sin a cos CI sine A o mate value, when o is at D, is =21, for the same reason the ultimate An A valuo of 1" is r and the ultimate values of cos 0, coos cos a, sin a 8' are each unity ; therefore we get —'21f.nt-1, or ; A a A r hence A, which determines the focus of the first set of refracted rays, is known ; and therefore also 8 + t, which is its distance from the second surface. Now, since the ray would traverse the same course if we sup posed it to commence at a, and proceed through a x u to A, it follows iu the same way that I la ra-1 from whence A' is known.
A' + r' If we neglect t as being small, we may eliminate -7,and thence obtain I 3 I 1 —+ —=.(ra-1)( — + —) ; the spherical aberrations may be found A' A r' r by a similar process to that we have employed for reflection, and the inverse or erect positions of the images ascertained by the like method.
When we have one side plane, we have only to suppose r infinite, and when concave to suppose r negative, and thus one general formula, by proper attention to the signs, may be made to apply to all forms of lenses.
There is a cause of aberrmtion for refracted light, which does not exist for reflected rays, and it is of more consequence in deforming and colouring images than all the effects of spherical aberration. The chro matic dispersion [Disreesioe] arises from the fact that all the coloured rays which compose solar or other light have different refractive indices for one and the same refracting medium ; hence the prismatic spectrum, which only consists of successive circular images of the sun, of the different colours of the rays, overlapping each other. This aberration has, by the successive labours of Dollond, Fraunhofcr, and others, been aucceasfully combated. [Orrics, PRACTICAL.] The possibility of correcting the aberration arising from the unequal refrangibility of rays of light in the refracting telescope, by means of a compound object-glass, permits the employment of object-glasses of a much larger diameter, in comparison with their focal length, than could otherwise have been tolerated. Now for a lens of given focal length, the spherical aberration increases rapidly with the aperture. It becomes therefore of importance to correct as far as this aberration by giving appropriate forms to the surfaces,or by combining together two or more lenses.
Writers on optics show that a curve-line by whose revolution about an axis there shall be described a surface which, being that of a refracting medium, will cause all rays incideut upon it, when they diverge from or converge to one point, to be refracted so as to converge to or diverge from one point is, in its most general form, of the fourth order : but when the radiant point is at an infinite distance from the refracting surface, as when it is at a celestial body, the form of the surface, supposing the density of the refracting medium to be greater than that of the medium which surrounds it and in which are the incident rays, is proved to be that of a spheroid : and rays falling on its convexity, parallel to the major axis of the spheroid, would, within the medium, converge accurately to the focus most remote from the place of incidence. If the refracting medium were less dense than that
in which are the incident rays, the surface would be that of an hyper boloid. The semi-transverse axis both of the spheroid and hyper boloid must be, to the excentricity,as the sign of the angle of incidence is to the sign of the angle of refraction : in the former case the refraction is from the surrounding medium into the spheroid ; and in the latter, from the concave surface of the hyperboloid into the surromiding medium. It follows therefore that if a meniscus lens denser than the surrounding medium have its anterior surface sphe roidal, and its posterior surface that of a sphere whose centre is at the further focus of the spheroid ; since the rays will then suffer no refraction in pressing through the posterior surface, it will be aplanatic. Also if the anterior surface of a medium be plane, so that the parallel rays incident perpendicularly on it may suffer no refraction in entering, and the other surface he part of a hyperboloid, the medium between the surface being denser than that which surrounds it, the plane lens thus formed will be aplanatic ; the refracted rays converging to the opposite focus of the hyperbola.
The form of the expression for aberration, when parallel rays are incident on a lens of moderate aperture having spherical surfaces, is such that the aberration cannot be made to vanish with any real values of the radii of those surfaces unless the index of refraction in the medium be equal to or less than But there is in nature no medium which has such a refractive index , therefore, failing in making spherical lenses strictly aplanatie, and it having been found impossible, hitherto, to form them with surfaces produced by the revolutions of conic sections, mathematicians have investigate] expres sions for the form under which, with a given refractive index, the aberration of the focus shall be a minimum ; see the article " Light " in the Encyclopmclia 3letropolitana' (art. 305), where the ratio between the radii of the surfaces, in this state, is given. From that ratio it is shown that, when the index of refraction is the lens should be of the double convex form,having the radius of the posterior surface six times as long as that of the anterior surface, or that which is nearest to the radiant point.