I30 



NATURE 



[JuLV 28, 1923 



Letters to the Editor. 



{The Editor does not hold himself responsible for 

 opinions cxptessed by his correspondents. Neither 

 can he undertake to return^ nor to cor res fond 7vith 

 the ivriters of rejected manuscripts intended for 

 this or any other part of Naiurk. No notice is 

 taken of anonymous communications] 



The Ouantum In Atomic Astronomy. 



The approach to the quantum by the path of 

 energy, though historically natural and probably 

 inevitable, is scarcely the simplest mode of presenting 

 it to students. So long as assumptions or guesses have 

 to be made, as a supplement to ordinary dynamics 

 when applied to events occurring in the interior of an 

 atom, it is best to make them nakedly, so as not to 

 cloak their character ; and then to let experience 

 justify them, and hope for subsequent theory to 

 explain them. This is a procedure after the manner 

 of Kepler. The following brief summary, though 

 inadequate as an exposition, is sufficient to indicate 

 the main points in what I imagine to be a slightly 

 clarified mode of presentation. 



Bohr assumed (virtually) that, in a family of 

 electrons revolving round a nucleus, the rate of sweep- 

 ing areas, rHdJdt or y*w constant for any one orbit, 

 proceeded discontinuously in arithmetical progression 

 from orbit to orbit. This supplied a kind of Bode's 

 law for the succession of satellite electrons, not at all 

 dissimilar from the actual rough succession of planetary 

 orbits round the sun provided that some of the possible 

 orbits may be left empty ; as they conspicuously often 

 are inside the atom. 



The recognised expression for twice the rate of 

 sweeping areas, for inverse-square motion round a 

 centre of force, is 



and this, multiplied by the mass of the revolving 

 particle, is its moment of momentum mpv, with p 

 the perpendicular on the tangent ; also called angular 

 momentum, mrHOjdt. 



Bohr's assumption is that in the atom this quantity 

 can only exist discontinuously in indivisible units or 

 atomic portions, say A, of which only integer multiples 

 are possible ; so that it equals «A. One would gladly 

 use the letter h for twice the rate of describing areas, 

 as usual, had not the symbol been otherwise mono- 

 polised, in this connexion, by a quantity which, though 

 approached differently, turns out on arrival to be 

 nearly the same. 



Our first equation, then, is that 



m jM.a{i -e*)=wA. 



The time period of an inverse-square orbit is well 

 known as 



= 27r V / — ; 



> u. 



and this is our second equation. 



So, combining these two equations, and ignoring 

 the excentricity e as an unimportant and provisional 

 detail, we get at once for the angular velocity in a 

 permissible circular orbit. 



2ir fj-'m" 



ll)= r^ = i 



T w'A" ' 



M being, as usual, the force intensity, or acceleration 

 at unit distance, namely in the electrical case, Ee/m, or 

 y'w*" For accuracy, ni should be interpreted through- 

 out as half the harmonic mean, Mm/(M + w), because 

 the revolution is round the common centre of gravity. 



NO. 2804, VOL. I 12] 



But, in accordance with Bohr's assumption 

 nA = mr*u ; so that nAw is energy, ntv*, or say 2VV'. 



Energy is therefore proportional to frequencv ; ai. 

 we can proceed to identify Aw with Planck's A . 

 and find that the relation between the introducech 

 constants is simply h = 2irA, because w= 2»»'. • 



Further, by remembering that whenever a partic ' 

 falls in towards an inverse-square centre of force ; 

 gains twice the energy which it can retain in a circul. 

 orbit (though no dynamical reason can be given f 1 ■ 

 its half-stopping and occupying such an orbit and 

 ejecting its surplus energy), we get for the energy 

 radiated, on Bohr's second assumption that radiati< 

 only occurs when electrons drop from orbit to orbi 

 the difference between injAwj and i«,Aw, ; or 



w.-w.=S'.U-;^.)- 



Whence Rydberg's spectrum - f requency-constan • 

 defined as the constant part of 3W/A, comes out in t; 

 alternative forms, 



N = 



fi'm' 2ir''E*e*tn 



4irA» • 



2A»A y»» 



of which the la^t seems to have some advantages. 



Oliver Lodge. 



The Resolving Power of Microscopes on Test- 

 plates for Microscopic Objectives. 



In letters published in Nature (September i, 192 1, 

 p. 10 ; February 16, p. 205, and May 27, 1922, p. 678) 

 on the above-mentioned subjects, I gave an estimate 

 of the limit of microscopic resolving power ; that is, 

 of the least distance which must exist between two 

 points in the focal plane if they are to appear as . 

 separate points in the image. I mentioned half a 

 wave-length of the illuminating light as its approxi- 

 mate value. I have now, however, reason to believe 

 that this is an overestimate and that o-7X is nearer 

 the mark. This is in agreement both with a re- 

 computation of the illumination near the image of 

 a point and with observations made on the test 

 plates. 



The image of a bright point in the geometrical focus 

 of a lens consists, as is well known, of a bright disc 

 surrounded by rings, the dark spaces between which 

 indicate the positions where the integral difference 

 of the optical length of the rays from any part of the 

 dark ring to the corresponding distance from the 

 geometrical focus is half a wave-length. 



In Fig. I let O be the geometrical focus and O7 the 



axis of the lens. Let SS' be a section of the spherical 

 wave surface which by the action of the lens is 

 converted into a second spherical surface with the 

 same axis and with its centre at the conjugate focus. 

 Let P be a point in the focal plane near O, and divide 

 the surface SS' into elementary zones by planes to 



