the curves mark, by their centres, the apertures corresponding to v=0·1, 0·2, and so on, up to v=3·0. The solitary circlets near v=1.0, 2.0, 3.0 belong to p=0, i. e. to the central or focal curve which is the usual Cornu spiral. It will be seen that the curve [1] deviates but little from the central one. The intensity, proportional to the square of distance from the origin, increases with v, or with the square of the aperture, first rapidly and then more and more slowly, and reaches its maximum a little beyond v1.2. Then it falls to a minimum which is considerably smaller than that maximum, and so on. The spiral character of the curve [1] would go on for over 3000 windings (v over 140) which, however, for any reasonable lens, would lie much beyond the limit-aperture (h=r/n); the spiral would reach its cusp, or first zero of Jo, much beyond that limit. The next curve, [10], is very soon deprived of its simple spiral character; the Bessel function vanishes already at v=1465, where k=x, and the curve has a cusp in the neighbourhood of which the stations v1.1, 1.2, etc. are more and more crowded. Then the curve emerges from the singular point in an elegant fashion and follows on a nearly circular path; it is drawn up to 30 only, but even beyond that it would never go far away from the spot. The intensity, for [10], reaches its maximum a little beyond v=27; the value of this maximum is only about of that of the curve [1]. The curve [20] has one cusp at v=0·3662, and another at v=1.929. It also is drawn up to 30. Finally, the curve [30] has in the interval studied as many as three cusps, one between v=0·1 and 0.2, another between 0.8 and 0·9, and yet another between 2.1 and 2.2. Neither the [30]-, nor the [20]-curve goes far away from the origin. The illumination is here very scanty, especially at v1.2 which is the most favourable aperture for the centre, and also for [1]. Other details can be seen from the drawing itself. Here but two more remarks: First, that it seems very doubtful whether there is at all a curve repassing exactly through the origin, i. e. whether there is at all a rigorously dark ring round the focus of the lens. (physically it is enough that the light becomes very weak already for [20]). And, second, that the best illumination at the focus and, at the same time, the best definition is reached a little beyond = 12. Opening the plano-convex lens so far as v2.0, for instance, would not only darken the focus, but make the intensity at [10] nearly as great as at the focus. = In the process of constructing the curves of fig. 3 many of the values of A, B needed have been calculated by quadratures; to make the set complete all others have also been calculated. These values of A, B, as defined by (24), are not by themselves interesting, and will not be given here; the square sums A2+ B2, however, proportional to the intensity, (23 a), may be useful. They are collected, together with those for a =0, in the following Table, which may be a good supplement to the P-curves themselves. The second column contains the sum of the squares of the Fresnelian integrals, the third etc. the values of A2+ B2, for v=0.1, 0.2 up to v 30, corresponding to a 2π.10-3/2 etc., that is, to the above curves [1], and so on. A2+ B2= v πυλ = е e1 2 Jo(avv) dv 2 those of fig. 3) are drawn as abscissæ and represent the intensity at The curves in fig. 4 (which could also be taken from directly according to this table, A2+ B2 as ordinates, and thus selected places of the focal plane Fig. 4. as a function of the aperture of the lens. It will be remembered that v is proportional to the square of the aperture, viz., by (20), My thanks are due to Mr. W. Widigjör for the execution of both sets of curves. to the centre or the focus, and to the usual Cornu spiral; that curve is well-known from the distribution of light outside the shadow of a semiplane; here its physical meaning is different: it represents the central intensity as function of (the area of) the aperture. It has its first, and highest, maximum at about v=1.23, as already mentioned. A number of isolated points below and near this curve represent the intensity at the circle a'= 20 п ✓ 1000 2π ✔1000 (i. e. p=λ, for the said example). These points are too near the first curve to be joined into a continuous line. The next curve, [10], corresponds to a'=(i. e. p=10x for the above example); its first, as well as its second, maximum is very flat. Its interesting feature is that it nearly reaches the focal curve at about v=1.9. The moral is obvious: at v=1.23 the definition as well as the light intensity are excellent, at v=1.9 very bad; at about v=23 much better 40π again. The next curve, [20], corresponds to a' = ✓1000' and the lowest, [30], to a'= 60π ✔1000 To form an opinion about the intensity as function of distance from the focus, at any fixed aperture 3.0, it is enough to read the above numerical table in horizontal rows (instead of columns). The corresponding curves would again corroborate the above remark, viz. that the greatest central intensity and the best definition of the image produced by a plano-convex lens, of refractive index n and curvature radius, are obtained at an aperture corresponding to about v1.23, and to be determined for any concrete case by (25). If, for instance, n=1.5 and λ= micron, then the best relative aperture is, in round figures, The last of these would be a very minute lens indeed. Practically, not going much below r=3 cm., the best relative aperture will not exceed. Opening the lens up to v 1.9 would spoil the central intensity and the definition considerably. The next best aperture, after e=1.23, would correspond to about v=23; the next favourable opportunity will lie a little beyond v=3. The marginal phase retardation 1.23, n=1·51; 512 or a little over is, for the best aperture v 3/8 of a period. Returning once more to the table on p. 45, notice that the v-values arrayed in the first column have themselves a simple meaning, their squares being proportional to the "normal" central intensity No which would correspond to a perfect wave. Thus the relative central intensity, which is the squared ratio of chord and arc of the Cornu spiral, is given by Io : No = (Ao* + B):v, i. e. simply by the figures of the second divided by the squares of those of the first column. Thus, for v=0.5, I。: No= 250' 247 and for v=1, or equal to a quarter-period, I: N。=8004, agreeing with Lord Rayleigh's result of 1879, and so on. The relative intensity at the focus decreases steadily, of course; the absolute intensity, due to the defective wave, increases only up to a certain maximum and oscillates in decreasing amplitudes round a limit corresponding to the asymptotic value A2+ Bo2. The central intensity by itself cannot, of course, inform us about the "definition." It so happens however that, in the case of the plano-convex lens at least, the best aperture, v 1.23, for the focal absolute intensity is also sensibly the best for the definition of the image. So much about the distribution of intensity in the focal plane of the lens. The intensity at points outside the focal plane will require but a few remarks, owing to the relative smallness of the axial intensity gradient (for small ) already hinted at. Consider, for instance, the points of the optical axis itself, that is, a'=a=0. Then, according to (22), the 1/2 arc element is dl= (dv, precisely as for the focal or Cornu-spiral itself, and, the sloping angle and the curvature of a B-curve, 1/2 where B=2π/ measures the axial coordinate of the point in question. Take, for instance, our previous example 1 n=1.5, r/λ= 106. Then, by (19), πυ B k=1000πv+B. *Phil. Mag. viii. p. 409. Rayleigh's figure, 0-8003, differs but insignificantly from the above one, |