380 FOURTH REPORT 1834. 



true or polarizing axes, and from which the forces which pro- 

 duce the phenomena of polarization and double refraction are 

 conceived to emanate. The polarizing force proceeding from a 

 single axis, is measured by the difference of the squares of the 

 velocities of the ordinary and extraordinary rays, and is supposed 

 to vary as the square of the sine of the angle which the direction 

 of the ray within the crystal contains with it ; and when two 

 such axes cooperate, it is assumed that the increment of the 

 square of the velocity, arising from their joint action, is equal to 

 the diagonal of a parallelogram whose sides are the increments 

 of the square of the velocity produced by each separately, and 

 whose angle is double of that formed by the two planes passing 

 through the ray and the axes*. From this hypothesis it followed 

 that two rectangular polarizing axes of equal intensity, and both 

 positive or both negative, compound a single resultant axis at 

 right angles to both. This axis is of the same intensity as the 

 component axes, but of an opposite character ; and, accordingly, 

 tliree equal rectangular axes of the same character balance each 

 other's effects, and have 7io resultant. Thus, then, the laws of 

 uniaxal crystals, as well as of singly-i'efracting media, are em- 

 braced in this hypothesis. The case of tivo resultant axes is 

 reducible to that of two unequal polarizing axes ; and it has been 

 shown to be a consequence of the rule that the difference of the 

 squares of the velocities of the ordinary and extraordinary rays 

 within the crystal, is proportional to the product of the sines of 

 the angles which the latter makes with the resultant axes. 

 M. Biot was led to the discovery of this beautiful law by ana- 

 logy!, and he afterwards observed that it was implicitly contained 

 in the law proposed by Sir David Brewster. 



The term " polarizing force" seems to have been adopted by 

 Sir David Brewster without any reference to the law which 

 governed the planes of polarization of the two pencils, — a law 

 which, in biaxal crystals, still remained unknown. In the case 

 of uniaxal crystals, it covild not fail to be observed, the plane of 

 polarization of one of the pencils contained the direction of the 

 ray and the axis ; while that of the other was a plane passing 

 through the ray at right angles to the former. Conceiving that 

 these planes, in biaxal crystals, must be symmetrically placed 

 with respect to the planes passing through the ray and the two 

 axes, M. Biot was led to the simple and elegant law — that the 

 plane of polarization of one of the pencils was that passing 



* " On the laws of Polarization and Double Refraction in regularly crystallized 

 Bodies," PM. rra»i. 1818. 



t " Memoire sur les Loisgenerales do la Double Refraction, &c.," Mem, Inst., 

 torn. iii. 



