I 



of the Optical Theory of Crystals, 343 



Or, in other words, the line of force is as much inclined to 

 the line of vibration as the ray is to the normal. 



Now the normal is perpendicular to the line of vibration, 

 and all four lines lie in one plane, 



.'. the ray is perpendicular to the line of force. Q. E. d. 



I may be allowed to conclude this long paper with a sum- 

 mary of some of the most remarkable consequences which 1 

 have extricated from Fresnel's hypothesis. 



1. The two meridian planes corresponding to any given 

 radius are perpendicular to each other*. 



2. So are the two corresponding to any given normal. 



3. Every meridian plane bisects the angle formed by two 

 planes drawn through the radius and the two prime radii. 



4. It also bisects the angle formed by two planes drawn 

 through the normal and the two prime normals. 



5. Each meridian plane contains one line of vibration and 

 the corresponding line of force. 



6. The ray is perpendicular to the line of force. All these 

 conclusions, except the fourth, are, I believe, original. 



The theory of external and internal conical refraction fol- 

 lows immediately as a particular consequence from the third 

 and fourth combined as already shown ; the same propositions 

 also enable us to draw a tangent plane to any point of the 

 wave-surface by mere Euclidean geometry. May not some of 

 these conclusions serve to suggest to physical inquirers the 

 question. Has the theory been started from the most natural 

 point of view ?t 



University College, Feb. 24, 1838. 



Note 2. — Investigation of the Wave-Sufface, 

 Since the appearance of the preceding parts, I have suc- 

 ceeded in completing the self-sufficiency of my method by 

 deducing the equation to the wave-surface from the expres- 

 sions given in Prop. (5.) for the angles between a front and the 

 principal planes in terms of its two velocities. If these angles 

 be CO, ^, 4/ , and the two velocities v^ v^^, we found 



{a'b^){a''-c'') 

 ^^ 



^ /(a^-v,^){a''-v.;' 

 cos cw = \ / ^^ '-L^ IL 



V {a'b^)(a^-c^) 



cos ^ = ./WE^nWE^ 



^ V (b^-.a^){(^-b^) 



* 1 have defined the meridian plane to be that which contains radius 

 vector and normal belonging to the same point. 



f This investigation supplies the step which Mr. Tovey was desirous 

 should appear in the Magazine. 



