252 FKESNEL ON DOUBLE REFRACTION. 



the other in the direction CE'. Now, according to the general 

 principle of small motions, these components are the absolute 

 velocities in the two systems of waves which result from the 

 decomposition of the first. If we suppose OO' and EE' at right 

 angles, as is the case for the directions of the ordinary and ex- 

 traordinary vibrations in a doubly-refracting crystal, the com- 

 ponent CO will be equal to the first absolute velocity multiplied 

 by cos i, and the component CE' to the same velocity multiplied 

 by sin ^. We are thus led to a very simple explanation of the 

 law of Malus, on the relative intensities of the ordinary and ex- 

 traordinary images by passing from the absolute velocities to the 

 vires vivce, which are proportional to their squares, cos'^ i and 

 sin^ i. But let us return to the components CO and CE'. If 

 we decompose them each into two others in the directions SS' 

 and TT', there will result for the former CO two velocities in 

 the directions CS and CT; and for the second CE', two com- 

 ponents acting in the directions CS and CT'. It is seen that in 

 the plane SS' the two resulting components act in the same 

 direction and are added to each other ; whilst they act in con- 

 trary directions in the plane TT', and must therefore be aflfected 

 with contrary signs ; which justifies the rule we have announced; 

 for what we have just said applies equally to all the points taken 

 on the ray projected in C, and therefore to the constant coeffi- 

 cient which multiplies all the absolute velocities of each system 

 of waves. This law, the enunciation of which may at first sight 

 have appeared complicated, is in substance, as we see, only a 

 very simple consequence of the decomposition offerees*. 



The principles just established with regard to the interference 

 of polarized ra3-s suffice for the explanation and calculation of 

 all the phsenomena of the colours of crystalline plates. We 



* I think it needless to give here the explanation of the fourth law of inter- 

 ference of polarized rays, which is a result of the present one, as I have shown 

 it to be in a Note joined to the Report of M. Arago, in the Annahs ile Chim'ie 

 et de Phijsique, torn. xvii. p. 101. This law consists in this, that the rays 

 which have been polarized at right angles, and are afterwards brought back to 

 the same plane of polarization, cannot present phsenomena of interference ex- 

 cept in so far as the primitive beam has received a previous polarization. Not 

 that they do not necessarily exert a mutual influence on each other as soon as 

 their vibratory motions are brought back to a common direction ; but the light 

 which has not recei.ed any previous polarization, and which may be considered 

 as the union of an infinity of systems of waves polarized in all directions, when 

 analysed by a rhomboid of calcareous spar after its passage across a crystallized 

 plate, produces at the same time in each of the two images opposite effects, 

 which mutually disguise each other (sc masquent), as may be easily deduced 

 from the law just explained, 



