396 



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



of normal sections perpendicular to the principal plane, we see that if b 

 be the greatest or least parameter they remain constantly positive. But 

 if b be the mean parameter, both expressions change sign twice, once in 

 passing through zero, and once through infinity, as changes from 

 0° to 90°. 



The radii of curvature become infinite together when 



tan 2 6- 



b 2 -c 2 ' 



(11) 



that is, when the plate is perpendicular to the optic axis. For a point in 

 the circular section the radius vanishes when 



tan 2 0: 



c 2 (a 2 -h 2 ) 

 a\b 2 -c 2 ) 



(12) 



that is, when the plate is perpendicular to the ray-axis. For a point in 

 the elliptic section the radius vanishes when 



tan 2 = 



a°-(a 2 -b 2 ) 

 c\b 2 -tf) 



(13) 



that is, when the plate is perpendicular to the normal to the elliptic 

 section at the point where the two sections intersect. 



A figure may make these changes clearer. Let xOz be a quadrant of 



Fig. 3. 



the plane perpendicular to the axis of mean parameter. Let BB' be the 

 circular, and AC the elliptic section, intersecting in E, PQ the common 

 tangent, EN a normal at E to the elliptic section. Conceive a plate cut 

 perpendicular to the plane of xz, its normal being inclined at the angle 6 to 

 Oz; and imagine 6 to change continuously from to 90° : and let p', 

 represent the radii of curvature in the secondary plane (xOz being deemed 



