FRESNEL ON DOUBLE REFRACTION. 289 



which represents the laws of elasticity of every medium whose 

 molecular groups have their axes of elasticity parallel, may be 

 cut in two circles by two planes drawn through its mean axis, 

 and equally inclined to each of the other two axes. In fact, I'e- 

 place the polar coordinates by rectangular ones in this equation, 

 which then becomes 



(^.2 + 2/^ + z'^f = a'cc'^ + U" i/2 + c2 ^2 ; 

 the circular section made in this surface may always be con- 

 sidered as belonging at the same time to the surface of a sphere 

 x^ -{■ y^ -X- z^ =■ r^ ; its circumference therefore will be found 

 at the same time in the cutting plane Z = A <r + B y, on the 

 surface of the sphere and on the surface of elasticity. Com- 

 bining the equations of these two surfaces gives 



r4 = a2.r2 + ly" y^ -V c^ z^ ; 

 substituting in this equation the value of z obtained from the 

 equation of the cutting plane, we have 



^2(a24- A^c^) +y2(j2^B2c2) + 2AB.c2.^y = r4. . (1.) 

 On substituting this value of z in the equation of the sphere, 

 we find for the projection of the same curve on the same plane 

 of^y, 



5^2 (1 + A^) + 2/2 (1 + B^) + 2 A B . .;p?/ = 7-2 (2.) 



Since the two equations (1.) and (2.) must be identical, we have 



1 + B2 _ AM^BV 2AB _ 2AB.c2 r^ ^ r^ 



FTA^ ~ a^ + A^c^' 1 + A2 ~ «2 + A^c^' 1 + A^ ~ a^^ A^ c^' 



The second condition can be satisfied only by A = or B = 0, 

 since otherwise it would be necessary to make c^ + A^ c^ = a^ + 

 A^c^, or a^—c^, constant quantities of which we cannot dispose. If 

 we suppose A = 0, we obtain from the first equation of condition 



B = + A / ^" ~ \ an imaginary quantity if [h) be the middle 



V c^ — b^ 



axis, since in that case the tw^o terms of the fraction placed 

 under the radical are of different signs. Hence if we suppose 

 a>h and b^ c,yie must make B = 0, whence we obtain for A 



the real value A = ± a / to b* 



V b^ — c^ 



B = indicates that the cutting plane must pass through the 



axis of y, or the mean axis of the surface of elasticity ; the two 



equal values with contrary signs which we find for A, that is to 



