Reflexion and Refraction. 129 



If we are asked what reasons can be assigned for the hypo- 

 theses on which the preceding theory is founded, we are far 

 from being able to give a satisfactory answer. We are obliged 

 to confess that, with the exception of the law of vis viva, the 

 hypotheses are nothing more than fortunate conjectures. These 

 conjectures are very probably right, since they have led to 

 elegant laws which are fully borne out by experiments ; but 

 this is all that we can assert respecting them. We cannot 

 attempt to deduce them from first principles; because, in the 

 theory of light, such principles are still to be sought for. It 



ternal supplementary angle. By producing the optic axes in the proper direc- 

 tions, we may always make the above plane (which Fresnel calls the plane of 

 polarization) bisect the internal angle. Supposing this to have been done for the 

 ray which was selected, put w and <a t for the sides PA and PA, of the spherical 

 triangle, and J/ for the contained angle APA t , Let s be the length of the wave 

 normal from the centre 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 wave surface ; and let a and c be the greatest and least semiaxes 

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



2 c 2 



tan e = - sin (w - &>,) sin ty. (ix.) 



. 2 



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

 same wave normal, and *, the length of the wave normal intercepted between the 

 centre and the tangent plane at the extremity of this ray, we shall also have 



a*-c z 

 tan e, = g- sin (o> + ,) cos i ^. (x.) 



If a ray is given in direction it will have two wave normals ; and then the 

 angles e, e,, which it makes with each normal, may be found from the formulae 



r 2 / 1 1 \ 



tan e = - ( - - ^ 1 sin ( - w/ ) sin \ i|/, 



where r and r t are the two radii of the wave surface which are in the direction of 

 the ray ; the spherical triangle P t AA 4 , of which the sides and contained angle are 

 expressed by the same letters as before, being now formed by producing the ray 

 and the two nodal diameters of the wave surface, until they intersect the sphere in 

 the points P t , A, A t . 



K 



