326 FRESNEL ON DOUBLE REFRACTION'. 



tained in the interior of this angle ; and consequently the pro- 

 jection of the diameter of the ellipsoid perpendicular to the plane 

 of polarization, which is normal to the trace of this plane, is 

 pjo-, J5_ necessarily found to be con- 



tained in the acute angle MAN 

 \if ° 



i" or M'AN' of the two circular 



^ '- ^ sections, since they are normal 



to the optic axes P P' and Q Q,' ; 

 therefore this diameter cannot 



3^1 .^^^!T^:::^4^:rr^ iV .-« "fi^st the surface of the ellipsoid 



outside of the tW'O parts whose 

 M^^T / i \ ^yT^ projections have for their limits 



MB'N'A and MBNA; butif 

 pf-- — iii--Q' from the point A as centre, and 



with radius equal to that of the 

 circular sections, a sphere be described, its surface will pass be- 

 neath that of the ellipsoid in these two parts. 



Hence none of the diameters of the ellipsoid projected in the 

 angular space MAN, M' A N' will be smaller than the diameter 

 M M' of the circular sections, which is equal to the mean axis 

 of the ellipsoid ; the length of the radii vectores corresponding 

 to this part of the surface has therefore for limits, on one side 

 the semi-major axis, and on the other the semi-mean axis. 



In the same way it might be shown that the length of the 

 radii vectores which give the measure of the velocities of the 

 second luminous beam, is comprised between the semi-mean 

 axis and the semi-minor axis. Now, in the case represented by 

 fig. 15, where the minor axis of elasticity divides the acute angle 

 of the two optic axes, and the major axis the obtuse angle, there is 

 a greater difference between the minor axis and the mean axis 

 than between this latter and the major axis, as we see by the 



expression — \ / y^ ^ for the tangent of the angle which the 



planes of the circular sections make with the major axis ; for 

 this angle being less than 45° by hypothesis, we have c^ {a^ — V^) 

 /. a^ {b^ — c^), or nearly {a — b) Lip — c), suppressing the com- 

 mon factors c {a + b) and a [b + c) as being sensibly equal. 



The reasonings we have entered into for the ellipsoid may be 

 applied just as well to the surface of elasticity, which gives by 

 the axes of its diametral sections the true directions of the lumi- 

 nous vibrations, and consequently those of their planes of polari- 



