298 FRESNEL ON DOUBLE REFRACTION. 



passing through its centre, that is to say, of a radius vector ; be- 

 tween «, /3 and X, Y, Z we have the following relations : — 



cos^X = z— ; — ^ . -c , cos- Y = , , o . ^3 , cos^ Z = 



Substituting these values of cos^ X, cos^ Y, cos^ Z in the above 

 equation, it becomes 



0^ (1 + a^ + |82) = e2 «2 _,. J2 _ ^2 ^. ^2^ 



This is also the polar equation of the surface of elasticity, but in 

 which the cosines of the angles X, Y, Z, which the radius vector 

 makes with the axes, have been replaced by the tangents («) and 

 (/3) of the two angles which its projections on the coordinate 

 planes x z, y z make with the axis of z. 



When the radius vector (o) attains its maximum or its mini- 

 mum, dv =■ 0; hence, on differentiating this last polar equation 

 of the surface of elasticity, we have for the equation of condition. 



The radius vector whose equations are x = uz, y = ^ z, being 



necessarily contained in the cutting plane z=^mx + n y, we have 



1 = TOa + w/3 ; 



an equation which gives by differentiation 



= m.da. + n.d^; 



d S ?/?■ 



whence ~ = ; substituting in the above differential equa- 



d a. n 



tion, we find 



0^ (a . « — /3 . m) = a^ .an — b^ . ^m. 



If we combine this relation with the equation 1 = mu + n^, we 



ff nd the following values for a and /3 : — 



(i^ — 0^) m „ _ {a^ — u^) n 



We shall observe in passing, that these expressions being of the 

 first degree, (a) and (|3) cannot have more values than (u^). Now 

 on substituting them in the place of (a) and (/3) in the equation 

 of the surface of elasticity, we find 



+ (a2-t;2) (i2_„2) ^ (^^) 



This equation being only of the second degree m ith regard to 

 (o^), can give only two values for it ; hence there are only two 



