } + A i h' = 0. (vn.) 



where h 1 denotes for one refracted ray what h denotes for the other, the value of h 

 being given by formula (27), and that of h' by the same formula with accented 

 letters. The angle of incidence, we may observe, has disappeared from the 

 equation. 



If, therefore, the laws of reflexion, which we have endeavoured to establish, are 

 consistent with each other, this last equation must be satisfied by means of the rela- 

 tions which the laws of propagation afford ; or rather, the equation must express a 

 property of the wave surface of the crystal, however strange it may be thought that 

 such a property should be derived from the laws of reflexion laws which would 

 seem, at first sight, to have no connexion at all with the form of the wave surface. 

 Now I have found that the equation (vn.) really does express a rigorous pro- 

 perty of the biaxal wave surface of Fresnel ; a very curious fact, which not only 

 shows that the laws of reflexion and the laws of propagation are perfectly adapted 

 to each other, but also indicates that both sets of laws have a common source in 

 other and more intimate laws not yet discovered. Indeed the laws of reflexion are 

 not independent even among themselves ; for the expressions (in.) and (iv.) in the 

 note on ordinary reflexion (page 101) have been deduced solely from the principle of 

 equivalent vibrations, and yet they satisfy the law of vis viva. Perhaps the next step 

 in physical optics will lead us to those higher and more elementary principles by 

 which the laws of reflexion and the laws of propagation are linked together as parts 

 of the same system. 



