982 Sir J. J. Thomson on a 



point but also upon the magnitude of the forces acting upon 

 the electron for some distance before it reaches the point. 



It is necessary, therefore, to make some simplifications in 

 the conditions. I shall suppose that at these low pressures 

 the negative ions are electrons, and that in consequence of 

 their great mobility as compared with that of the positive 

 ions, the current is carried by the electrons. This assump- 

 tion makes a great simplification in the equations which 

 represent the distribution of the electric force. 



The case we shall consider is that when the discharge 

 passes between two parallel plates at right angles to the axis 

 of x so that the electric force X is everywhere parallel to oc 

 and depends only upon this co-ordinate. Let n and p be 

 respectively the number of electrons and positive ions per 

 unit volume at a place fixed by the co-ordinate x measured 

 from the cathode ; then, if e is the charge on an electron, 



g=4^( ra -p) (1) 



If i is the current through unit area and w the average 

 velocity of the electrons parallel to x, 



i=zicne (2) 



If q be the number of ion's produced per second per cubic 

 centimetre at x, a. the coefficient of recombination, then 7 

 since wn is constant, 



q = unp (3) 



Thus equation (1) may be written as 



ax \^we a i J 



We have now to consider the relation between the velocity 

 of the electron and the electric force. If we suppose that 

 the energy imparted to an electron while it is describing 

 a free path is taken from it at its next collision with a 

 molecule, the average velocity of the ions at any point will 

 be proportional to the electric force at that point. This 

 result will not, however, be applicable if the free path of the 

 ion is long enough to be comparable with lengths which -are 

 within the scope of our observation ; and to assume it would 

 make impossible the explanation of any structure whose 

 length-scale was not large in comparison with the mean free 

 path. Again, the assumption that after each collision the 

 ion starts off with regard to energy just as it would if it had 

 not previously been under the action of the electric force is 



