ioo On the Laws of Crystalline 



It may not be amiss to apply our general ' rules to the case 

 of ordinary reflexion and refraction. Suppose then a polarized 

 ray to fall on the surface of an ordinary medium. Draw a plane 

 through the incident transversal and the refracted ray ; this will 

 be the plane of polarization of the refracted ray, and it will 

 intersect the reflected wave plane in the reflected transversal. 

 The refracted transversal will be the diagonal of a parallelo- 

 gram, whose sides are the other two transversals; hence we 

 have the relative lengths of the transversals, and thus every- 

 thing is determined.* 



* This construction was mentioned at the meeting of the British Association in 

 Dublin. See the Keports of the Association, or London and Edinburgh Phil. Mag. 

 vol. vii. p. 295. The following is an extract from the Paper which I read at that 

 meeting : 



" The formulae given by Fresnel for the same purpose will be found to agree 

 exactly with this rule, in determining the positions of the planes of polarization ; 

 and his expression for the amplitude of the reflected vibration is also in accordance 

 with our construction. But the coincidence does not hold with regard to the am- 

 plitude of the refracted vibration, though the vis viva of the refracted ray is the 

 same in both theories. 



" Now it is very remarkable that if we alter the hypotheses of Fresnel where 

 they are at variance with the preceding principles, we shall, from his own equa- 

 tions of condition, deduce formulae agreeing in every respect, even as to the ampli- 

 tude of the refracted wave, with the construction which we have accounted for in 

 a different way (i. e. by using the relation among the pressures instead of the law 

 of vis viva}. The requisite alterations are two in number. First, the vibrations 

 are to be supposed parallel to the plane of polarization, and not perpendicular to it, 

 as Fresnel conceived ; and secondly, the density of the ether is to be considered the 

 same in both media, from which it follows, that the corresponding ethereal masses, 

 imagined by Fresnel, are to each other as the sine of twice the angle of incidence 

 to the sine of twice the angle of refraction. Substituting in Fresnel's equations of 

 condition this value of the ratio of the masses, we obtain the formulas which I am 

 inclined to regard as correct." 



The equations spoken of in this extract are those which arise from the prin- 

 ciple of vis viva, and from the equivalence of vibrations parallel to the separating 

 surface of the two media. But it is worth while to observe, that when the vibra- 

 tions are all in the same direction, that is, when the light is polarized perpendicular 

 to the plane of incidence, the very same formulae will come out from Young's re- 

 markable analogy of the two elastic balls, one of which impinges directly on the 

 other, 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 pola- 



