FRESNEL, ON DOUBLE REFRACTION. 26/ 



unity of these differential displacements; I call a, b, c the three 

 components along these axes of the force excited by the displace- 

 ment parallel to the axis o^ x; a' b' c' the three components of 

 the force excited by the displacement parallel to y ; and lastly, 

 a" b" c" the components of the force excited by the displacement 

 parallel to z. 



To obtain the force which results from a small displacement 

 equal to unity, along any other direction whatever making angles 

 X, y, Z with the axes of x, y, z, we must first, in accordance 

 with the preceding theorem, take on these axes the statical com- 

 ponents of the displacement, which will be respectively cos X, 

 cos Y, cos Z, and determine the forces separately produced 

 by each of these displacements ; then calculate the resultant of 

 all these forces. 



Now to obtain the components of the force produced by the 

 displacement along the axis of x equal to cos X, we must mul- 

 tiply successively cos X by the coefficients a, b, c, since they 

 represent the components of the force excited by a displacement 

 equal to unity, and because, as we are here considering only 

 very small variations, the forces developed are proportional to 

 the lengths of these differential displacements ; so that the com- 

 ponents of the force resulting from the displacement cos X are 



parallel to x y z 



a . cos X, b . cos X, c . cos X. 

 Similarly, the components of the force produced by the displace- 

 ment cos Y along the axis of y are 



parallel to x y z 



a! . cos Y, b' . cos Y, c' . cos Y. 



And the components of the force excited by the displacement 

 cos Z, which takes place along the axis of z, are 



parallel to x y z 



a".cosZ, 6".cosZ, c".cosZ. 



Adding together those components whose directions are along 

 the same axis, we have for the total components 



parallel to a? . . . a . cos X + a' . cos Y + a" . cos Z. 



parallel to y . . . b . cos X 4- i' . cos Y + b" . cos Z. 



parallel to z . . . c . cos X + c' . cos Y + c" . cos Z. 

 These components determine the magnitude and direction of the 

 total resultant. 



