From the truths demonstrated in the two preceding sections, it is manifest that those rays which are inci dent near the axis of a lens, or of a spherical surface, are converged to a different focus from those which fall at a distance from the axis. This error, or deviation of the rays from a mathematical point, is called the Spheri cal Aberration.
As the convergence of the incident rays to one focus is essentially necessary to the perfection of images formed by optical instruments of all kinds, it has been proposed, as we have already seen in the History or Optics, to use lenses generated by the revolution of the ellipse or hyperbola. In compound lenses, whose dia meter is considerable, the accurate convergence of the rays may be effected by a proper adjustment of the sur faces of spherical ldrises, as will be afterwards shown ; but this mode of producing an exact convergence of the rays to a single focus is impracticable in small lenses, such as are used in microscopes and eye-pieces ; and therefore we have no doubt but that the ingenuity of opticians will again be turned to the construction of el liptical and hyperbolical lenses, the theory of which we •hall now proceed to explain.
A meniscus has the property of refracting all con verging rays if its first surface is convex, or all diverg ing rays if its first surface is concave, to a single focus, provided the distance of the radiating point from the centre of the first surface is to the radius of the first surface, as the sine of the angle of incidence is to the .
sine of the angle of refraction.
Let MADN, Fig. 12. be a meniscus, and QABC, Ql.A.'BiC rays converging to the point C. Take the point F, so that CE : EA = Sin. Incidence : Sin. Re fraction, and from F, Nvith any radius FB less than FA, describe the circle MBB'N, the rays QA Q'A, and all Jthers, will be refracted accurately to the point F, so that the lens is entirely free of sfiherica/ aberration. Since Sin. Incidence: Sin. Refraction=CE : EA=Sin. CAE: ACE, and since CAE is equal to the angle of incidence, ACE must be equal to the angle of refrac tioo, that is, to EAC. But the angle AEC is common to the two triangles, ACE, AFT, consequently the tri angles are similar, and CE : EA=EA : EF ; and as the threc first terms of this analogy are constant, EF is also constant, and all the refracted rays will meet the axis in the same point F. But as these rays converge
to I: the centre of the spherical surface MBN, they will fall perpendicular upon that surface, and will therefore pass on to without suffering any farther change in their direction.
When diverging rays fall under similar circumstances upon the concave surface of a lens, the proposition is proved in the very same manner.
If a lens has its anterior surface part of a prolate spheroid formed by the revolution of an ellipse, whose greater axis is to the distance between its foci as the sine of incidence is to the sine of refraction, and if the other surface is concave, and part of a sphere whose centre is the farther focus of the spheroid, a pencil of rays, parallel to thc axis of the spheroid, will be re fracted, so as to converge accurately to the farther focus of the spheroid.
Let PDQ, Fig. 13. be. thc lens, whose anterior sur face PDQ is part of a prolate spheroid, generated by the revolution of the ellipse PDQK about it major axis DK, having II and I for its foci, and let is posterior surface PQ, placed at any distance between D and 1, have I for its centre, the rays AB, MD, parallel to the axis DK, will be refracted accurately to the focus I. Having joined IIB, IB, draw EBC, touching the gene rating ellipse in B, and through B draw GBL perpen dicular to EBC, and cutting DK in N. Since by the nature of thc ellipse, (CONIC SECTIONS, Prop. viii ) HBC — IBE, and NBC = NBE, we have IIBN — INB, and by GEOMETRY, Prop. XVII.
But Sin. ABG is the Sine of incidence, consequently iin. IBL is the Sine of refraction, and BI is the ray refracted by the speroidal surface PDQ. sincc PQ is a spherical surface, whose centre is I, the ray BI will fall perpendicularly upon it, and will therefore Tass on to I without suffering any farther refraction.
In the very same manner it may bc shown, that every other ray of a pencil, parallel to AB, will be re fracted to the same point I.