52 Mr. Mac Cullagh on the Laws of 



reflected transversal uniradial in its direction ; and that the total (or actual) 

 reflected transversal will be the resultant of the two partial ones. 



elude that it will be preserved by their resultants. Here then is a test of the consistency of our 

 theory ; for we are bound to show that the law of vis viva is not infringed by the adoption of the 

 principle in question. Now it is easy to see that, whatever be the two directions in which the inci- 

 dent transversal is resolved, the final results will always be the same ; because, taking the component 

 in each of these directions separately, the reflected and refracted transversals belonging to it must be 

 obtained, in the first place, by the help of a resolution performed in the uniradial directions. We 

 need not, therefore, consider any case but that in which the resolution is uniradial throughout. 



The incident transversal being denoted by T , let T^ be the reflected transversal determined by 

 the rules given in the text ; and let the uniradial components of the former be r,, ■/,, while those of 

 the latter are r,, t*,. Then will 



T,''= riH^,' + 2r,T',cos(9 — 9',), 



where the signification of 9 , 6\, fl,, fl'j is the same as in the text. The vis viva of one refracted ray is 

 ^i(f,' — T-j*)) and that of the other is m^(r'^' — r'^') ; therefore the vis viva of both refracted rays is 



a quantity which ought to be equal to 



and consequently the equation n^ 



r^r\cos{9-9\)=r,r\co,[6-6\) (v.) 



ought to be true. This equation, by help of the expressions (6) for r,, -i",, and the like expressions 

 for T'j, t'j, becomes 



sin(.,+gsin(.,+g(l+tan9,tan9',) = sin(,,-,^)sin(,_,'^)(l-|-tanfl3tan9'3); (vi.) 

 which again, by substituting the values (13) and the other similar values, is changed into ^.'l/''^' /U'^rti,^ 

 sin (;^ + ig{cos(jj— ('j) + cotane2COtan9'J+A + A'=0, (vii.) 



where h' denotes for one refracted ray what h denotes for the other, the value of h being given by 

 formula (27), and that of A' 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 relations 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 (vii.) really does express a^gm-ous property of the biaxal wave surface 





