Plane-Polarized Light. 445 



function of direction round a centre M'hich is proportional to the 

 reciprocal of the square of the diameter of an ellipsoid of small 

 excentricities, so that the range of variation of the function is 

 small as compared with its amount, then any function of that 

 function, whose range of vai'iation is small also, may be repre- 

 sented approximately by the reciprocal of the square of the dia- 

 meter of another ellipsoid, having its centre and the directions 

 of its axes the same with those of the first. 



The square of the \'elocity of jiropagation of transverse vibra- 

 tions is proportional to the transverse elasticity of the medivmi 

 divided by the mean density ; that is, by the sum of all the 

 vibrating masses in unity of volume. That sum is the sum of 

 the masses of the nuclei, added to the masses of atmosphere with 

 which they ai'e loaded. The atmospheric load of each nucleus 

 depends on, or is a function of, the density of the atmosphere 

 adjoining the nucleus along the line in Avhich the latter vibrates. 

 The mode of distribution of the atmospheres depends on the 

 attraction of the nuclei upon them, and therefore on the mode 

 of arrangement of the nuclei. The mode of arrangement of the 

 nuclei, when it is symmetrical and uniform, may be expressed by 

 means of their mean intenmls. 



The mean interval of the nuclei is a function of direction, of 

 such a natui'e that its three ^'alues for any thi'ee orthogonal 

 directions being multiplied together give a constant result, viz. 

 the space occupied (not filed) by one nucleus, or the quotient 

 of a given space by the number of nuclei contained in it. Hence 

 the sum of the values of the logarithm of the mean interval for 

 any three orthogonal directions is a constant quantity ; and that 

 logarithm, therefore, is proportional to the reciprocal of the 

 square of the diameter of an ellipsoid, whose three axes may be 

 called the axes of atomic distribution. Therefore the mean in- 

 terval, the atmospheric load of the nuclei, and the square of the 

 velocity of propagation, for a given direction of transverse vibra- 

 tion, are all functions of the reciprocal of the square of the dia- 

 meter of an ellipsoid, and have maxima and minima corresponding 

 to its three axes, which are those of atomic distribution. 



Now in all known crystalline media the range of variation of 

 these quantities for different directions is very small compared 

 with their amount. Therefore each of them may be approxi- 

 matelij represented by the reciprocal of the square of the diameter 

 of an ellipsoid whose axes are parallel to those of atomic distri- 

 bution. 



If, then, the directions of viliration in a given crystal which 

 correspond to the greatest and least velocities of transmission 

 arc known, let these directions (which arc at right angles to each 

 other), and a third direction at right angles to them both, be 

 taken for the axes of an ellipsoid, the lengths of those axes being 



