LIGHT. 243 LIGHT. Therefore v = /", as a negative quantity; and O' i3 on the same side of the lens as at a distance / from the lens. This point is called a "principal focus;' and there is evidently another one on the other side of the lens at the same distance from it. Therefore all rays parallel to the axis on one side of the lens diverge after passing through the lens as it fiom a point at a distance / from the lens on the same side as were the parallel rays. For this reason a lens of this kind is called a •diverging' one. Similarly, a ray on one side of the lens pointed toward the principal focus on the farther side emerges from the lens parallel to the axis. Fiu-ther, a ray through the centre of the lens remains parallel to itself. These principles enable one to trace at least three rays leaving a point not on the axis and thus to find its image, and to draw images of any object. Drawings are given of a few special cases. Fig. 1-2. It is evident from the general formula that !; will be positive, i.e. there will be a real image if u is negative and numerically less than /. i.e. if O is on the opposite side of the lens from that on which the rays enter, and between the lens and the principal focus. This is the case when the entering rays are converging toward the point O, as shown in Fig. 11: under these con- ditions would be called a 'virtual' source. Again, as is seen from Fig. 12, parallel rays falling on the lens emerge as if coming from that point in the focal plane on the same side as that from which the rays come where the ray through the centre of the lens meets the plane. If two thin lenses of focal lengths f, and f, are put close together with their axes coinciding, they form a lens whose focal length, /, is given by the formula 1 = 1+1. If the two lenses are at a distance d apart, the focal length of the combination / is given by the formula If the lens is not thin, but so thick that this fact must be taken into account, it may be proved that there are two planes perpendicular to the axis, either in the lens or near it, which are so placed that if u is the di?»lance from the source O to one plane and c that from the image O' to the other, the same formula as be- fore applies. These planes are called the 'prin- cipal planes.' They have other important proper- ties also; but for a full discussion of thick lenses and their combination reference must be made to some treatise on optics. Lenses and systems of lenses as used in micro- scopes, telescopes, photogiaphic lenses, etc., are subject to the following — among other — imper- fections: ( 1 ) typherical Aberration. — An oblique ray from a point on the axis docs not pass through the same focus as does a homocentric pencil. (2) Curvature of Field. — The image of a large plane figure perpendicular to the axis will not be formed on a plane, but on a curved surface. (3) Uistortion. — The image of a large object, e.g. a building, is not similar in all its parts to the parts of the object: the magnification may be different and a rectangular portion of the object may appear in the image with curved edges. (4) If the image of a small plane area perpen- dicular to the axis is formed by a lens which is large compared with the object, the image is not a small plane area perpendicular to the axis unless Abbe's 'sine formula' is satisfied. If the system satisfies this condition, it is called apla- na I ic. (.5) Chromatic Aberration. — Owing to the fact that the index of refraction differs with the color of the light, there are different focal lengths for rays of different color: and, further, if the lenses are thick, the magnification is different also. See AcnROM.VTiSM. Many of these imperfections can be avoided by a suitable choice of material for the lenses, by proper curvature for the lens-surface, and by using suitable combinations of lenses. Lens combinations which are free from certain errors are called special names. A photographic lens free from distortion of field is called 'ortho- scopic:' one free from curvature of field and from chromatic and spherical aberrations is called •anastigmatic:' a microscopic objective free from secondary spectra of chromatic aberration and aplanatic for several colors, 'apochromatic' Several facts should be borne in mind in re- gard to an oi)tical combination: (1) The intrinsic brightness of any surface cannot be increasi»d by any optical me.-ins. The natural brightness of any object — that is. the light receiveil per square centimeter of the image on the retina of the eye when looking directly at the object— is as great as the brightness when looking at the object through a telescope or microscope. (2) The bright- ness of a star or point-source may. however, be increased bv using a telescope, because by means of it more light is brought into the eye; the increase is in general proportional to the ratio of the area of the object-glass of the telescope to that of the iris of the eye. (3) No amount of magnification will enable one to 'resolve' two
