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The theoretical curve is obtained from Clapeyron's relation

L= (V_v)T dp


to show the change of latent heat with temperature. (Since the values of V and др are not known with the same accuracy as L, the slope of the ar curve only has been taken from this relation. This is sufficient for the purpose of reducing the results.)

It will be noticed that the more accurate methods of measurement give values which show clearly the agreement with the thermodynamical result.

On comparing this with the author's results, which are measured directly in mean calories, it follows that

538.5 x 4.188=538-88 J,

i. e.,

J=4.1850002 joules per mean calorie.

It will be noticed that this is in exact agreement with the values obtained by the earlier experimenters-Schuster and Gannon, Reynolds and Moorby, Callendar and Barnes,-and with the values deduced on separate occasions (from all data then available) by Griffiths and by Barnes; on the other hand, this value and those obtained by the more recent experimenters are certainly too discordant to be reconciled.

This agreement between the values given by the classical experiments and the value given by a method so different as the present substantially increases the probability that the value of the Mechanical Equivalent lies between 4·184 and 4.185 joules per mean calorie.

V. Light Distribution round the Focus of a Lens, at various Apertures. By L. SILBERSTEIN, Ph.D., Lecturer in Natural Philosophy at the University of Rome *.

Bibliographic and Introductory.


HE distribution of the intensity of light in the neighbourhood of a caustic has been studied by Sir G. Airy as early as in 1838 (Camb. Phil. Trans. vol. vi.), for the case, however, of an unlimited beam only. Some of the effects of spherical aberration of limited beams upon the c intensity and the definition of the image have been investigated by Lord Rayleigh in 1879 (Phil. Mag. vol. viii. pp. 403-411). The chief problem considered by him relates to a beam of cylindrical waves of rectangular section, their aberration being assumed proportional to the cube of the lateral coordinate . The solution is reduced to the evalua


tion of an integral of the form fcos (ax+b)da. Availing himself of the numerical results of the mechanical quadratures recorded in Airy's paper, Lord Rayleigh calculates and draws three intensity curves for the focal plane (loc. cit. p. 406), corresponding to the case of no aberration (b=0), and to those in which the marginal aberrations amount to

and period. The practically important consequence drawn from the aspect of these curves is that "aberration begins to be distinctly mischievous when it amounts to about a quarter-period." In the next case studied, that of symmetrical aberration proportional to 4, Lord Rayleigh calculates, by the aid of a series, the intensity at the central point only. Passing, finally, to beams of circular section, he limits himself again to the calculation of the central intensity, viz. in the case of axially symmetric aberration. proportional to the fourth power of the distance, and finds that, as in the preceding cases, aberration begins to be prejudicial when it mounts up to a quarter of a period. This result has since become widely known, having been incorporated into the Enc. Brit.† and several English text-books. In 1884 Lommel investigated the distribution of illumination in the diffraction image of a point given by a circular aperture ‡. The problem in this case reduces to the evaluation of integrals of the form

SJo(ax) cos (bx2)xdx, SJ(ax) sin (bx2)xdx

*Communicated by the Author.

+ Cf. Lord Rayleigh's article on "Diffraction of Light" in the 11th ed. of Enc. Brit. vol. viii. pp. 238-255.

E. Lommel, Bayer. Akad. d. Wiss. vol. xv.


which are developed into power series of the upper limit or of its reciprocal. Lommel's results have been adapted, in 1891, to the problem of pin-hole photography by Lord Rayleigh whose paper, besides a theoretical and experimental discussion of the subject, gives also five curves exhibiting the distribution of light round the centre of the image corresponding to different apertures. As will be seen later on, the typical phase aberrations of wave-surfaces emerging from lenses differ in kind from those involved in the pin-hole problem. Investigations aiming directly at a diffractional treatment of the images produced by lenses were undertaken, in 1893, by R. Straubel, whose papers are quoted in Winkelmann's Handbuch' (1906, vol. vi. p. 106), but unfortunately are not accessible to the writer, and a little later by K. Strehl in a very attractive book entitled 'Theorie des Fernrohrs auf Grund der Beugung des Lichts' (Barth: Leipzig, 1894)†. The earlier part of the work being dedicated to preparatory matter, Strehl investigates in Chap. V. and VI. the intensity along the optical axis and in the focal plane of an aplanatic object-glass, and since this is materially the same problem as that of a circular aperture treated by Lommel (loc. cit.), Strehl bases himself upon Lommel's results, as had already been done by Lord Rayleigh, and enunciates a number of theorems on the general features of the light distribution for the case in question. The effects of "spherical aberration" are treated in Chap. VII., where the intensity formula is developed for the case in which the emergent wave is an ellipsoid of revolution; the series developments (pp. 62-63) are very complicated, and it does not appear that they could conveniently be applied to concrete numerical calculations. They enable Strehl, however, to enunciate some general theorems about the symmetry relations of the diffraction effects associated with spherical aberration of the said kind, and an important conclusion on the true measure of the mischievous effect of spherical aberration (p. 65). In a sense, Strehl is right in declaring that by his investigation "t problem of spherical aberration is completely solved." So

