FRESNEL ON DOUBLE REFRACTION. 269 



forces, and first those of the former A x A Q. If we represent 

 by X, Y, Z the angles made by the straight line A P M with the 

 axes of X, y and ^r, A B being equal to unity, by hypothesis, 

 A Q = cos X, and the differential force in the direction A M is 

 represented by A . cos X ; its components therefore are 

 parallel to . . a? y z 



A.cos^X., A.cosXcosY, A.cosXcosZ. 

 Let us now find what are the components of the second differ- 

 ential force B X B Q acting along B Q. Since A B = unity, 

 B Q, = sin X . , and this force is represented by B . sin X. I 

 decompose it in the first place into two others in the directions, 

 one of B A and the other of B P* perpendicular to B A . ; the 

 first component, which is parallel to the axis of x, is equal to 

 B . sin X X cos A B Q, or — B sin^ X, and the second has for 

 its value B sin X x sin A B Q, or B sin X cos X . I resolve 

 this second component into two other forces in the directions 

 EB and FB, that is parallel to the axes of?/ and z.; the first 



B F" 



will be equal to B sin X cos X . x j^-p, and the second to 



T^=;^v.«.v BF , ,BE cosY ,BF cosZ 

 B.smXcosX X ^^3: but g^ = _^ and ^ = _^; 



hence the values of the components parallel to y and z become 

 respectively B cos X . cos Y and B . cos X . cos Z. We have then 

 for the three components of the second differential force, 

 parallel to ... o-' y z 



— B sin^ X, B cos X cos Y, B cos X cos Z. 

 Adding together the parallel components of the two differential 

 forces, we find for the total components, 

 parallel io . . x y ^ 



Acos^X-Bsin^X, (A + B)cosXcosY, (A + B)cosXcosZ. 

 If we now suppose the material point A to be displaced along 

 the axis of y by a quantity equal to unity, we shall find in the 

 same manner the following components, 

 parallel to . y x z 



A.cos^Y-Bsin^Y, (A + B)cosX.cosY, (A + B)cosY.cosZ. 

 And for a similar displacement along the axis of ^ we should have 



* ['rhere is evidently a mispiint of P for M in the original, in this pasc— 



TUANS.] 



