230 



UNDULATORY THEORY OF LIGHT. 



fast as tho analyzer. Wlien the analyzer is in azimuth 90° from its original 

 position, the dark radius is in azimuth 180° from its original position; when 

 the analyzer has completed half a revolution, the dark radius has made a whole 

 one. This singular fact is, however, easily explained. The analyzer suppresses 

 that ray on whose plane of polarization it is crossed; and the planes of polariza- 

 tion of the rays in this small circle are, at opposite ends of every diameter, at 

 right angles to each other. If QAPA' he the small 

 circle of tangency, Q heing the point of contact with 

 the circular section, or the foot of the normal, we have 

 seen that the molecular movement from any other 

 point of contact in this plane must be toward Q ; that 

 is, the movement at q must be in the direction ^Q, 

 ]. that at r in the direction ?Q, that at^ in the direction 

 ,pQ, &c. Suppose the direction of molecular move- 

 ment in the analyzer to be AA, parallel to cQa. 

 The vibrations in the ray at Q have the same 

 direction. The analyzer allows that ray therefore 

 freely to pass ; but it is crossed on the ray at P 

 F'S- 70 whose direction of vibration is PQ. The radius, CP, 



will therefore be dark. Now let the analyzer take a position in azimuth A' A', 

 parallel to a' a', tangent at r. Draw the diiimeter rCp. Draw (^sq perpendic- 

 ular to rG. It will be parallel to a'a' , and it will be the direction of vibration 

 of a ray at q. The analyzer in the position A'A', is therefore in harmony 

 with the ray q, and it is crossed on a ray whose molecular movements are 

 in Qr, at right angles to Q^' Join Qr, Q/», rq. The angle (^pr is 

 equal to the angle Q^r, since both stand on the same arc, rQ. And the 

 triangle rqs is similar to the triangle Q^r, the one being right angled by 

 construction, and the other because it is inscribed in a semicircle. The triangle 

 rqs is therefore similar and also equal to rQ.s% or the arc Q^ is twice the arc Qr. 

 Draw the diameter qp', and join Q/;-'. The angle q^j)' is a right angle ; conse- 

 quently the molecular movements at p', being in the direction ^Q, parallel to 

 Cr, and at right angles to ^-Q, or to a'a' or A'A', will be suppressed. In turning 

 the analyzer in azimuth, therefore, from AA to A' A', or through an azimuth, 

 measured by AA' = Qr, the dark ray has advanced through an azimuth 

 P^'— Q2'=2Q/-, which was the point to be proved. 



In the equation of the wave surface, if b be the mean axis, and we make 

 ,-;=i, we shall have, after reduction — 



{a^x'+hhfJr^rz'—a''h'){x^+if+z''— !/)=(). [57.] 



If b and c remain unequal, and we put b=^a, the equation is — 



(a^jj + aV+r^c^— aV)(x2 + 7/2 + ;~2_^2)^0. [58.] 



These are both equations of a spheroid and a sphere, touching each other at 

 the poles. The first is that of an oblate spheroid circumscribing the sphere, 

 and answers to the case of a negative crystal of one axis. The second is that 

 of a prolate spheroid circumscribed by the sphere, and answers to the case of a 

 positive crystal. The case of quartz, so remarkable on other accounts, is pe- 

 culiar also in the fact that the two nappes of its wave surface are not in contact 

 anywhere. The ellipsoid is entirely within the sphere, and there is no direction 

 either of equal wave or of equal ray velocity. 



These equations suggest the geometrical relations between the surflxce of 

 elasticity and the wave surface. The larger diameters of the one are at right 

 angles to the larger diameters of the other, and the smaller have the same rela- 

 tion. For crystals of one axis, the surface of elasticity is an ellipsoid of revo- 

 lution. If its form is prolate, it generates an oblate wave ; if it is oblate itself, 

 th'-' wave is prolate. 



The causes of varying elasticity of the luminiferous ether within crystals are 



