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of Mr. Mac CuUagh's briefer paper, which have appeared to me to 

 deserve a peculiar and special attention. I mean the geometrical 

 elegance of the principal enunciations, and the philosophical cha- 

 racter of the interspersed remarks. 



As a specimen of the former, I shall select the theorem of the 

 polar plane. When light in air is incident on a doubly-refracting 

 crystal, it may be polarised in such a plane, that one of the two 

 refracted rays shall disappear ; and then the one refracted vibration 

 which corresponds to the one remaining refracted ray, must (by 

 the hypotheses or laws already mentioned) be the resultant of the 

 one incident and one reflected vibration ; and consequently these 

 three vibrations must be contained in one common plane, which 

 plane it is therefore an object of interest to assign a simple rule 

 for constructing. In fact, the refracted vibration is known, in 

 direction, from the laws of propagation of light in the crystal, and 

 the hypotheses already mentioned ; if, then, we know how to draw 

 through its direction the plane just now referred to, we should only 

 have to e:^amine in what lines this plane intersected the incident 

 and reflected waves, in order to obtain the direction of the incident 

 and reflected vibrations, and afterwards (by the rules of statical 

 composition) the relative magnitudes of all the three vibrations, or 

 the relative intensities of the incident, reflected, and refracted lights. 

 Now Mr, Mac Cullagh shows, that the desired construction can be 

 deduced from the properties of the doubly refracting medium or 

 wave, as follows : Let ot, op represent in length and in direction 

 the velocity of the refracted ray, and the slowness of the re- 

 fracted wave; so that, by what has been before supposed, the 

 refracted vibration ov is perpendicular to the plane top ; then, 

 if a plane be drawn through the vibration ov, parallel to the 

 line TP, this plane, which Mr. Mac Cullagh calls i\ie polar plane 

 of the ray ot, will be the plane desired ; that is, it will contain the 

 incident and the reflected vibrations, if these be uniradial, or, in 

 other words, if they have such directions, or correspond to such 

 polarisations, as to cause one of the two refracted rays in the crystal 

 to disappear. 



Many elegant geometrical corollaries are drawn, in the Essay; 

 from this theorem of the polar plane ; but I shall only mention one, 



