Mechanism of Radiation. 423 



making a constant angle with this line. The direction-cosines 

 of the axis of the doublet at time t are each of the form 



a + b cos (wt + 77), 



so that the component of the doublet along any axis is of the 

 form 



A (a + b cos (wt + 7})) cos pt, 



and this may be replaced by three simply harmonic doublets 

 of frequency 



p—iv, p, p+w 



Hence the radiation in question may be represented by three 

 simple harmonic terms of frequencies 



p—w, p, p + ic. 



Analysed in a spectroscope, this radiation will show bright 

 lines at the points corresponding to the frequencies p, p + w, 

 and (in the more general case) p±2iv, &c. In adding up 

 the radiation from all the vibrators, we must suppose w to 

 vary as we pass from vibrator to vibrator, so that the whole 

 radiation may be supposed to consist of two parts. 



(a) A bright line at the point of the spectrum correspond- 

 ing to the frequency p. 



(ft) A band of light, symmetrical about the point p, and of 

 which the width depends upon the mean rotation of the 

 vibrators. 



§ 3. The complete spectrum is, of course, found by adding 

 up the radiation corresponding to every degree of vibratory 

 freedom. The condition that the resulting spectrum shall be 

 a pure line-spectrum, is that that part of the spectrum which 

 has just been denoted by ft shall be imperceptible. This 

 condition can be fulfilled in two different ways : — 



(i.) The intensity of the bauds of light may be very small 

 in comparison with the intensity of the bright lines. 



(ii.) The breadth of these bands may be so small that they 

 are indistinguishable from the lines. 



§ 4. Let us begin by the consideration of the former 

 alternative. The aggregate brightness of a band bears to 

 that of a line a ratio comparable with unity except when the 

 radiation emitted by a vibrator is approximately the same in 

 all directions. Now spherical symmetry of this kind would 

 be at variance with the fundamental suppositions of the 

 undulatory theory, for spherical symmetry could only be 

 obtained by supposing the radiation to be specified by a single 

 vector, and this vector to be radial at every point ; whereas 

 the undulatory theory is such that two vectors are required 



2F2 



