iti the Theory of Double Refraction. %bS 



angle; be so assumed that 



2 2 5 « ,;. (r) 2 sin* (-g-) A s' A^/' | 



The same values will also render equal to zero the quantity 



The equation 



+ {f-h^) (f—a^) n2 = 0, 



from which Fresnel deduces the equation to the wave-surface, 

 may be obtained without employing the method of Fresnel, 

 by which it is shown incidentally that the quantity v^ is a 

 maximum or minimum. Fresnel supposes the constitution of 

 the medium to be such, that when the force is resolved into 

 two components, 



1. In the direction of the displacement, 



2. In a direction perpendicular to that of the displacement, the 

 second component is also perpendicular to the plane of the 

 wave : the direction of the displacement remains unaltered. 



Let I X •\- my -\- nz =■ Q be the equation to a plane par- 

 allel to the plane of the wave, /, m and n denoting the same 

 quantities as in Mr. Smith's paper in the Cambridge Trans- 

 actions, vol. vi. p. 87, and in Mr. Sylvester's paper in the 

 Lond. and Edinb. Phil. Mag., 1837, vol. xi. p. 463. 



As this plane is also parallel to the direction of displacement, 

 I cos X + m cos Y + « cos % = 

 cos^J^+ cos^ r+ cos^;2=l 

 w* = a^ cos2 X+ h^ cos^ Y ->r <? cos» %. 



If ^/ {a!^ cos^ X + &* cos^ Y -ir d^ cos* %} = f 



the cosines of the angles which the resultant or line of force 

 makes with the axes are 



a* cos X W- cos Y c^ cos % 



if the resultant is contained in a plane parallel to the plane 

 I' X -f- ml y + n'z = 



a^ V cos A' + Z»2 m' cos Y ■\- c^ n cos 2=0. 

 If this plane is also parallel to the direction of displacement, 



/' cos X + m' cos Y + n' cos Z — 



m 



' ( n'^ /»2\ rkf^e, V «,/ 



_ K'-c') COS X JL- ^ {a^-b^)cosX 



I' {b^-~c') cos Y V "" (c^- 6*) cos 2* 



2A2 



