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The constructions to which the author has been led by 

 his theory are extremely simple, and may be explained most 

 easily by referring to a paper which he has already pub- 

 lished in the Transactions of the Academy, vol. xvii. pp. 251, 

 252. To avoid circumlocution, he uses the term transversal, 

 to denote a right line parallel to the plane of polarization of 

 a ray, and perpendicular to the direction of the ray itself. 

 When the transversal is spoken of as a finite magnitude, its 

 length is understood to be proportional to the amplitude 

 of the vibrations in the polarized ray. Let o (as in the 

 place just referred to) be the point of incidence on the crys- 

 tal, and ot, ot', the directions of the two refracted rays, the 

 points t, t', being on the wave-surface. Corresponding to 

 the points T and t' on the wave-surface, there are two other 

 points, p and m, on a second surface which is reciprocal to the 

 wave-surface. The points p and m are derived from the 

 points t and t' by an easy rule which is given in the place 

 before cited. Now if we wish to find in what direction the 

 incident ray must be polarized in order that the ray ot' may 

 disappear, let us draw, through the point o, a plane a 

 perpendicular to the plane otp, and parallel to the right 

 line tp, which joins the corresponding points, T, p. This 

 plane a will intersect the planes of the incident and re- 

 flected waves in two right lines, which will be the transversals 

 of those waves ; so that if the incident ray or wave be pola- 

 rized parallel to the first intersection, the reflected ray will 

 be polarized parallel to the second intersection, and there 

 will be only a single refracted ray ot. A right line 

 drawn through the point o, perpendicular to the plane 

 otp, will lie in the plane A, and will be the transversal of the 

 refracted ray ot ; and if, measuring from the point o, the 

 lengths of the three transversals represent the amplitudes of 

 the respective vibrations, the transversal of the refracted ray 

 ot will be the diagonal of the parallelogram, whose sides are 

 the transversals of the incident and reflected rays. The prob- 



