OPTICS 803 In the preceding statement it has been supposed for simplicity that the lens conies to a sharp edge. If this be not the case we must take as the thickness of the lens the difference of the thicknesses at the centre and at the cir cumference. In this form the statement is applicable to concave lenses, and we see that the focal length is positive when the lens is thickest at the centre, but negative when the lens is thickest at the edge, To determine practically the focal length of a convex lens we may proceed in several ways. A convenient plan is to set up a source of light Q (fig. 14) and . a screen q at a A j C I A distance ex ceeding four times the focal length, and to observe the two positions of the lens A, A at which the source is in focus upon the screen. These positions are symmetrically situated, and the distance between them is observed. Thus Xow so that J_ 1 AQ+A 2 f= . 7 = i
AQ + Aj From the measured values of- Q^ and A A , / can be deduced. If A and A coincide, the conjugate foci Q and </ are as close as possible to one another, and then/= |-Q /. The focal length on a concave lens may be found by com bining it with a more powerful convex lens of known focus. Aberration of Lenses. The formula (1) determines the point at which a ray, originally parallel to the axis and at but a short distance from it, crosses the axis after passage through the lens. When, however, the ray considered is not quite close to the axis, the point thus determined varies with the distance y. In the case of a convex lens the ray DH (fig. 15), distant HC ( = y) from the axis, A crosses it after refraction at a point F which lies nearer to the lens than the point F determined by (1), and corre sponding to an infinitely small value of y. The distance F F is called the longitudinal aberration of the ray, and may be denoted by Sf. The calculation of the longitudinal aberration as dependent upon the refractive index (/*) and the anterior and posterior radii of the surfaces (r, s) is straight forward, but is scarcely of sufficient interest to be given at length in a work like the present. It is found that - i v i _J s A / 8 W <. r* / /, s, and /being related as usual by (1). The first question which suggests itself is whether it is possible so to proportion ? and s that the aberration may vanish. Writing for brevity R, S, F respectively for r~ l . s~ l , f~ l , and taking G = M _ so that - s = ( G /^) - R we s et Since M >1, both terms are of the same sign ; and thus it appears that the aberration can never vanish, whatever may be the ratio of r to s. Under these circumstances all that we can do is to ascertain tor what form of lens the aberration is a minimum, the focal length and aperture being given. For this purpose we must suppose that the nrst term of (4) vanishes, which gives J .(5). The corresponding value of - s is .(6); so that ~s : r = 44 ,"- _ 2 a (7). In the case of plate-glass /x = 1 5 nearly, and then from (5), (6), (7) Both surfaces are therefore convex, but the curvature of the anterior surface (that directed towards the incident parallel rays) is six times the curvature of the posterior surface. By (3) the outstanding aberration is -s 15 ?/ 2 -TIT ..(8). The use of a plano-convex lens instead of that above determined does not entail much increase of aberration. Putting in (3) s=oo , and therefore by (1) r = %f we get This is on the supposition that the curved side faces the parallel rays. If the lens be turned round so as to present the plane face to the incident light we have rx>, sf, and then ..(10), nearly four times as great. For a somewhat higher value of /J. the plano-convex becomes the form of minimum aberration. If s=oo in (6), 4+/x-2/x 2 = 0, whence ^=1 69. If fj. be very great, ve see from (5) and (6) that r and s tend to become identical with/. For the general value of /j. the minimum aberration corresponding to (7) is by (4) g* ?/ _ M4/*~ 1) M-M The right-hand member of (11) tends to diminish as JJL increases, but it remains considerable for all natural substances. If ti = 2, Pencils. Hitherto we have supposed that the axis of the pencil coincides with the axis of the lens. If the axis of the pencil, though incident obliquely, pass through the centre of the lens, it suffers no deviation, the surfaces being parallel at the points of incidence and emer gence. In this case the primary and secondary foci are formed at distances from the centre of the lens which can only differ from the distance corresponding to a direct pencil by quantities of the second order in the obliquity. Hence, if the obliquity be moderate, we may use the same formulae for oblique as for direct pencils. The consideration of excentrical pencils leads to calcula tions of great complexity, upon which we do not enter. Chromatic Aberration. The operation of simple lenses is much interfered with by the variation of the refractive index with the colour of the light. The focal length is decidedly less for blue than for red light, and thus in the ordinary case of white light it is impossible to obtain a per feet image, however completely the spherical aberration may be corrected. From the formula for the focal length we see that Sf . / 1 1 */* 1 ^ = -8/j.l I = r -7 /- r s / ft- 1 / so that or the longitudinal chromatic aberration varies as the focal length and as the dispersive power of the material com-
