Crystalline Reflexion and Refraction. 



55 



Then will e = 0, and £'= the arc Ee. Putting and & for the spherical angles 

 Aoi and Aei, we shall easily see that 6^-=^ e-j-T'BQ.'', and e'^=i6'-\- 90°, if weconceive 

 the point A and the positive axis of z to be both on the upper side of the plane 

 XOY. And if w' denote the arc Ae, while b and a respectively express the 

 reciprocals of the principal indices, ordinary and extraordinary, the law of 

 Huyghens, for the double refraction of uniaxal crystals, will give us 



where 



tan E = — 2 — sm w cos w , 



s^= ^ = b'-\- (a'-b') sin^. 



-X- 



Observing these relations, we have, from (34), 



tan03= — cos(tj-4-0*^^^' 

 for the ordinary ray; and from (35) we get 



(36) 

 (37) 



(38) 



tane', = — cos(t, - 1' J cotan 6'— (a^— 5') 

 tan e\ =cos(<, + I'J cotan 6' — (a* — S') 



sinw cosw sin't 



sin0'sin(t,+ t'„)' 



sino) cosw sm I, 



(39) 

 4 ' » ^-.' 



sins' sin(t, — t'j,) ' 



for the extraordinary ray. 



The four preceding equations determine the uniradial directions ; and the / 

 following equation, '^T^ 



cos(<,+ Otan0+cos(t,+ t'^)cotane'— (a'— i') rf =0, (40) 



I 



sin0'sin(tj — ('2) 



obtained by putting tan 0, = tan 0',, is that which determines the polarising 

 angle. 



In making use of this last equation to deduce the law of the polarising angles 

 in various positions of the axis of the crystal, we shall confine ourselves to the 

 case in which the reflexion from the crystal takes place in air, because the angle 

 *i — 's ^'1^ ^^^^ ^6 considerable, and the quantities cos(<j-j-tJ and cos(i^-\-i'^) will 

 be small, so that it will be easy to arrive at approximate results. For we shall 

 , have, in the first place, / '^A^^/j^ 



'■^jfo .£f^ /-/-/- ^,^^^ ^.,//^/^ ^ £,. 



