Reflexion and Refraction. 127 



at the second surface of a crystal ; but these must be reserved 

 for a future communication. It would be easy, indeed, to write 

 down the algebraical solutions resulting from our theory; but 

 this we are not content to do, because the expressions are rather 

 complicated, and, when rightly treated, will probably contract 

 themselves into a simpler form. It is the character of all true 



single refracted ray E(\} emerging from the crystal in a direction parallel to R\. Put 

 TI, TS, Tg, and T,,, T',,, T(I) for the transversals of the rays in the order in which they 

 have been named. As the transversal TZ is supposed to be given4n magnitude, the 

 lengths as well as the directions of r\ and TS can be found by the, construction in 

 page 97. 



Now, the direction of TS being changed, and its magnitude retained, let the ray 

 -Z?3 be turned directly back, so as to be incident again on the crystal, and to suffer 

 reflexion and refraction at the first surface. Then the two refracted rays which it 

 gives will be parallel to R tl , JR' ti , and their transversals will be equal and parallel 

 to T,,, T',,. The reflected ray which it gives will coincide with Hi ; and the re- 

 flected transversal, when compounded with TI, will furnish a resultant equal and 

 parallel to the emergent transversal T(i). 



Thus the constructions, which have been given for the first surface, may be 

 made available for the second surface, and every question relative to crystalline 

 reflexion may be solved geometrically by means of the polar planes. 



The foregoing rule was not, properly speaking, deduced from theory. I first 

 formed a clear conception of what the rule ought to be, and then verified it for the 

 simple case of singly-refracting media, and finally proved it for doubly-refracting 

 crystals. The truth of the rule, in crystals, depends upon the truth of the three 

 following equations : 



sin (i tl -f t'J { cos (i,,-i',,) + cotan 0,, cotan 0',, } +h ll + h' it = Q, ^ 

 sin(i 3 - t j { cos (i' a ' -M J ~ cotan #2 cotan 0,, } + ^ 2 - A^O, (vin.) 



sin (t 2 - i' J {cos (t a + /) - cotan 62 cotan 8' tt } + h z - h' = 0, 



in which the notation is intelligible without any explanation. The first equation 

 is the same as equation (vn.) already noticed ; and the other two differ from it 

 only in appearance, the change in the signs being occasioned by a change in the 

 relative position of the rays. 



"When the reflexion is total, I suppose we may follow the example which 

 Fresnel has set us in the case of ordinary media. The general algebraic expres- 

 sion for each reflected transversal will then become imaginary ; and by putting it 

 under the form 



- 1 sin*), 



we shall have T for the reflected transversal, and * for the change of phase. 



