To those mathematicians who have investigated the theory of the refracting telescope, it has often, says Mr. Herschel, been objected, that little practical benefit has resulted from their speculations. Although the simplest considerations suffice for correcting that part of the aberration which arises from the different refrangibility of the different coloured rays, yet in the more difficult part of the theory of optical instruments which relates to the correction of the spherical aberration, the necessity of algebraic investigation has always been , acknowledged; although, however, the subject is confessedly within its reach, a variety of causes have interfered with its successful prosecution, and the best artists are content to work their glasses by empirical rules. In the investigations detailed in this paper, the author’s object is, first to present, under a general and uniform analysis, the whole theory of the aberration of spherical surfaces; and then to furnish practical results of easy computation to the artist, and applicable, by the simplest interpolations, to the ordinary materials on which he works. In pursuing these ends he has found it necessary somewhat to alter the usual language employed by optical writers;—thus, instead of speaking of the
focal length
of lenses, or the
radii of their surfaces
, he speaks of their
powers
and
curvatures
; designating, by the former expression, the quotient of unity by the number of parts of any scale which the focal length is equal to; and by the latter, the quotient similarly derived from the radius in question. After adverting to some other parts of the subject of this paper, more especially to the problem of the destruction of the spherical aberration in a double or multiple lens, and to the difficulties which it involves, Mr. Herschel observes, that one condition, hitherto unaccountably overlooked, is forced upon our attention by the nature of the formulæ of aberration given in this paper; namely, its destruction not only from parallel rays, but also from rays diverging from a point at any finite distance, and which is required in a perfect telescope for land objects, and is of considerable advantage in those for astronomical use: 1st, The very moderate curvatures required for the surfaces; 2nd, That in this construction the curvatures of the two exterior surfaces of the compound lens of given focal length vary within very narrow limits, by any variation in either the refractive or dispersive powers at all likely to occur in practice; 3rd, That the two interior surfaces always approach so nearly to coincidence, that no considerable practical error can arise from neglecting their difference, and figuring them on tools of equal radii.
On the aberrations of compound lenses and object-glasses