478 Prof. Challis on the Theory of Double Refraction 



The equations given by equating to zero the right-hand sides of 

 these equalities are respectively satisfied by # = and y = 0, 

 showing that the elasticity in the direction of the axis of z is a 

 maximum or minimum. The same is evidently the case with 

 respect to the other two axes. As these results have been ob- 

 tained independently of the forms and values of the functions <fi 

 and/, it appears that every crystal which satisfies the assumed 

 law of symmetrical atomic arrangement has three axes of maxi- 

 mum or minimum elasticity at right angles to each other. 



Next let the coefficients of elasticity in the directions of the 

 three axes of x, y, oo be respectively ef, e 2 2 , e 3 2 , and such that 

 the force which is produced by a given relative displacement of 

 the atoms in the direction of an axis is equal to the coefficient 

 x displacement. We have now to find the coefficient of elasticity 

 for any direction making the angles a, ft, 7 with the axes. For 

 this purpose we shall regard, as heretofore, the crystalline medium 

 as consisting of discrete atoms held in positions of stable equili- 

 brium by attractive and repulsive forces, and shall assume that, 

 according to the law of the coexistence of small vibrations, each 

 atom can perform independently simultaneous oscillations in dif- 

 ferent directions in obedience to simultaneous disturbances. On 

 this principle we may consider an actual displacement (Bv) in the 

 given direction to be the resultant of the three displacements 

 Bv cos a, Bv cos /3, Bv cos 7 in the directions of the axes. Now 

 these displacements, by hypothesis, give rise to forces in the 

 directions of the axes equal to e, 2 x Bv cos a, e 2 2 x Bv cos /5, 

 e 3 2 x Bv cos 7. And it has already been explained that the forces 

 which these expressions represent are due to condensations of 

 the crystal considered as a continuous substance, and are not in 

 kind different from the forces we are concerned with in ordinary 

 mechanics. On this account these forces, and also the force due 

 to the original displacement, take effect wholly in the directions 

 of the displacements. For the same reason, the three forces in 

 the directions of the axes may be resolved in the direction of the 

 original displacement, and the sum of the resolved parts, viz. 



(ej 2 cos 2 a + e£ cos 2 j3 + <? 3 2 cos 2 7) x Bv, 

 being the total force in that direction, must be equal to. the force 

 due to the original displacement. If, therefore, e 2 be the coeffi- 

 cient of elasticity for the given direction, we shall have 



e 2 = £j 2 cos 2 u + e£ cos 2 /3 + e 3 2 cos 2 7. . . . (/3) 



Reverting again to the equation (a), it is important to remark 

 that for a ray having a given value of \ in vacuum, which for 

 brevity we shall call a ray of given colour, the product \/j, is con- 

 stant ; that is, if X be the value of X in vacuum, we shall have 

 for the same colour in a medium \ =\/jb. The reason for this 



