8 Mr. O'BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT 



case of oblique incidence, where the vibrations take place parallel to the plane of incidence. 

 The laws, which the directions of the normal rays obey, are curious, and have not been noticed 

 before so far as I am aware ; nor indeed can I perceive that these rays have been taken into 

 account in a satisfactory manner by writers upon this subject. 



In this section I have shewn that if we take the equations of connection in their simplest 

 form, Fresnal's formulse wiU result from them on two suppositions ; first, that normal waves 

 are propagated very slowly compared with transversal waves ; and secondly, that normal waves 

 are propagated with the same, or nearly the same, velocity in vacuum and in transparent media. 

 The former hypothesis seems to me to be very improbable, for it is very difficult to conceive 

 a stable medium in which normal waves are propagated more slowly than transversal. I may 

 observe here, that M. Cauchy's equations and results are obtained by assuming the truth of 

 this hypothesis, (see his Exercices for 1840, p. 135), and appear, on tiiis account, liable to 

 objection. 



In Section IV. I have shewn that Fresnal's formulae may be applied, without making any 

 vafue use of the symbol v — 1, to the case of Total Internal Reflexion, and that he was 

 fully justified in the very remarkable interpretation be put upon his formulae in this case. 



In Section V. I have shewn that normal waves will never produce any sensible effect on 

 the eye by producing transversal vibrations, provided the velocity of propagation of normal 

 waves be either very great, or very small, compared with that of transversal waves. 



In Section VI. I have attempted to prove, from well established experimental laws, that 

 polarized light consists of vibrations at right angles to the plane of polarization. 



In Section VII. I have briefly shewn how we must proceed when the equations of con- 

 nection are not taken in their simplest form, in which they are used in Section iii. 



Lastly, in Section VIII. I have obtained expressions which apply to substances of high 



refractive power, such as the diamond, and from which I have deduced results in exact 



accordance with the experiments of Mr Airy. These expressions are different from those of 



Mr Green, which certainly cannot be correct, since they give (see Cambridge Transactions, 



/3' 1 fi^ 1 . . 



Vol. VII. p. 23,) ^ = more than — , for plate-glass ; and — = more than - , for diamond : which 



results are utterly at variance with experiment. The fact is, Mr Green's original mistake 

 respecting the constants {A) and (J5), mentioned above, obliges him to suppose that the index 

 of refraction is the same for normal and for transversal waves, and this makes his results true 

 only for substances of very low refractive power ; for instance, they are quite at fault in the 

 case of common plate-glass, both as regards the intensity and the rotation of the plane of a 

 polarized ray. If v is put = ;a in ray result, it agrees with Mr Green's, which confirms the 

 correctness of what I have just stated, 



SECTION I. 

 Preliminary Observations. 



Before we proceed to the direct investigation of the laws of Reflexion and Refraction, we 

 shall make a few observations, which will be found useful hereafter. 



(1.) Let a, /3, 7 be the small displacements at any point {xyz) of a wave propagated 

 with a normal velocity («) \ p, q, s the direction of the cosines of v, and V the actual velocity 



of the vibrating particle, i.e. the resultant of the velocities -^, ~- , -~ ■ 



at at at 



