260 FRESNEL ON DOUBLE REFRACTIO.V. 



forces which act along FM and along EM are each represented 

 by <p { x^w'^ + y'^). Moreover, the sine of the angle FMB is 



equal to — and its cosine to — r== ; therefore the two 



components of the force acting along FM are, parallel to x, 



X I 



• <I> ( v^.r- + y'^), or .r . \J/ . {x'^ + if-), and parallel to 



s/ x^ + y^ 



y 



y • ■ g I • ^ •\s/ x^ + y-) or ?/ . vp . (jr^ + ?/2), if we take for 

 \ X -V y 



the positive direction of the forces parallel to the axes of coordi- 

 nates, that in which each of these two components acts. Simi- 

 larly, the two components of the action exerted by the molecule E 

 are respectively — x .<]> [x^ -{- y'^) and y . '\> . [x"^ -\- y^) ; that is to 

 say, they only differ from the former in the sign of {x). Now, 

 to calculate the small quantities by which these components are 

 altered in consequence of the displacement of the point M, we 

 must differentiate their expressions with respect to x ; we find 

 thus, for the differentials of the components of the force F M, 



parallel to .x" [\|/ (,2?^ + ?/^) + 2x~^' {x^ + ?/^)] dx, 



parallel to y 2xy .^' {x^ + y'^) dx. 



The expression for the force E M differing only from that for the 

 force F M by the sign of x, we may obtain at once the variations 

 of its components by simply changing the sign of x in the two 

 preceding expressions, without changing, be it understood, that 

 of the small displacement dx, which takes place in the same 

 direction for both forces. Now, by the mere inspection of these 

 formulae, it is seen that the differential of the component parallel 

 to X will preserve the same sign, and will therefore be added to 

 that of the force FM, whilst the differential of the component 

 parallel to y will be subtracted from the corresponding variation 

 of the other force, and will destroy it. There results, therefore, 

 from the small displacement of the point M along AB, a force 

 parallel to the same line, and which tends to bring back this 

 point towards its position of equilibrium. 



Therefore if, the point M remaining fixed, the superior portion 

 of the medium be slightly displaced parallel to AB (which comes 

 to the same thing), the point M will be pushed in the direction 

 AB, as well as all the other molecules of this layer, which will 

 therefore be urged throughout its whole extent to slide along its 



