510 



Prof. J. D. Everett. 



[Jan. 22, 



axis of the lens is taken as axis of z, the axis of y being in and the axis 

 of x perpendicular to the plane of symmetry. The equations of the 

 twelve emergent rays thus found enable us to plot twelve points of the 

 section made by a plane at any given distance z ; and through these a 

 curve can be drawn by hand. 



5. These sections are, however, inclined at about 45° to the emergent 

 rays, and are about midway between the axes of z and y. To remedy 

 this inconvenience, the equations of the rays are transformed to new 

 axes (of 7] and (), the axis of {"being midway between the axes of z and 

 y, and coinciding with the original direction of the incident beam. 

 Sections perpendicular to this direction are found by assigning different 

 constant values to(, and a "direction-curve" is drawn, which is a 

 section of a cone whose generators are parallel to the emergent rays. 



6. Harmonic reduction is applied to the direction-cosines ; and 

 harmonic expressions containing either two sines without cosines or two 

 cosines without sines are found to give a remarkably close representa- 

 tion of the facts. 



7. Each ray of the emergent pencil intersects two other rays. One 

 set of intersections are in the plane of symmetry, and are the inter- 

 sections of rays symmetrically placed with respect to this plane. These 

 intersections constitute the secondary focal line, which is absolutely 

 straight, and lies in the production of the straight line drawn through 

 the centre of curvature of the convex face of the annulus, parallel to 

 the rays in the glass. These may be called " left-and-right " inter- 

 sections. 



8. The other set of intersections constitute the primary focal line. 

 They may be described as "up-and-down" intersections, inasmuch as 

 the plane of any pair of intersecting rays is at a small inclination to the 

 vertical — that is, to the plane of symmetry ; whereas the plane of a 

 pair which intersect in the secondary line is perpendicular to the plane 

 of symmetry. Each intersection involves an inversion of. relative 

 position of the two rays concerned ; and the combined effect of the 

 " up-and-down " inversion at the primary line, and the subsequent 

 " left-and-right " inversion at the secondary, is to cause a distant section 

 to be an inverted image of the annulus. 



9. In the " up-and-down " intersections, the pairing of the rays 

 follows an unsymmetrical law. Each of the rays between 0° and 79° is 

 paired with a ray between 180° and 79°; — a fact which I discovered 

 empirically in my original calculations. The exact law of pairing has 

 since been detected by Professor A. E. H. Love. It is, that the chord 

 joining a pair passes through a fixed point, namely the point in which 

 the plane of the annulus is cut by the secondary line. See §§ 23,. 24, 26. 



10. The first rays to intersect are those from 0° and 180° ; and their 

 intersection is the vertex of the primary focal line. This line is 

 approximately a parabola, lying in a plane which recedes with a 



