FRESNEL ON DOUBLE REFRACTION. 283 



two successive displacements of a single molecule along similar 

 directions ; and we may apply to the complex displacements re- 

 sulting from luminous waves the principles before demonstrated 

 for the case where one molecule is disturbed from its position 

 of equilibrium, whilst all the others remain fixed. 



This being established, let us take the three axes of elasticity 

 of the vibrating medium as coordinate axes, and denote by 

 o^, h^, c^ the elasticities put into play by vibrations parallel to 

 the axes of x, y, z, so that the corresponding velocities of pro- 

 pagation, which are proportional to the square roots of the elas- 

 ticities, are represented by a, b, c: we propose to determine 

 the elastic force resulting from vibrations of the same nature, 

 but parallel to any other direction whatever, making with these 

 axes the angles X, Y, Z. I take as unity the amplitude of 

 these vibrations, or the constant coetficient of the relative dis- 

 placements of the parallel strata of the medium ; for in order to 

 compare the elasticities, it is necessary to compare the forces 

 resulting from equal displacements. This coefficient being equal 

 to 1, those of the components parallel to x, y and z will be 

 cos X, cos Y, cos Z. We know besides that these forces will 

 have the same directions, according to the characteristic pro- 

 perty of axes of elasticity. 



Hence, denoting by (/") the resultant of these three forces, we 

 shall have 



/= \/a4 . cos2 X + 6" . cos2 Y + c4 . cos^^ Z ; 



and the cosines of the angles which tliis resultant makes with 

 the axes of x, y, z, will be respectively equal to 



o^.cosX fi^.cosY c^.cosZ 



~^' ~T~' ~J^' 



We see that in general this resultant has not the same direction 

 as the displacements which have produced it. But we can 

 always decompose it into two other forces, one parallel and the 

 other perpendicular to the direction of the displacements. If 

 the second force be found at the same time normal to the plane 

 of the wave, it will no longer have any influence on the propa- 

 gation of luminous vibrations, since according to our fundamental 

 hypothesis, the luminous vibrations are performed solely in the 

 direction of the surface of the waves. Now we shall take care 

 to reduce to this case all calculations relative to the velocities of 



