698 BELL SYSTEM TECHNICAL JOURNAL 



tion of "electric charge" along the imaginary string, relatively to its 

 centre at the origin. If it should turn out zero, there would be equal 

 amounts of charge to left and to right of the centre; if it should turn 

 out positive or negative, there would be more charge to the right of 

 the centre than to the left, or more to the left than to the right; if it 

 should turn out variable, if for instance the coefficient of the cosine- 

 term should differ from zero, there would be a surging of the charge 

 to and fro across the origin. 



The functions /,(x) have been written down in equation (155), near 

 which it was shown that they are alternately even and odd functions 

 of x; /o, /2, /4 • • • are even, /i, /s, /s • • • are odd. Their squares are 

 consequently even, the products of their squares by x are odd, functions 

 of x; and the first two integrals in the expression (205) for M vanish. 



As for the integral | xfifjdx, its integrand is an odd function of x if 



i and j are both even or both odd, and in either case it vanishes; so 

 that if two wave-patterns corresponding both to even-numbered or 

 both to odd-numbered Stationary States coexist, there is no surging 

 of charge to and fro, and the electric moment of the system remains 

 constant. If i is even and j odd, or vice versa, the conclusion is not 

 so immediate. It follows however from the general properties of the 



J 'I CO 

 xfifjdx always vanishes 

 — oo 



unless i and j differ by one unit, so that in every case but this the 

 electric moment is continually zero. This leads us to the rule: 



// two modes of vibration i and j coexist, the electric moment of the 

 ''imaginary string'' representing the linear harmonic oscillator varies 

 sinusoidally with the frequency (vi — vj), if and only if i = j ± I; 

 otherwise the electric moment is and remains zero. 



This may be interpreted as meaning physically that radiation 

 occurs only when two "adjacent" states — two states for which the 

 quantum-number differs by one unit — coexist; that transitions are 

 possible only between adjacent states. 



This is a Principle of Selection. It is the principle of selection 

 predicted for the linear harmonic oscillator in the earlier versions of 

 atomic theory, and sustained by observations on those features of 

 band spectra which are attributed to simple-harmonic vibrations of 

 molecules. 



Thus in the case of the linear oscillator, the idea of interpreting the 

 square of the amplitude of the "^-vibrations as density of electric 

 charge is twice successful. When the oscillator is in one of its sta- 



28 Courant-Hilbert, Methoden der math. Physili, p. 76. 