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Phil. Mag. vol. xxxi. (1891) pp. 87-99.

After that Strehl has published several papers in Zeitschr. f. Instrumentenkunde for 1895-98 which I have not been able to consult. However, to judge from Winkelmann's quotation, the ground covered by these papers is essentially that of Strehl's book. Winkelmann (loc. cit. p. 403) quotes also, in connexion with the diffractional theory of the telescope, Ch. André's "Etude de la diffraction dans les instruments d'optique" (Paris, 1876), without, however, describing the contents of this paper, which was published in Ann. sc. de l'école norm. supérieure, vol. v.

fact it is, in its essence. None the less it seems desirable to treat problems relating to concrete lenses with all numerical (or graphical) details, and to express the results in terms of the attributes of the given lens or lens system.

It is precisely the object of the present paper to give a fully worked out example of this kind, as a part of investigations undertaken at the instance of Messrs. Adam Hilger in connexion with their Lens Interferometer which exhibits ad oculos, through its "contour map," the phase retardation of all the elements of an originally plane wave produced by the passage through a given lens. The example selected for the present purpose relates to the simplest possible lens, viz. the plano-convex lens, traversed by a beam of finite circular section along the optical axis. It has seemed that, owing to its extreme simplicity, it may be the best to show the reader a practicable and easy way of dealing with more complicated telescopic objectives.

To complete the above bibliographic sketch we have still to mention that the remaining chapters of Strehl's work are dedicated to the diffractional aspect of astigmatism and coma which are treated on similar lines as spherical aberration, to cylindrical waves, etc. These subjects, however, are beyond the scope of the present communication. Lastly, we have to mention a more recent paper by James Walker (Proc. Phys. Soc. London, vol. xxiv. 1912, pp. 160164) in which the subject of Strehl's Chapter VII., viz. the intensity due to a rotationally ellipsoidal wave, is again taken up. Here the expression for the intensity is developed into a complicated double series (cf. last line of the paper quoted) which, although mathematically unobjectionable, does not seem convenient for actual calculation *.

It must be kept in mind that for physical applications hardly more than two significant figures in the final light intensity are required. Under these circumstances the method of mechanical quadratures or a graphic method, analogous to that of the Cornu spiral, seems by far the most convenient. Although laborious for very accurate work, it certainly becomes very handy when only the said degree of precision is aimed at. It will be explained and applied in what follows.

It has occurred to me that some of Walker's intermediate formulæ, as for instance that on top of p. 163, would easily yield a more "commodious" expression than is the final one.

General Formula.

Let the portions of the surface of a fixed sphere, of centre O and radius R, which we will take as our reference sphere, be the seat of monochromatic luminous oscillations of constant amplitude a, but of different phases 7,

a cos

Given the distribution of n over s, find the intensity of light at the centre O and at points P near O. Let ds be an element of the reference surface, r its distance from the point P in question, e the light velocity and λ=c7 the wave-length, in vacuo. Then the usual way of applying Huyghens' principle gives, for the luminous vibration at P,




a 1

- S sin (nt.




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where n=2π/T. Let N (fig. 1) be the pole of angular coordinates, i. e. ON the axis, 0 and the pole distance and the longitude of an element ds. Let op be the longitude

Fig. 1.

2r-n) ds,


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of the point P, further its distance from the axis and o its axial distance from O, away from N. Then, neglecting the squares of p/R, σ/R,


ρ = R [1 − 2 sin 0. cos (4-4p) + R Phil. Mag. S. 6. Vol. 35. No. 205. Jan. 1918.



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