PRINCIPAL SECTIONS OF THE WAVE SURFACE. 227 



The lines drawn tlirough the centre. and the points N and N' are however 

 the optic axes ; for it is equality of ray velocity which makes an optic axis. 

 But it is not true that the two rays whose velocities in ON are equal, can spring 

 from the same incident ray. Herein there is an important difference between 

 crystals of one axis, and those of two. In crystals of one axis, when it is pos- 

 sible for two rays Avhose planes of polarization are transverse to each other, to 

 have a common path and common velocity, they both proceed, or may proceed, 

 from the same original ray. 



This is not so in crystals of two axes ; and what is more, no single incident ray 

 of common light, in this class of crystals, can give a single refracted one ; for 

 there are no common points of tangency, in which both nappes may be met by 

 the same plane. 



If a tangent plane be drawn to the wave surface parallel to one of the circular 

 sections of the surface of elasticity, it will take the position of AD, DB, &c., in 

 the figure ; and will be tangent at once to the ellipse and the circle in the principal 

 section through the axes.* If, then, (in the same figure,) AB represent a refracting 

 surface, and N'C a ray of common light incident at 0, in such a manner as to 

 take the direction CQ'" within the crystal, for the nappe whose section is cir- 

 cular, it will yield another ray, OP'" for the nappe whose section is elliptical. 

 These two rays will be polarized in planes transverse to each other. The 

 directions of their respective molecular movements, and therefore the positions 

 of their planes of polarization, may be inferred from the following considerations. 



The circular form of the section QQ'Q"Q'", shows that the velocity of the 

 rays belonging to that section is equal in all directions. The molecular move- 

 ments must therefore be affected by a constant elasticity. Their directions must 

 accordingly be invariable. In order that these directions may remain invariable, 

 while a ray moves as a radius vector in the plane QQ', &c., they must be 

 perpendicular to this plane, or parallel to h. Accordingly the ray CQ'" is 

 polarized in the plane of the section. The other ray, OP"', is polarized at right 

 angles to the plane of the section. 



The radius, OQ'", of the circular section is normal to the tangent plane AD. 

 For the angle CQ"'A. is a right angle, by the property of the circle. And the 

 wave surface on opposite sides of the plane of the section is symmetrical. The 

 molecular movements of' the ray OP'" are, therefore, in the plane, which, 

 passing through the ray, is normal to the tangent plane. Or, if we draw a line 

 joining the point of contact with the foot of the normal from the centre, this 

 line will be the direction of molecular movement in the ray. 



The proposition just stated may be generalized, and extended to all rays. In 

 the case of OQ'", the point of contact and the foot of the normal coincide; and 

 any line drawn through Q'" fulfils the required condition, leaving the direction 



*■ The truth of this statement may easily be shown thus : Suppose ordinates to XX', ZZ', 

 to be drawn Irom P' and Q'. Let x and 2 represent the ordinates from P', and x' and z' 

 those from Q'. It is evident that the angle at C, where the tangent BC intersects the axis 



J>Q 2 z' 



of X, which we will put =a, will have for tanwnt ;:— 7. Also, that the same tanirent =— . 



' ^ '^ CC x' — X 



Put CC:=/i, BC=/c'. Then, by the property of the ellipse, we have — 

 kx=^c-, kx'=^b-, k'z'^^h", Ii'z=^a'. 

 Hence, k{x'—x)=b''—c''; and k'(z—z')=a^—b-^. 

 Dividing the second of these equations by the first, member for member, we obtain — 



k'(z—z') a"—b'^ „ a-—b" 



— ^ =- ; or tau-a=:: , ; and tana 



k{x' — x) b- — c^ b- — C" 





But this (equation [48]) is the tangent of the inclination of the circular section of the surface 

 of elasticity to a, the axis of greatest elasticity, which is the axis of x. It follows that a 

 plane which, being normal to the section through the inosculating points of the wave surface, 

 is tangent at once to the ellipse and the circle in that section, is parallel to one of the circular 

 sections of the sm'face of elasticity. 



