42 Mr. Mac Cullagh on the Laws of 



plane is most easily ascertained, for it is then nothing more than the plane of 

 polarisation of the common theory. For example, if we take the ordinary ray of 

 a uniaxal crystal, its polar plane will pass through the ray itself and the axis 

 of the crystal. Of course in an ordinary medium the polar plane and the plane 

 of polarisation are synonymous. 



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

 reflexion and refraction. Suppose then a polarised 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 polarisation of the refracted ray, and it 

 will intersect the reflected wave plane in the reflected transversal. The re- 

 fracted transversal will be the diagonal of a parallelogram, 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 Reports 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 formula; given by Fresnel for the same purpose will be found to agree exactly with this 

 rule, in determining the positions of the planes of polarisation ; and his expression for the am- 

 plitude of the reflected vibration is also in accordance with our construction. But the coincidence 

 does not hold with regard to the amplitude 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 equations of condition, deduce formulae agree- 

 ing in every respect, even as to the amplitude 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 polarisation, and not j)erpendicular 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 equa- 

 tions of condition this Value of the ratio of the masses, we obtain the formulae which I am inclined 

 to regard as correct." 



The equations spoken of in this extract, are those which arise from the principle 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 vibrations are all in the same direction, that is, when the light 

 is polarised perpendicular to the plane of incidence, the very same formulae will come out from 

 Young's remarkable analogy of the two elastic balls, one of which impinges directly on the other 



