Mechanism of Radiation. 439 



normal displacement of the ideal atom as a sum of these 

 displacements. Hence the forces acting upon any element 

 A / o£ the ideal atom when this atom undergoes any one of its 

 normal displacements will be very approximately equal to the 

 forces acting upon the ion A when the real atom undergoes 

 the same displacement. 



But the ion A is very nearly in the position corresponding 

 to that occupied by the element A', and the total mass of the 

 element A' is equal to the mass of the ion A. Hence the 

 integral mass-displacement of the element A! is very nearly 

 equal to the mass-displacement of the ion A. 



It follows that the free vibrations of the ideal atom V will 

 be very approximately reproduced, as regards both frequency 

 and displacement, in the real atom I, provided only that we 

 select the imaginary law of force for the ideal atom in the 

 manner indicated above. It is easy to see that this imaginary 

 law of force must approximate at infinite distances to the limit 

 of the real law at infinite distances, namely the ordinary 

 electrostatic law. Hence, so long as we make no assumptions 

 about this imaginary law except that at infinite distances it 

 approximates to the ordinary electrostatic law (the assumption 

 which was made in §§ 12, 17), we shall be justified in sup- 

 posing that every quantitative result obtained for the free 

 vibrations of the ideal atom will be very approximately a 

 reproduction of a similar result for the real atom. 



§ 23. It follows that there must be a line of the real spec- 

 trum in the neighbourhood of every line of the ideal spectrum. 

 Confining our attention to the lines of a single spectrum- 

 series of the ideal spectrum, we see that the lines must be 

 approximately reproduced, line for line, in the real spectrum, 

 until we reach such large values of n that the distance 

 between successive lines becomes comparable with the error 

 of our approximation : after this it will be impossible to trace 

 oorresponding lines. This last result is, from another point 

 of view, obvious : when the order n of an harmonic becomes 

 comparable with the number of ions on a great circle of an 

 atom, it is impossible to distinguish a displacement of the real 

 atom given by an harmonic of order n from one given by an 

 harmonic of different order. 



§ 24. For one kind of spectrum-series which oecurs in the 

 ideal spectrum (equation (25), p. 436), the value of p 2 at the 

 head is equal to (p*(l + %i) +<t 2 ^ 2 )I(t. When the transition- 

 layers are supposed to be infinitely thin, this may be regarded 

 .as a continuous function of r. Hence the heads of these series 

 will form a continuous band of light. 



This, however, is not true for lines of the spectrum-series 

 other than the head, since it is only for n = co that p enters 



2 G2 



