o/Tresnel's Optical Theory of Crystals. 541 



and in like manner 



v t v„ for the sake of neatness are left unexpressed in terms of i { i n . 

 This is the simplest form by which the position of the lines 

 of vibration can be denoted. 



Corollary. 



From the last proposition it appears that 

 cos e, sin i 



Hence we may construct geo- 

 metrically for the two planes of 

 polarization. 



Let I K be the projections of 

 the two optic axes on a sphere, E 

 the projection of the normal to the 

 front, P the projection of one line 

 of vibration ; then 



cosPK sin K E 

 cos PI sin I K 



Draw F E G the e of which 

 P is the pole, meeting P K, P I 

 produced in G and F. 



Then cos P K = sin K G, and cos P I = sin I F, 

 sin K G _ sinKE 

 sin F I sin I E 



sin K G sin I F 



sin K E " sin I E 

 sin K E G = sin I E F 



•\ K E G = I E F or 180 - I E F. But P E F = P E G 



.-. E P bisects either the <e I E K or the supplement to it. 



These two portions of E P give the two planes of polariza- 

 tion. The construction is the same as that given in Mr. Airy's 

 tracts, and originally proposed, I believe, by Mr. Maccullagh. 

 [To be continued.] 



