138 Prof. J. J. Thomson on the Velocity of 



passing through the origin. The curve given by Paschen 

 (Wied. Ann. xxxvii. p. 69) for the relation between spark- 

 length and potential when the electrodes were spheres 

 1 centim. in radius is very nearly straight, and would corre- 

 spond to the above formula if K' were between 600 and 

 700 volts. 



If R. is the electromotive intensity required to produce a 

 spark of length /, we see from the above since R = Y/l that 



E= T + *. 



Thus the electromotive intensity required to produce a spark 

 increases as the spark-length diminishes ; in other words, the 

 electric strength of a thin layer is greater than that of a 

 thick one. 



The curve on this hypothesis representing the relation 

 between the electromotive intensity and the spark-length is a 

 rectangular hyperbola ; the curves given by Dr. Liebig (Phil. 

 Mag. [5] xxiv. p. 106) for air, hydrogen, carbonic acid, and 

 coal-gas seem to approximate to hyperbolas. 



The distance between the striae is proportional to the time 

 taken by the atoms in the Grotthus chain to recombine ; this 

 time will be greater the greater the distance between the 

 atoms ; if we assume that it is proportional to the distance 

 travelled by an atom between two collisions, the distance 

 between the striae will be proportional to the mean free path, 

 and therefore inversely proportional to the density: thus we 

 may write in equation (1) \ = (5/p, where p is the density of 

 the gas. With this substitution equation (1) becomes 



V = K'+^ (2) 



Paschen's observations on the electric strength of air, 

 hydrogen, and carbonic acid at various pressures seem fairly 

 accordant with this formula ; they show, however, that K' is 

 not quite independent of the density but increases slowly 

 with it. Jf we are dealing with sparks so long that the 

 second term on the right-hand side of equation (2) is large 

 compared with the first, then the spark-potential for the same 

 gas depends only upon the product lp. Paschen's experi- 

 ments seem to show that this law holds with great accuracy ; 

 it would, however, be interesting to have experiments with 

 smaller values for lp than those used by him. 



In the preceding equations we have supposed the field to 

 be uniform and the striae of the same length ; if the field is 



