" I would premise that the plano.concave form for the correcting flint lens has in that quality a strong recommendation, particularly as it obviates the danger of error which otherwise exists in centring the two curves, and thereby admits of correct workmanship for a shorter focus. To cement together also the two surfaces of the glass dimin ishes by very nearly half the loss of light from reflexion, which is con siderable at the numerous surfaces of a combination. I have thought the clearness of the field and brightness of the picture evidently increased by doing this ; it prevents any dewiness or vegetation from forming on the inner surfaces ; and I see no disadvantage to be anticipated from it if they are of identical curves, and pressed closely together, and the cementing medium permanently homogeneous.
" These two conditions then, that the flint lens shall be piano concave, and that it shall be joined by some cement to the convex, seem desirable to be taken as a basis for the microscopic object-glass, provided they can be reconciled with the destruction of the spherical and chromatic aberrations of a large pencil.
" Now in every such glass that has been tried by me which has had its correcting lens of either Swiss or English glass, with a double convex of plate, and has been made achromatic by the form given to the outer curve of the convex, the proportion has been such between the refractive and dispersive powers of its lenses, that its figure has been correct for rays issuing from some point in its axis not far from its principal focus on its plane side, and either tending to a conjugate focus within the tube of a microscope, or emerging nearly parallel.
" Let A s (fig. 13) be supposed such an object-glass, and let it be roughly considered as a plano-convex lens, with a curve A D C running through it, at which the spherical and chromatic errors are corrected which are generated at the two outer surfaces; and let the glass be thus free from aberration for rays FnEe issuing from the radiant point u E being a perpendicular to the convex surface, and n D to the plane one. Under these circumstances, the angle of emergence ga E a much exceeds that of incidence F Di, being probably nearly three times as great.
" If the radiant is now made to approach the glass, so that the course of the ray rnEo shall be more divergent from the axis, as the angles of incidence and emergence become more nearly equal to each other, the spherical aberration pro duced' by the two will bo found to bear a loss proportion to the opposing error of the single correcting curve A C a ; for such a focus therefore the rays will be over-corrected.
"But if F still approaches the glass, the angle of incidence continues to increase with the in creasing divergence of the ray, till it will exceed that of emergence, which has in the meanwhile been diminishing, and st length the spherical error produced by them will recover its original proportion to the opposite error of the curve of correction. When F has reached this point e(at which the angle of incidence does not exceed that of emergence so much as it had at first come short of it), the rays again pass the glass free from spherical aberration.
" If r be carried from hence towards the glass, or outwards from its original place, the angle of incidence in the former case, or of emer gence in the latter, becomes disproportionately effective, and either way the aberration exceeds the correction.
"These facts have been established by careful experiment : they accord with every appearance in such combinations of the piano convex glasses as have come under my notice, and may, I believe, be extended to this rule, that in general an achromatic object-glass, of which the inner surfaces are in contact, or nearly so, will have on one side of it two foci in its axis, for the rays proceeding from which it will be truly corrected at a moderate aperture ; that for the space between these two points its spherical aberration will be over-corrected, and beyond them either way under-corrected.
" The longer aplanatic focus may be found, when one of the piano convex object-glaeses is placed in a microscope, by shortening the tube, if the glass shows over-correction ; if under-correction, by lengthening it-, or by bringing the rays together, should they be parallel or diver gent, by a very small good telescope. The shorter focus is got at by sliding the glass before another of sufficient length and large aperture twat is finely corrected, and bringing it forwards till it gives the re flexion of a bright point from a globule of quicksilver, sharp and free from mist, when the distance can be taken between the glass and the object.
The longer focus is the place at which to ascertain the utmost aperture that may be given to the glass, and where, in the absence of spherical error, its exact state of correction as to colour is seen most distinctly.