FRESNEL ON DOUBLE REFRACTION. 277 



of the vibrating medium, which were originally situated on the 

 straight line ADC; this curve may be represented, as we have 

 seen, by the equation 



y = 2b.sm2 7r (-j.sin27r (t—-J, 



which becomes y = 2 b . sin 2 ■tt A — ) when the molecules arrive at 



the limit of their oscillation ; at this moment their velocity is 

 nothing; and we may consider it as the origin of motion for the 

 ensuing oscillation, which must result from the accelerating 

 forces tending to bring Y'lg:. 6. 



back the molecules into ly 

 their relative positions 1 

 of equilibrium. Let m j 

 and m' be two material ■ . b 



points very near to and a!^^"!'^! ""^--^. ^p ^ 



equally distant from the /'j P- p V'' » ~\^ 



molecule M ; denote by ! 

 dx the constant length i 

 ofthe interval/? P or Py, j 



comprised between two consecutive ordinates. The difference be- 

 tween the ordinates M P and 7n'p' is the quantity by which the 

 point M is displaced from its primitive position relatively to the 

 molecules comprised in the plane drawn through m' perpendicu- 

 larly to the axis A C of the curve ; hence the accelerating force 

 exerted on M by this stratum of the medium, in consequence of 

 this displacement, is pi-oportional to m' p' — M' P. If we consider 

 the molecules comprised in the plane passing through the point m 

 and perpendicular to A C, their action on M resulting from their 

 relative displacement will also be proportional to the extent of 

 this displacement MP— mp, but will act in the contrary direc- 

 tion to that of the other accelerating force ; so that the resulting 

 action of these two equidistant strata on the molecule M will be 

 proportional to the difference of the two relative displacements, 

 or to (Py, if the distance Mp or M.p' is very small with regard 

 to the length of an undulation*. 



* In the note on the dispersion of light, placed at the end of the first part of 

 this memoir, I have examined the mechanical consequences which result from 

 the supposition that the mutual action of the molecules one on the other extends 

 to sensible distances relative to the length of an undulation ; for the present I 

 confine myself here to the more simple case treated by geometers, who have 



