FRESNEL, ON DOUBLE REFRACTION. 265 



the direction of the force has varied. If therefore we refer the 

 original force and the new one to two rectangular directions, M R 

 and M S, the differential in the direction M R will only arise 

 from the small increase C P of the distance, and will be propor- 

 tional to C P, whilst the differential in the direction M S will 

 result solely from the small change of direction of the force, and 



MP. 



will be proportional to rjr^, or simply to M P, the distance M N 



remaining the same ; thus the first differential may be repre- 

 sented by A X C P, and the second by B x M P, A and B being 

 two factors which remain constant so far as the action exerted 

 by the same molecule N is concerned. 



Let us consider as yet only the particular action of this mole- 

 cule, and suppose M to be displaced successively in three rec- 

 tangular directions, and by quantities equal to the pi'ojections of 

 M C on these three directions ; through the point M draw a 

 plane perpendicular to M N, which will cut that of the figure, 

 that is the plane N M C, in the straight line M S. The dis- 

 placement M C has produced the two differential forces A x C P 

 and B X M P, the former in the direction M R and the second 

 in the direction M S. The displacements on the three rectan- 

 gular directions situated any how in space will likewise produce 

 each a differential force parallel to M R, with another force per- 

 pendicular to this line, and comprised also in the normal plane 

 M S drawn through the point M ; the former will be obtained 

 by multiplying by the same coefficient A the distance of the 

 new position of the molecule from the normal plane, and the 

 second by multiplying by the same coefficient B the distance of 

 M from the foot of the perpendicular dropped from this new 

 position on the normal plane. Next let us find the resultant of 

 three differential forces parallel to M R which have the same 

 coefficient A, and the resultant of three differential forces con- 

 tained in the normal plane which have B for their common coef- 

 ficient. The displacements in question being the projections of 

 the displacement M C on the thi-ee rectangular directions which 

 have been chosen, the sum of their projections on the direction 

 M R must be equal to C P; and consequently the resultant of 

 the three differential forces parallel to M R will be equal to 

 A X C P, that is to the force produced by the displacement 

 M C in this direction. It is easy to see in the same way that 

 the resultant of the three differential foi'ccs coniprised in the 



