270 FRESNEL ON DOUBLE REFRACTION. 



parallel to . . z x y 



Acos^Z-B.sin^Z, (A + B)cosX.cosZ, (A + B)cosY.cosZ. 

 The simple inspection of the components of the dift'erential 

 forces excited by these three small displacements, shows that the 

 displacement parallel to x gives in the direction of the axis of y 

 the same component as the displacement parallel to y produces 

 in the direction of the axis of x; and gives in the direction of 

 the axis of z the same component as the displacement parallel 

 to z produces in the direction of x ; and lastly, that the com- 

 ponent parallel to z of the force excited by the displacement 

 along the axis of?/ is equal to the component parallel to y of the 

 force excited by the displacement along the axis of z ; that is to 

 say, generally, the component parallel to one axis produced by 

 the displacement along one of the two others is equal to that which 

 results parallel to this latter from a similar displacement parallel 

 to the former axis. 



This theorem being demonstrated for the individual action of 

 each molecule M on the point A, is consequently proved also for 

 the resultant of the actions exerted by all the molecules of a me- 

 dium on the same material point; hence there exists always 

 between the nine constants a, b, c, a', b', d, a", b", c" the three 

 following relations : — 



b = a<, c = a", c' = b"; 



which reduces the number of arbitrary constants to six. 



We may then in general represent as follows the components 

 of the three forces resulting from three small displacements 

 equal to unity, and operated successively along the axes of x, y 

 and z : — 



For the displacement along the axis of x, — 



Components a, h, g. 



Parallel to ^lo^y, z. 



For the displacement along the axis of y, — 



Components b, h,f. 



Parallel to y,^"^^' 



For the displacement along the axis of z, — 



Components <^) Oif- 



Parallel to z,x,y. 



Thus the three components of a similar displacement in any 

 direction whatever, making with the axes of x, y and z angles 

 respectively equal to X, Y, Z, will be — 



