i io On the Laws of Crystalline 



sin 2 ( 2 tan 



tan ft = cos (ii - 2 ) tan ft -r- j. ; , . , , 



t (34) 

 a fl sin 2 ti tan E 



tan ft = - cos (ii + i a ] tan ft + : , 



cos ft sin (d- t 2 ) J 



These formulae give the uniradial directions, or the positions 

 of the incident and reflected transversals, when the sole refracted 

 ray is that with which we have been occupied. The like direc- 

 tions, when the other ray exists alone, will be given by the 

 formulae 



n, / , N , & sinY 2 tan * 



tan i = cos Ui - 1 2 ) tan 6 2 + 7^7 --;, N ' 



cos 2 sin (u + 1 2 ) 



> (35) 



* O / 1 / 



BIT*** TQTI c 



. frf s / \ J_ /!/ O1JJL t 2 tCtU. c 



tan = - cos ftj + 1 2 ) tan 2 + f . 



cos 2 sm (ti - t l ) J 



where all the quantities, except i l9 which remains the same, are 

 marked with accents, to show that they belong to the second 

 refracted ray. 



The uniradial directions having been found by these equa- 

 tions, the relative magnitudes of the uniradial transversals are 

 determined by equations (6). When the incident transversal 

 is not uniradial, it is evident, as we said before, that it may be 

 resolved* in the two uniradial directions ; that each component 



* That, if an incident transversal be resolved in any two directions, the reflected 

 and refracted transversals arising from it will be the resultants of those which would 

 arise from each of its components separately, is a principle which appears very 

 evident, insomuch that we can hardly suppose it to be untrue, without doing 

 violence to our physical conceptions. Nevertheless, it is necessary to prove that 

 this principle is not contrary to the law of vis viva ; for though the vis viva may be 

 preserved by each set of components (as it is when these are uniradial), yet we 

 cannot therefore conclude 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 incident transversal 

 is resolved, the final results will always be the same ; because, taking the compo- 

 nent 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 per- 



