FRESNEL ON DOUBLE REFRACTION. 281 



ments, be it understood, are supposed on the other side of the 

 plane. 



If all these displacements, instead of increasing with the 

 distance, wei'e equal to S, the elasticity put in play would be the 

 same as in the case where, the medium remaining immovable, 

 the molecules only comprised in this plane had slided by 

 the small quantity 8. It will be moreover remarked that, 

 if there were only one of these molecules displaced from its 

 position of equilibrium, the direction of the plane in question 

 would have no influence on the force to which it woidd be 

 subject. 



Call this force F ; it is the sum of the actions exerted on the 

 molecule remaining fixed by all the strata of the medium. Now, 

 to pass from this case to that with which we occupied ourselves 

 in the first place, it would be necessary to multiply the action of 

 the first stratum by zero, that of the second by 1, that of the 

 third by 4, that of the fourth by 9, &c. Since in this case 

 the first stratum has not changed its position, the second is dis- 

 placed by the quantity S, the third by 4 S instead of S, the fourth 

 by 9 8, and so on : we should have, besides, the same progression 

 whatever were the direction of the plane of the wave. Hence 

 we must always multiply the individual actions of the strata 

 situated in the same rank by the same numbers, in order to take 

 into account the extent of their displacements ; moreover, the 

 coefficients, which depend on the distance of each stratum from 

 the fixed molecule, will also be the same at equal distances, 

 supposing, as we have done, the molecular actions to diminish 

 in all directions according to the same function of the distances ; 

 consequently the total numerical series by which F must be 

 multiplied to obtain the elastic force which results from the un- 

 dulatory movement, will remain constant for the different direc- 

 tions of the parallel strata, or of the plane of the wave, and this 

 force will depend only on the mere direction of the molecular 

 displacements. 



Application of the preceding principles to media where the Axes 

 of Elasticity preserve the same direction throughout their whole 

 extent. 



If this principle be admitted, the theoretical probability of 

 which I have just shown, and whose accuracy I have besides 

 verified by very precise experiments on the velocities of light in 



