﻿The Binding of Electrons by Atoms. 193 



is steered with increasing velocity down towards the middle, 

 where the cars can pass and repass without difficulty. 



To avoid a deep hole in the sink in the middle, the profile 

 can change to the parabola of a forced vortex, where 



v= M , n = 2, y=^-, NG=p, A.=iGV = iSP. 



On a horizontal circle of this track of one lap to the mile, 

 NP = 840 feet ; described in two minutes at 30 miles an 



NP 



hour, NV=60 feet, and cot 0= ^= = 14, a slope of 4°. 



Raise the speed to 60 miles an hour on this track, 

 NP = 420, NV = 270, feet, and the slope is nearly 30°, the 

 circuit of two laps to the mile made in 30 seconds. 



At a speed limit of 90 miles an hour, NP = 280, NV = 540, 

 feet; round a circle of three laps to the mile, on a slope of 

 over 62°. The surface could then change to a paraboloid, 

 with a flat area in the middle, where a car could come to rest. 



XIII. The Binding of Electrons by Atoms. By J. W. 

 Nicholson, F.R.S., Fellow of Balliol College, Oxford*. 



ACCORDING to the quantum theory of atomic structure 

 and of the emission of line spectra, the paths of the 

 electron in the atom vary according to the particular co- 

 ordinates used in the process of quantizing the separate 

 momenta. Thus in the simple case of a hydrogen atom, 

 containing a nucleus and one electron, we may use either 

 spherical polar or parabolic coordinates, and the admissible 

 orbits are entirely different in the two cases. Yet the final 

 values of the atomic energy are the same, and consequently 

 each method yields the same theoretical spectrum. It has 

 been suggested that there is in fact, in every case, only one 

 type of coordinates which can be used, when all the modi- 

 fying circumstances, such as the variation of the mass of the 

 electron with speed, are taken into account. The only pro- 

 blems yet solved are those in which the separation of 

 variables, after the manner of Jacobi, can be effected, and 

 the contention is in fact that there is, in every case, only one 

 set of coordinates which allows this separation, when non- 

 degenerate cases of the motion are discussed. 



But it is generally believed that the atomic energy is in 

 all cases determinate and definite. We shall show, in the 



* Communicated by the Author. 

 Phil. Mag. Ser. 6. Vol. 44. No. 259. July 1922. O 



