of Electrons, and the Spectrum of Canal Rays. 671 



other hand, in a radiating gas the number of ions is only a 

 small fraction of the number of atoms present, and on this 

 account collisions between ions are less frequent than between 

 atoms. For this reason assumption (C) does not seem less 

 probable for ions than for atoms or molecules. 



We shall now extend the results of § 1 2 to the case of a 

 group of a large number of similar systems, which we may 

 treat as independent of each other, in accordance with 

 assumption (0) ; in other words, we shall neglect the 

 radiation during collision in comparison with the radiation 

 during the intervals between collisions. The only role of 

 collisions is assumed to be to produce the disturbances, 

 wholly or in part, to which the radiation is due. We wish 

 to decide how much of the radiation is due to the collisions, 

 how much to radiation pressure. 



§ 14. Let us consider the systems contained in any small 

 element of volume. Just as in the kinetic theory of gases,. 

 so in our case we suppose this element to contain a very 

 large number of systems, although its linear dimensions are 

 small compared with distances usually employed in experi- 

 mental work. We must first sum the expressions (14). . .(22) 

 for all systems having velocities between U and U-j-cZU, 

 but differently orientated; afterwards we must sum for all 

 velocities and directions of velocities. 



In the first place, we cannot generally assume that alt 

 directions of polar axes (a, b, c) are equally probable, because 

 the direction of the velocity U is a singular direction. In 

 fact, in the case of a bundle of canal rays, since the bundle 

 as a whole is equivalent to an electric current, there is a 

 magnetic field inside it, which causes all the systems of 

 electrons to turn their axes towards the lines of magnetic 

 force, that is, away from the direction of motion. Let us 

 assume that the chance that the axis (a, b, c) of our system 

 may lie within the elementary solid angle dco is Ydwj^Tr. 

 The distribution function F depends on (a, 5, c) and on U ; 

 write tan %=cjb. We may expand F in a series of spherical 

 surface harmonics of the form 



F = 2 2(A is cos 5% + B 2 - s sins % )(l-rt 2 f . P<*>(a), (23) 



t=0s=0 



where ~P ( - S \a) stands for d 8 Pi(a)/da 8 . 



In the particular case mentioned, of the canal rays, the 

 part of F due to the magnetic force consists of harmonics of 

 the first order. Other harmonics may very well occur in 

 consequence of collisions. 



