280 FRESNEL ON DOUBLE REFRACTION. 



tinuing to subdivide in this way the vibrating medium by parallel 

 planes infinitely near and equidistant, in proportion as they are 

 further off from the first, the molecules which they contain are 

 found further removed from their original position relatively to 

 the material point which we are considering ; but this effect is 

 more than counterbalanced by the enfeebling of the forces re- 

 sulting from the increase of distance, and it ceases to be sensible 

 at a certain distance, which, without being probably altogether 

 to be neglected with regard to the length of an undulation, can 

 only be but a very small fraction of it. Whatever be the law 

 according to which the molecular forces vary with the distance, 

 it is natural to suppose that this law remains the same for the 

 same medium in all directions. I do not mean by this to say 

 that the molecules situated at the same distance from the mate- 

 rial point exert upon it in all directions equal repulsions ; but 

 only that these repulsions, though unequal, vary in the same 

 manner with the distance. 



Admitting this hypothesis, which is very probable from its 

 simplicity, we may conclude from it, I think, that the elasticity 

 put into play by the small displacements of the molecules does 

 not change so long as the direction and the extent of these dis- 

 placements remain the same at the same distance from the plane 

 of the wave, whatever besides may be the direction of this plane. 



Suppose, in fact, that the molecular displacements are always 

 parallel to the same direction ; and consider two different planes 

 di'awn through this direction, which shall represent successively 

 the surface of the wave in two different situations. Subdivide 

 the vibrating medium into infinitely thin and equidistant strata, 

 first parallel to the former plane, and afterwards parallel to the 

 second ; call 8 the small quantity by which the second stratum 

 or the second row of molecules becomes displaced relative to 

 that which is contained in the plane of departure ; the molecules 

 originally situated on straight lines perpendicular to this plane, 

 now form curved lines in consequence of the undulatory move- 

 ment; and the displacements are sensibly proportional to the 

 squares of the distances from the plane of depai'ture in those 

 strata sufficiently near to exert an appreciable action. Hence 

 4 S will be the quantity by which the molecules of the third row 

 will become displaced relatively to those of the plane of de- 

 parture; and in the same way 9 S, 16S, &c. will be the rela- 

 tive displacements of the succeeding strata. Similar displace- 



