364 Dr. H. C. Pocklington on Rotatory 



Hence A and the two vectors in [] are coplanar, so that 



+ yQ(S^X + ft)S7\)l=0. 



Expanding and dividing by — S«/3X, the imaginary parts 

 vanish, and we have 



(i +i} X *~Kft +^) s - y ^ v f3A + S^/3A + S 2 7 A = 0. 

 Solving this quadratic, 



^ 2 +g= {S.VaXVjSX ± >SV**\N*ft\-\*&y\\l\*, (6) 



the equation of the index-surface. The right-hand side is 

 independent of the tensor of A, and the equation gives the 

 length of the radius vector in terms of its direction. Since 

 X 2 is negative, the quantity under the radical sign is always 

 positive, and cannot in general vanish. 



Thus the index-surface consists of two sheets which 

 nowhere meet, hut only approach each other at the optic 

 axes. If 7 is small the surface there is approximately a 

 hyperboloid of two sheets. Near the axis the surface de- 

 viates from Fresnel's surface by small quantities of the first 

 order, while at a distance from the axis the deviation is of 

 the second order. 



4. The wave-surface is the polar reciprocal of the index- 

 surface with respect to a unit sphere about the origin as 

 centre. The interesting parts of the surface lie near to the 

 optic axes. We can obtain the general shape of the surface 

 there by drawing a cross-section of the index- surface by the 

 plane containing the optic axes and obtaining the polar re- 

 ciprocal of this by construction. This curve is shown in the 

 diagram. The cuspidal edge of the wave-surface corresponds 

 to inflexions on the outer sheet of the index- surface. If y 

 is small, FG and BC are nearly flat and lie close together, 

 and the intensity of their illumination is small. If a point- 

 source of light exists at one face of a plate cut perpendicular 

 to the axis, and the light emerges at the opposite face into an 

 isotropic medium of nearly the same refractive index, the 

 wave-front is only slightly altered. The parts EF and CD 

 with HG and BA approximately belong to a wave that has 

 two focal curves, one a ring, the other a line perpendicular to 

 it. On examining the emergent light by a lens, a ring or a 



