GEOM Prop. III. QE : FE = Dp : Ep.
Hence E p, the distance from the focus, bears a con stant ratio to D p, the distance from the directrix PQ, and consequently p is a point in a conic section, whose focus is E, and whose directrix is PQ. See Coxic SEe T1ONS.
We have already seen, in the preceding section, that rays passing near the centre of a lens have a different focus from those which pass near its circumference, and that incident rays can only be refracted to a mathe matical focus by means of lenses having the form of the conic sections, or the compound figures already de scribed. As it is necessary, however, in practice to use spherical surfaces, it becomes of importance to deter mine the exact amount of spherical aberration under different circumstances, and to explain the methods by which it may be diminished or corrected by opposite re fractions, DEF. Let AB, Fig. 6. be a lens receiving the paral lel rays NA, EP, F the focus of the central rays EP, and f the focus of the marginal rays NA ; then having drawn through F the line FG, perpendicular to the axis ECF of the lens, F f is called the Longitudinal Spherical Aberration, and FG the Lateral Aberration. AB is called the ifperture, and AC the of thc lens.
When parallel homogeneous rays are incident upc:: a spherical surface, the longitudinal aberration is to the versed sine of the semi-aperture, as the square of the sine of refraction is to the rectangle of the sine of in cidence, and the difference of the sines of incidence ant! refraction very nearly.
Let ACB, Fig. 6. be a spherical surface, separating two refracting media, E its centre, C its vertex, AC its semi-aperture, AP the sine of the semi-aperture, PC the versed sine of the semi-aperture, and F the focus of parallel rays infinitely near the axis. Let the ex treme marginal ray NA be refracted in the direction AG, crossing CE in f, which will therefore be the fo cus of extreme parallel rays. Join EA, and having pro longed GA towards M, let fall the perpendiculars ENI, EN upon the lines AN, AM ; upon f as a centre, with the radius f A, describe the arch AD. The angle EAN is the angle of incidence, and EANI or EA f the angle of refraction, and EN and ENI are the sines of inci dence and refraction, which we shall call n and in Then, because the triangles E f NI and A f P arc si milar, we shall have Ef :f A orf D=ENI : AP, or EN ; Lut hy Prop. XIII EM : EN = EF : FC. Hence, Ef:f D = FC.
By GEOME1RY, P1'01). VI.
Ef : E f—f D EF—FC, that is, Ef : ED=EF : EC, and by GEOMETRY, Prop. II. and EF : E f=EC : ED, and by GEOMETRY, Prop. V.
Er —Ef: EF=EC—ED: EC ; that is, by substitution, Ff : EF=DC : EC, and by GEOMETRY, PrOp. III.
F f : DC= EF : EC=ENI : EN=rn : nz—n But since the versed sines PD, PC, of small arches AD, AC of unequal circles that have the same right sine AD, are reciprocally proportional to their radii Df, EC, we have