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to the analysis of a pencil thus composed, after it has suffered re- 

 flexion at different angles of incidence ; and for this purpose makes 

 the plane of reflexion bisect the right angle formed by the plane of 

 polarization of the two pencils composing the beam. As the angle 

 of incidence diminishes from 90, these latter planes are gradually 

 turned more and more towards the plane of reflexion ; so that the 

 angles they form together become more and more acute, until the 

 incidence becomes that of the angle of complete polarization, in the 

 particular medium from which the reflexion takes place ; in that case, 

 the planes become parallel, and the whole of the beam is then com- 

 pletely polarized. As the angle of incidence still further diminishes, 

 these planes become again inclined to one another, but in contrary 

 directions. Thus the total polarization of the reflected pencil at the 

 polarizing angle is effected by the turning round of the planes of 

 polarization of one half of the light from right to left, and of the 

 other half from left to right, each through an angle of 45. But 

 when the pencil is only partially polarized, each plane of polarization 

 has been turned round in opposite directions, from an inclination of 

 45, to one either less or greater than this. The light has, in this 

 case, suffered a physical change of a remarkable kind ; for it now 

 constitutes neither natural nor polarized light, but something inter- 

 mediate between both. It is not the former, because its planes of 

 polarization are not rectangular ; nor the latter, because they are 

 not parallel. The examination of a pencil of this description by a 

 doubly-refracting medium, which was the test employed by those 

 who conceived the polarization to be complete in one portion while 

 the remaining portion was wholly unpolarized, does not afford the 

 means of deciding this question ; for the results, as the author shows, 

 would be the same in either case. By applying the law of repar- 

 tition of light, when doubly refracted, between the ordinary and ex- 

 traordinary rays discovered by Malus, namely, that it follows the 

 duplicate ratio of the sine and cosine of the angle of inclination to 

 the principal section of the crystal, we obtain the same expression 

 for the intensities of both rays, whichever of the hypotheses we adopt. 

 But the proof which a single reflexion is unable to afford is supplied 

 by examining the results of a succession of reflexions. We are thus 

 furnished with means of comparing with great precision the deduc- 

 tions from theory with the results of experiment, and of establishing 

 the correctness or fallacy of both hypotheses. This investigation is 

 pursued by the author in the body of the paper, in considerable de- 

 tail. He finds the formula given by Fresnel, expressing the law of 

 change in the plane of polarization by reflexion, to be perfectly con- 

 formable with observation, the results of which it also expresses with 

 great accuracy when applied to rays which have undergone partial 

 polarization. He controverts the accuracy of a proposition advanced 

 by Arago, that equal proportions of light are polarized at equal an- 

 gular distances from the angle of complete polarization. As the in- 

 dex of refraction differs for the different colours of the spectrum, 

 the polarizing angle will be different in each. In bodies of high 



