298 Dr. Beer on the deduction o/FresnePs construction from 



otherwise they may be situated with regard to the axes of the 

 crystal ; that, on the contrary, the velocity of oscillations which 

 are parallel to a principal axis are dependent on the position of 

 the plane of the waves. From this it would follow, that the 

 vibrations of the extraordinary ray in a crystal of one axis are 

 perpendicular to the said axis ; that those of the ordinary re- 

 fracted ray, on the contrary, lie in a principal section. One 

 might be disposed to believe that the proof is hereby furnished, 

 that in the case of plane-polarized light the plane of oscillation 

 and the plane of polarization coincide, and thus to regard the 

 other view, according to which these planes stand perpendicular 

 to each other, as refuted. This apparent proof is, however, illu- 

 sory. Proceeding from the formula? of Cauchy, the last-men- 

 tioned view, which is known to be that to which Fresnel adhered, 

 may be established, if other quantities be neglected than those 

 usually fixed upon, but which we are equally entitled to neglect. 

 I shall show this in the following ; and by reference to figures 

 in space, will endeavour to make this subject plainer than would 

 be possible if we confined ourselves to the geometrical expression 

 of the final result alone. I am induced to publish the present 

 paper by the reading of a memoir from W. J. M. Rankin e in 

 the June Number of this Magazine, in which the question as to 

 the direction of vibration in plane-polarized light is handled, 

 and to which the following in some measure attaches itself. 



In his Memoire sur la Dispersion de la Lumiere, Cauchy has 

 shown that in every homogeneous medium, to every plane P 

 and definite length of wave \ an ellipsoid E belongs, which he 

 has named the ellipsoid of polarization, with the aid of which it 

 is easy to give a simple expression to the laws according to which 

 waves of the length \, whose planes are parallel with P, proceed 

 through the said medium. In general only three plane-polarized 

 waves proceed parallel with P ; their directions of vibration run 

 parallel with the three axes of E ; and if we express the halves 

 of these axes by A, B and C, then the velocities of these thi'ee 

 waves respectively are 



: iV' ^ 27rA* 27r*B ^"^ 27r*C* 



Supposing the centre of the ellipsoid E to be the origin of a 

 system of orthogonal coordinates, the said origin being conceived 

 to coincide with any molecule of aether, it will be expressed by 

 an equation of the following form : 



Lx^ + Ml/ + N^2 + 2?ys + 2Q/VZ + 2Ry<j= 1 . 



Between the coefficients of this equation on the one side, and 

 the constitution of the sether, as well as the direction of the 



