666 BELL SYSTEM TECHSICAL JOVRKAL 



By integrating (41) from one of these values to the other aiui douhHiig 

 the result, we get the period of the revolution 



T='2Tr\/ma\/eE. (42) 



J2. Motion of an Electron in n Central Field Differing Slightly from an 

 Inverse-square Field 



Suppose we modify the atom-model composed of a nucleus and an 

 electron by imagining that the force exerted by the one upon the 

 other varies not exactly, but very nearly, as the inverse square of 

 their distance apart. For instance, one might imagine that the force 

 varies as r^'""; or that the nucleus acts upon the electron with an 

 attraction equal as heretofore to eE/r'-, plus an additional attraction 

 (or repulsion) varying inversely as the cube of the distance. In any 

 such case the potential cnerg>' of the atom-model would not be quite 

 equal to —eEr; there would be an additional term f{r). In the 

 case of an inverse-cube field superposed upon an inverse-square field, 

 the expression for the potential energy would be 



V=-eE r-C,r- (43) 



The second term on the right hand side will be much smaller than 

 the first, at and only at distances much greater than 2C/eE; but 

 l)y imagining C sufficieiitlN' small, we can arrange to have the inverse- 

 cube field very much smaller than the inverse-square field, over all 

 the region in which tlic orbit of the electron is likcK' to lie; and this 

 is all that matters. 



The orbit of the electron may be described, in all these cases in 

 which the force deviates very slightly from an in\erse-square force, 

 as an ellipse precessing in its own plane. That is to say : an ellipse 

 of which the major axis swings at a uniform rate around the nucleus 

 as if around an axle perpendicular to its own plane — as though the 

 electron were a car, running around and around an elliptical track, 

 quite unaware that the track itself is endowed with a revolving motion 

 of its own. (Or, in other and more sophisticated words, the orbit 

 (jf the electron is an ellipse stationary in a coordinate-system revolving 

 around the nucleus at a uniform rate). Such an orbit is known as 

 a "rosette," and a part of a rosette is shown in Fig. 3. 



Another way of describing the important feature of liiis orinl is 

 to say that the two coordinates r and of the electron in its orbit 

 (referred to O as origin and OP as the direction = (), in Fig. 3), 

 while the)' arc both periodic, do not have the same period. While r 

 is running throuj^h its entire cycle from r„^^^ to r,„,„. and back again, 



