aG 



as in an 



Electrical Field. 



535 



determine Bj and B 2 uniquely unless we impose some relation 

 between B, and B 2 . The most likely seems Bj = — B 2 . Jf 



the potential 

 determined. 



For the ge 

 take the form 



is given at two points then /3 and B are 

 leral discussion of the solution we may then 



cosh 



ehx + a 



= sn (\a? + fl, k), 



where a, A, j3, k are supposed known. We get 



cosh ehx + « = 2 sn a (\* + /3,k)- 1. 



Now cosh ehx + a is proportional to the matter density of 

 free atoms. Further the density of the molecules is a func- 



and the first integral is 



tion of (|*Y 





a*/ 



8tt 



1 + 



h_ 



2 m 



B 



h_ J}, 2 



2»iN 





r-w*M 



2 cosh ehx + a ~ 



Thus in general the matter density of the gas is periodic. 

 The distance between points of equal density is given by d 

 where 



^M'4,£)( 



1 + 



A 11: 



2m N,N 2 , 



where a> and w! are the complete periods of the elliptic 

 functions and m and m 1 are the least integers which make d 

 real. 



Sinh ehx + a, which is proportional to the electrical density, 

 is also periodic in the same period. Where the function 

 sinh ehx + a vanishes we have an equal number of free posi- 

 tive and negative atoms. At such a place there is most 

 chance of recombination. It is probable that such recombi- 

 nation gives rise to luminosity. Jf the points of maximum 

 matter density coincide with the points of least electrical 

 density, then the above calculation would indicate that we 

 should have very well defined planes of maximum luminosity. 



The planes of minimum electrical and maximum matter 

 density will not, however, in general coincide. Thus, though 

 we should still have planes of maximum luminosity, they will 

 not be so well defined. 



These considerations suggest that we have something very 

 closely related to the condition of things in a striated vacuum- 

 tube. 



