394 Prof. G. G. Stokes on the Foci of Lines [June 21, 



I proceed now to the case of a plate cut in any direction perpendicular 

 to one of the principal planes, to which I propose to limit myself , merely 

 observing that the leading features of the most general case have already 

 been noticed. 



The principal plane perpendicular to the plate being a plane of optical 

 symmetry, the proper directions for the cross lines are parallel and per- 

 pendicular to that plane. Let the plane of symmetry be the plane of 

 xz, without necessarily implying thereby that the axis of y is that of mean 

 parameter, and let 6 be the inclination of the normal to the plate to the 

 axis of z. The section of the wave-surface, which I assume to be that of 

 Fresnel, by the principal plane being a circle and an ellipse, the formulae 

 for the foci of a line perpendicular to the principal plane will be the same 

 as for a uniaxal crystal, inasmuch as the relation which, in the case of a 

 uniaxal crystal, subsists between the radius of the circle and one of the 

 semiaxes of the ellipse is not involved in the formulas. Por light polarized 

 perpendicularly to the principal plane, then, the apparent index for a 

 line parallel to y is given by (2), while for the other pencil it is simply 

 b~ x or fx. 



To find the foci for a line lying in the plane of symmetry, we must 

 have recourse to the wave-surface itself, and not merely to its principal 

 section. We have to find the radius of curvature at any point in the 

 principal section for a normal section perpendicular to the principal 

 plane. 



Let P be a point in the principal section, VN the normal at P, M a 

 point in PN near P, and through M draw MQ parallel to y, cutting in 

 Q the sheet of the wave-surface to which P belongs. Then the limit 

 of MQ 2 -^2 PM, as M moves up to P, will be the radius of curvature 

 required. 



Taking the equation of the wave-surface under the form 



(x 2 + y 2 + z 2 )(a 2 x 2 + by + c 2 z 2 ) - a\ b 2 + c 2 )x 2 

 - b\c 2 + a 2 )y 2 - c\a 2 + b 2 )z 2 + dW = 



(5) 



let x, 0, z be the cordinates of P, and x + lx, y,z + $z those of Q. Substi- 

 tuting in (5), which, by hypothesis, is satisfied by the coordinates x, 0, z, 

 observing that dx, Sz are small quantities of the order y 2 , and omitting 

 small quantities of the order y*, we find 



{ a V + c V + a\x 2 + z 2 -b 2 -c 2 )} 2xdx -| 

 + {a 2 x 2 + c 2 z 2 + c\x 2 + z 2 -a 2 -b 2 )}2zh I 

 + { a V + c 2 z 2 + b\x 2 \-z 2 - a 2 - c 2 ) }y 2 = J 



Let PM=p, and first suppose P to lie in the circular section. Then 

 a? 2 + z 2 = b 2 , x=b sin 0, z=b cos 0. Also, as the normal coincides with the 

 radius vector, 



