FRESNEL ON DOUBLE REFRACTION. 2/1 



Parallel to a? . . .a. cos X + A . cos Y + ^ . cos Z = p. 



Parallel toy . . . b .cosY + h .cosX +f. cosZ = q. 



Parallel to ^ . . . c . cos . Z +^. cos X +/. cos Y = r. 



I now proceed to show that there exists always a direction for 

 which the resultant of these three components coincides with 

 this very direction of the displacement itself; that is to say, that 

 real values may be given to the angles X, Y, Z such that the 

 resultant of the three components shall make with the axes of 

 X, y and z angles respectively equal to X, Y and Z ; or, in other 

 terms, such that these three components shall be to one another 

 in the same ratio as the quantities cos X, cos Y, cos Z. 



To find the direction which satisfies this condition, I shall 



substitute for the three unknown quantities cos X, cos Y, cos Z 



(which are reduced to two by the equation 



1 = cos2 X + cos^ Y + cos2 Z) 



the tangents of the angles which the projections of the straight 



line on the planes x z and y z make with the axis of z, in order 



to be able to decide as to the reality of the angles from that of 



the values of the trigonometrical lines given by the calculation. 



Let then x = mz and ^ = w ^ be the equations of the straight 



cosX cosY ., ,, , 



hne : we have m = ^, n = ^; now the three above com- 



' cos Z cos Z 



ponents, which I shall represent by p, q and r, must be to one 



another in the same ratio as the quantities cos X, cos Y, cos Z, 



in order to satisfy the condition just mentioned. 



_,- , , r P cosX q cosY 



We have therefore - = ^ = m, ^ = ^ = n; ov, put- 



r cos Z r cos Z 



ting for p, q, r their values, 



cos X , cos Y 



a.cosX + A.cos Y+^r.cosZ _ ' cos Z ' cos Z ^ 



c.cos Z+^.cos X + /".cos Y ~ cosX „ cosY 



^"^^^ cosZ ^''•cosZ 



, cos Y , cos X 



/ !>.cosY + ^.cosX+/.cosZ ^ "cosZ"'" 'cosZ"^-^ 

 e.cosZ+^.cosX+/.cosY ~ cos X „ cosY 



m = 



And 



cos Z * cos Z 



Or lastly, 



am + hn + a 

 c + gm +fn 



(I.) 



