o/* Fresnel's Optical Theory of Crystals. 77 



We may, in the same way, find the inclination of the tan- 

 gent plane to either of the prime radii, and to the plane which 

 contains them both, in terms of i^ and i^; the former by a re- 

 markably elegant construction ; but the final expressions do 

 not present themselves under the same simple aspect. 



If we call 4> the angle between the ray and the front, we 

 may still further reduce by substituting for r^ its values in 

 terms of i^ ^^^ and we shall obtain 



2 (c^ - a^) 

 cot <f> = — ^ ' 



ea.tan^^4-^'+fl^cot^^ 



ysi„(. + '^±^')si„(.-'-^). 



cosec <y . cosec ^^^ 



And if TTy 'n^^ be the inclinations of the normal to the two prime 

 radii, it may be shown that 



cos iTy = cos ^ sm <^ 4- sm <^ cos ^^ sm -~ 

 cos TT/y = cos ^ sm <y + sm ^ cos *^y sm ~ 



Cor. (1.) For imiaxal crystals -^ = 90 and </ + i^^, so that 



tan of the inclination of normal to radius vector 



= r* . ( —2 -^ J sin 2 6 for one point, 



and = for the other. 



Cor. (2.) For every point in the circular section which passes 



through the poles sin -^ = 0, and for the other two circular 



sections i/ ± i^ = or 180°. 



.*. Every point in the three circular sections is an apse. 



Cor. (3.) When a nearly — c —^ —^ is very sm.all; and 



.*. the normal and radius vector very nearly coincide. 



Cor. (4<.) Referring to the last figure we see that O' N' bi- 

 sects the angle R' O' Q'. Now R' O, Q' O are respectively 

 perpendicular to the planes passing through O and the optic 

 axes ; and therefore the meridian plane as we may term it, i. e. 

 the plane containing both the ray and the normal, always bi- 

 sects the angle formed by the two planes drawn through the 

 ray and the two optic axes. 



