Crystalline Reflexion and Refraction. 43 



The reason of this construction will be evident, if we consider that, in an 

 ordinary medium, the polar plane is the same as the plane of polarisation ; and 



supposed previously at rest, the masses of the balls being to each other in the ratio of the ethereal 

 masses mentioned above. And, perhaps, this consideration affords the simplest possible explanation 

 of Brewster's law relative to the polarising angle ; for, as there is no reflected motion when the balls 

 are equal, the whole velocity of impact being communicated to the ball that was at first quiescent, 

 so there is no reflected vibration 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 vibratioti 

 then passes into the refracted ray. In general, if » , (^ denote the angles of incidence and refraction, 

 the masses of the imaginary balls will be as sin2( to sin2i^ ; and, if the velocity of the original 

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



sin2t| — sin2(2 tan(t| — ij) 



sin2ij4-S'n2ij ■ tan(i,-|-t2)' 



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



2sin2i sin2t, 



(i-) 



(ii.) 



sin2tj-f-sin2ij sin(ij+(^)cos(i, — i^)' 



for the velocity communicated to the other ball. These expressions, (i.) and (ii.), are the same as 

 the values of t^ and r^, which we should deduce from equations (1) and (2), on the next page, by 

 supposing r^ to be unity, and the angles 6^, 9^, 9^ 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 polarised in that 

 plane, the incident, the reflected, and the refracted transversals are to each other as sin(( -(- '2)5 

 sin(i —I,), and sin2(j 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 



sin(t,— ij) 



sin(<,4-g 



and the refracted transversal is 



sin2i. 



(iii.) 



(ir;) 



8in(«,4-ij) ■ 



It has been already observed, that our theory differs from that of Fresnel with regard to the 

 magnitude of the refracted transversals. The expressions (ii.) and (iv.) must, in fact, be multiplied 



G 2 



