FBESNEL ON DOUBLE UEFRACi'iON. 253 



might therefore limit here the development of these considera- 

 tions whose special object it is to give the theoretical demonstra- 

 tion of the rules for calculating the tints of crystalline plates. 

 We think however that it will not be useless to point out here 

 some of the most simple consequences of these principles. 



I suppose that a beam of polarized rays falls perpendicularly 

 on a crystalline plate situated in the plane of the figure. Let, 

 as before, PP' denote the direction parallel to which the vibra- 

 tions of the incident beam are performed ; OO' and EE' those 

 of the vibrations of the ordinary and extraordinary beams into 

 which it is divided after having penetrated into the crystal- 

 Suppose this crystalline plate to be sufficiently thin, that there 

 may be no sensible difference of route between the two emergent 

 beams, or that it has such a thickness that the difference of 

 route may contain a whole number of undulations, which comes 

 to the same thing ; all the points taken on the ray projected in 

 C, for example, are simultaneously urged in the two systems of 

 waves by velocities which correspond to the same epochs of the 

 oscillatory movement ; they will have therefore at each point of 

 the ray the same ratio of intensity, that namely of the constant 

 coefficients of the absolute velocities of the two systems of waves ; 

 therefore their resultants will be parallel, and will all be projected 

 in the direction PP', since the components are all, two and two, 

 in the ratio of cos i to sin i. Thus the light arising from the 

 union of the two emergent beams will still be polarized, since 

 all its vibrations will be performed in parallel directions, and its 

 plane of polax'ization will be the same as that of the incident 

 beam. 



Suppose now the difference of route of the ordinary and 

 extraordinary beam, on emergence from the crystal, to be a 

 semi-undulation, or an uneven number of semi-undulations; 

 this is as if, the difference of route being nothing, we were to 

 change the sign of all tlse absolute velocities of one of the two 

 systems of waves ; thus, the velocity which urges the molecule 

 C at a given instant, in the first beam pushing it from C towards 

 O for example, that which is caused by the second beam, instead 

 of pushing this molecule from C towards E', as in the preceding 

 case, will push it from C towards E ; so that the I'psultant of 

 these two impulses, instead of being directed along C P, will 

 have the direction of a line situated on the other side of C O, 

 and making with this latter an angle equal to the angle (?) con- 



