SOME CONTEMPORARY ADVANCES IN PHYSICS— X 99 



field is superposed, the problem is scarceh- more difficull; if an external 

 electric field is superposed, it is dii'ticult but soluble — proxided always 

 that the mass of the electron \)v supposed inxariable. If the mass of 

 the electron is supposed to vary with its speed as the theory of rela- 

 tivity requires, and as certain experiments suggest, the problem 

 remains soluble — proxided that no outside influences act. I'or all 

 these cases the orbits which x-ield the obserxed energy-xalues haxe 

 been traced; and certain features have been shown to be common 

 to all of these "permitted" orbits, and to no others, so that by these 

 features the permitted orbits are set apart from all the rest. Inxersely, 

 anyone xvho is told xvhat these features are, and who is sufficiently 

 adept in dynamics, can trace all the orbits which display them and 

 calculate the energy-\-alues for these orbits and so predict the energy- 

 values of all of the Stationary States of an atom consisting of a nucleus 

 and a single electron. Such orbits are known as quantized orbits. 

 The rules xvhereby they are set apart from all the multitude of orbits 

 not permitted are the Quantum Conditions; xvhich some one, it is to 

 be hoped, will some day succeed in deriving from a general Principle 

 of Quantization. 



The most general way of phrasing these conditions is difiicult to 

 grasp, and the more intelligible ways are not the most general. The 

 most general conditions yet formulated are not adequate for all 

 cases; the completely adequate principle is yet to be discovered. 

 For the purposes of this summary and of most of what follows, a 

 very limited expression of the Quantum Conditions xvill be sufificient. 

 In the Second Part of this article it was proved that the permitted 

 orbits of an electron of invariable mass revolving in an inverse-square 

 field such as is supposed to surround a bare nucleus, are certain 

 ellipses. It was further stated, without proof, that if the electron of 

 invariable mass revolves in a central field which deviates slightly 

 from an inverse-square field, then the permitted orbits are certain 

 "rosettes" or precessing ellipses — each orbit may be traced by im- 

 agining an ellipse revolving steadily in its own plane around the source 

 of the central field (I will say the nucleus) at one of its foci. All 

 the orbits are rosettes; the permitted orbits are certain rosettes which 

 are distinguished from the others by a distinctixe feature. One xvay 

 of expressing this feature involves the angular momentum p^ and 

 the radial momentum pr of the electron. In terms of the mass m 

 of the electron, its distance r from the nucleus, and its angular velocity 

 d(t)'dt about the nucleus, these quantities are by definition : 



pr = m{dr/dt), p^ = mr-id^/dt). (1) 



