266 FBESNEL ON DOUBLE REFRACTION. 



normal plane is equal to B x M P. In fact, they are expressed 

 by the same coefficient B multiplied by the projections of the 

 three rectangular displacements on this plane; hence, to find 

 their resultant consists in finding the statical resultant of these 

 three projections considered as representing forces. Now, in 

 this point of view, the three rectangular displacements are the 

 statical components of the displacement M C, and consequently 

 their projections on the normal plane M S the statical compo- 

 nents of M P, which is therefore their resultant ; so that the re- 

 sultant of the three differential forces contained in the normal 

 plane is directed along M P, and represented by B x M P, that 

 is it is equal in magnitude and in direction to the differential 

 force arising from the displacement M C comprised in the same 

 normal plane. 



Therefore, finally, we find the molecule M urged by the same 

 differential forces, whether we make it undergo the small dis- 

 placement M C, or, supposing it successively displaced in three 

 rectangular directions and by quantities equal to the statical com- 

 ponents of M C in these directions, we find the resultant of the 

 forces produced by these three rectangular displacements. 



This principle, being true for the action exerted by the mole- 

 cule N, is equally so for the actions exerted by all the other 

 molecules of the medium on M ; so that we may rightly pro- 

 nounce that the resultant of all the small forces arising from the 

 displacement M C, or the total action of the medium on the mo- 

 lecule M after its displacement, is equal to the resultant of the 

 forces which would be separately produced by three rectangular 

 displacements equal to the statical components of the displace- 

 ment M C. 



Seco7id Theorem. 



In any system whatever of molecules or material points in 

 equilibrium, there exist always for each of them three rectan- 

 gular directions, along which every small displacement of this 

 point, by slightly changing the forces to which it is subject, pro- 

 duces a total resultant whose direction coincides with the line of 

 displacement itself. 



To demonstrate this theorem, in the first place I refer the 

 various directions of the small displacements of the molecule to 

 three rectangular axes, arbitrarily chosen, as axes of x, y, z. I 

 suppose that the molecule is displaced successively along these 

 three directions by the same small quantity, which I take as the 



