56 FRESNEL ON THE COLOURS PRODUCED IN 



ratus is applicable to the fluid. If, then, o and e represent the 

 numbers of ordinary and extraordinary undulations in the fluid, 

 and i the angle which the primitive plane of polarization makes 

 with the principal section of the rhomboid of calcareous spar 

 that serves to develope the colours, we obtain, as a general ex- 

 pression for the intensity of the luminous vibrations in the ordi- 

 nary image, 



F. V 2+2*^°^ [2i— 2 7r(e— o)], or F. cos [i— 7r(e — o)], 



F- being the intensity of the incident pencil; and for the extra- 

 ordinary image, 



F . sin [i—ir (e — o)]. 



These formulae have been calculated for the case in which the 

 axis of the crystalline lamina inserted between the two glass 

 parallelopipeds was to the right of the first plane of double re- 

 flexion ; they apply consequently to those fluids the particles of 

 which have their principal section to the right of their plane of 

 entrance. In the opposite case, the formulfe become 



F . cos \i + T( {e — o)] for the ordinary image, and 

 F . sin [i + v {e - o)] for the extraordinary image. 



M. Biot observed that the angle through which the principal 

 section of the rhomboid of calcareous spar must be turned, in 

 order to cause the disappearance of the same kind of rays of the 

 extraordinary image, was proportional to the length of fluid tra- 

 versed. This remarkable law is an immediate consequence of 

 the preceding foi'mulae. In reality, the kind of rays in question 

 w'ould cease to exist in the extraordinary image when we have 

 i + n {e—o) = o, or i = i "■ (^~o) ; the upper signs correspond 

 to the case in which the particles have their principal section to 

 the right of their plane of entrance, and the lower signs to the 

 contrary case. But e and o are proportional to the distance tra- 

 versed in the fluid ; consequently the angle i must also be pro- 

 portional to it. 



If it be supposed that e^ o, the first value for i will be posi- 

 tive and the second negative. The angles having been reckoned 

 from left to right in the calculations, w^e must conclude from^ 

 these values for i, that in the first case the light rotates from I 

 left to right, and in the second from right to left, using the lan- 

 guage of M. Biot, which is the simplest mode of expressing the 

 appearances of the phaenomenou. If, on the contrary, we suppose 



