Reflexion and Refraction . I o I 



The reason of this construction will be evident, if we con- 

 sider that, in an ordinary medium, the polar plane is the same 

 as the plane of polarization ; and that, when there is only one 

 refracted ray, the three transversals lie in the polar plane of 

 that ray, according to the general remark with which we set 

 out. We now proceed to show that the theorem asserted in 

 this remark is a consequence of our hypotheses, and we shall 

 afterwards deduce a few results which may be readily compared 

 with experiments. 



Let us suppose then that the direction of the incident trans- 



rizing angle ; for, as there is no reflected motion when the halls are equal, the whole 

 velocity of impact heing communicated to the hall that was at first quiescent, so 

 there is no reflected vihration when the ethereal masses are equal ; that is, when 

 the sine of twice the angle of incidence is equal to the sine of twice the angle of 

 refraction, or when the angles of incidence and refraction are together equal to a 

 right angle. The whole of the incident vihration then passes into the refracted 

 ray. In general, if t lt t a denote the angles of incidence and refraction, the masses 

 of the imaginary halls will he as sin 2^, to sin 2t 2 ; and, if the velocity of the ori- 

 ginal impact be taken for unity, the common theory of the collision of elastic bodies 

 will give 



sin 2t, - sin 2t, tan (i. i.) 



. _ i _ ? Qp . _ ^ 1 2/ /j \ 



sin 2^ + sin 2i a tan (^ + i a )' 



for the velocity retained by the impinging ball after the impact ; and 



sin 2^ + sin 2 a sin(( 1 + Jcos(( 1 -j a )' '' 



for the velocity communicated to the other hall. These expressions (i.) and (n.), 

 are the same as the values of r 3 and r 2 , which we should deduce from equations (1) 

 and (2), on the next page, by supposing T X to be unity, and the angles 6\, 62, 63 to 

 be right angles. The general construction given in the text will lead to the same 

 results, if we find from it the limiting ratios of the transversals, on the supposition 

 that their directions approach each other indefinitely, and ultimately coincide in a 

 right line perpendicular to the plane of incidence. 



When the transversals are all in the plane of incidence, or when the light is 

 polarized in that plane, the incident, the reflected, and the refracted transversals 

 are to each other as sin (t^ + ij, sin (*!-,), and sin 2t 2 respectively; because each 

 transversal is proportional to the sine of the angle between the other two; and, in 

 the present case, the angle between any two transversals is equal to the angle 

 between the corresponding rays. Hence, taking the incident transversal for unity, 

 the reflected transversal is 



*! n <''-''); (in.) 



