Crystalline Reflexion and Refraction. 67 



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 theories 

 that the more they are studied, the more simple they appear to be. And we may 

 add, that a close examination of such theories always meets with its reward, in 

 the unexpected* consequences which present themselves to view. Nothing can 



that the sheet in which that note is found was printed ofif before I had obtained the result announced 

 in the subsequent note, p. 52. Various delays occurred while my paper was going through the press ; 

 and 1 took advantage of them to increase its value, by appending notes on some of the questions 

 which I had overlooked or omitted in the first consideration of the subject. 



* As an instance of this, it may be mentioned, that the conclusion arrived at in the note, p. 52, 

 was wholly unexpected. And in verifying the equation (vii.), an unexpected and useful theorem 

 was obtained; for it became necessary to find a manageable expression for the tangent of the angle b 

 which the wave normal makes with the ray. This expression is wanted in applying the formulae (34) 

 and (35) to biaxal crystals, and therefore I shall make no apology for introducing it here. 



Having described a sphere concentric with the wave surface, let the wave normal OP and the 

 two optic axes (which are the nodal diameters of the index surface,) be produced from the centre 

 O to meet the sphere in the points P,, A, A,, respectively, thus marking out the angles of a spheri- 

 cal triangle P,AA,. The same wave normal may belong to two different rays ; and if we select one 

 of these rays, its transversal must lie in a plane drawn through the wave normal, and bisecting either 

 the internal angle AP^A^ of the spherical triangle, or the external supplementary angle. By pro- 

 ducing the optic axes in the proper directions, we may always make the above plane (w hich Fresnel 

 calls the plane jofpolarisation) bisect the internal angle. Supposing this to have been done for the ^ 



ray which was selected, put vj and tu, for the sides P,A and P^A, of the spherical triangle, and \|/ for 

 the contained angle AP^ A^. Let * be the length of the wave normal from the centre O to the point 

 where it intersects the tangent plane applied at the extremity of the ray, that is, applied at the 

 point where the ray meets its own nappe of the wa^^e surface ; and let a and c be the greatest and 

 . least semiaxes of the ellipsoid which generates the wave surface. Then we shall have 



tansrr ^^^ sin(m — aj,)sin^\{/. (ix.) 



And it is now manifest that if e^ be the angle which the other ray makes with the same wave normal, 

 and s, the length of the wave normal intercepted between the centre and the tangent plane at the 

 extremity of this ray, we shall also have 



at c« 



ta"«/=-271~sin('«4-"'/)cosi^{;. (x.) 



If a ray is given in direction it will have two wave normals; and then the angles g, e,, which it 

 makes with each normal may be found from the formula j. 



