Striated Electrical Discharge, 247 



new ph} T sical considerations will have to be introduced, which 

 were not allowed for in forming this equation. 



§ 2. Our analysis seems to indicate that there is only one 

 explanation possible which is consistent with the main facts 

 of the theory of conduction by ions, and this, stated briefly, 

 is somewhat as follows : — 



Thomson's equation gives a solution for the case of steady 

 motion which not only is mathematically possible, but is the 

 only solution which satisfies the assumed conditions through- 

 out. This steady motion, however, is, under certain circum- 

 stances, unstable, just as the steady motion to which the 

 hydrodynamical equations lead for trie case of liquid flowing- 

 through an orifice is unstable. And, just as the liquid is 

 known in nature to break up into a succession of small drops, 

 so the discharge in a vacuum-tube breaks up, under suitable 

 conditions, into a succession of small discharges, each dis- 

 charge forming a single striation. 



A mathematical explanation of this may be given in the 

 following way : — 



If we suppose that Thomson's equations are absolutely true 

 over all ranges of the variables, then there is only one form 

 of discharge possible, and this is non-luminous. When, 

 however, the volume-density of the ions becomes indefinitely 

 great, it is obvious that these equations can no longer hold 

 without modification. It is impossible to get an infinite 

 volume-density of electricity in nature, and this fact is not 

 contained in the equations. We are therefore compelled to 

 admit the existence of new forces which were not allowed for 

 in the original equations ; these forces are negligible until the 

 volume-density of the ions reaches a certain large limit, but 

 they then come into action, and are sufficient to prevent the 

 volume-density from ever actually becoming infnite. These 

 forces may be of the nature of a repulsion between pairs of 

 ions which have approached indefinitely near to each other, 

 or, if we imagine the ions to be bodies of finite size, they will 

 consist of the forces of elasticity which prevent a collection of 

 such bodies from being compressed into an indefinitely small 

 space. 



But, without making any assumptions about the nature of 

 these forces beyond those laid down in the sentence in italics, 

 it can be shown that the existence of such forces makes 

 possible a second type of solution, which is essentially different 

 from that considered by Thomson, and that this solution, inter- 

 preted physically, predicts the existence of a system of 

 striations in the middle of the tube, which may be expected 



S2 



