1813.] Imperial Institute of France. 927 



The theory of crystaHization does not determine the lengths of 

 the sides opposite to these angles. M. Haiiy has determined 

 them so as to represent the secondary forms as simply as possihle^ 

 and has conceived them to be to each other in the ratio of 12 to 

 13. The axis of double refraction has no symmetrical relation 

 to such a parallelogram ; hut if we triple the side 12, allowing 

 the other to remain constant, the axis of double refraction will- 

 then pass along the greater diagonal of this new parallelogram, 

 and will form an angle of 16° 13' with the side 36. 



This direction being known, M. Biot exposed thin plates of 

 sulphate of lime, under an incidence perpendicular to a polarized 

 ray ; and analysed the light transmitted, making use successively 

 and indiscriminately of a rhomboid of Iceland crystal, or of the 

 reflection of a mirror. He observed two coloured images, as 

 M. Arrago had announced, and he ascertained that they possessed 

 the following characters : 1. A part of the incident light, v/hicli 

 we shall call E, is polarized by the plate ; the remainder, O, 

 preserves its primitive polarization. 2. The colour of the portion 

 polarized by the plate is the same in what azimuth soever its 

 axis is placed, relative to the plane of polarization of the ray. 

 3. When the light is analysed by means of a rhomboid of Ice- 

 land crystal, the principal section of which is directed according 

 to that plane, the ordinary image given by the rhomboid is con- 

 stantly a mixture of the two colours O and E. The extraordinary 

 image is always of the colour E ; and the separation of the two 

 colours is complete when the axis of the plate forms an angle of 

 45° with the plane of polarization of the ray. 



M. Biot attempted at first to represent these phenomena by 

 the same formulas that Malus had given for the intensity of the 

 pencils given by the rhomboids of carbonate of lime. He found 

 that this law would not apply. He endeavoured to dis-cover the 

 niodiiications which it must undergo, and by multiplying his 

 observations every way he found the two formulas following, 

 which represent all the phenomena. Let us suppose that the 

 axis of the plate makes an angle i with the plane of polarization 

 of the incident ray; let us suppose, likewise, that the transmitted 

 light is analyzed by means of a rhomboid of calcareous spar, the 

 principal section of which makes an adgie a with the same plane. 

 Let us call E the intensity of the portion of incident light which 

 the plate polarizes, and let O be the complementary portion 

 which preserves its primitive polarization. If we denote by F^, 



F^, the intensities of the two ordinary and extraordinary pencils, 



observed across the rhomboid, we shall have 



== O cos/^ a -f E C0S.2 (2 i — a), 



F = O sin.® a + E sin."- (2 i — a). 



If we wish to analyse the transmitted light by making use of 

 p 2 



