On the Laws of Crystalline Reflexion. 85 



taken up from reading a little Paper, by M. Cauchy, in the 

 Bulletin des Sciences Mathematiques for July 1830 ; and by 

 combining such a relation with the three conditions afforded 

 by my own law of equivalent vibrations, I had actually obtained, 

 for the polarizing angles in different azimuths, a formula (that 

 marked (5) in my former Paper), which I found to agree very 

 well with Sir David Brewster's experiments, and which M. 

 Seebeck has found to agree still better with his own. 



The formula for the polarizing angle is obtained by equat- 

 ing two values of the deviation ; and it is remarkable that the 

 very same formula comes out in my present theory, although 

 the values of the deviation are entirely different. Referring, 

 for brevity, to the notation of my former Paper, I find, for the 

 case of a uniaxal crystal, 



tan j3 = cos (i + $) tan 0, (a) 



o' I- r\ & i 2 , ;M sin iA' cos i// sin 2 ^ 



- tan 3 = cos- (i + <b } cotan + (a 2 - b 2 } . ,., . ?- (b) 



sin v sm (* - $} 



These equations (a) and (b} are to be substituted for equa- 

 tions (2) and (3),* which are the equations that M. Seebeck found 

 to be at variance with his experiments. 



By means of formula (5), equation (a) becomes 



TOT 



= y sin 2 q sin (p - 0), . .' (c) 



from which the deviation in any azimuth may be readily calcu- 

 lated. The azimuth (as M. Seebeck reckons it) begins when 

 = 0, and p is then positive. This formula (c) perfectly represents 

 the experiments of M. Seebeck on Iceland spar. The corre- 

 sponding expressions for biaxal crystals may be easily deduced, 

 and will be given in a Paper which I am preparing to lay 

 before the Royal Irish Academy. 



At the time of my last communication I was not aware that 

 the case in which the plane of incidence is a principal section of 

 the crystal (or the azimuth = 0) had been solved by M. Seebeck, 

 and that formula (7),f which I regarded as my own, had been 

 obtained by him long before. 



* Supra, p. 79. t Supra, p. 80. 



