Gases by the Motion of Negatively charged Ions. 215 



equation 8, we may simplify the notation by using a x and X l 



for - and — . The new variables denote the values of a and 



V P 



X when the pressure is one millimetre. When X x is greater 



than 400, the tangent to the curve fig. 7 tends to become 

 parallel to the axis of x when x x is increased. This shows 

 that the value of a x approaches a maximum for large forces. 



The greatest value of a x as obtained from the experiments 

 at *1 71 millim. pressure is 19. When this value is reached, 

 the increments o£ a for large increases of force are small, so 

 that the number of collisions ft cannot much exceed this 

 value. In order to represent the curve by an equation 

 similar to equation 7, in which the coefficients /3p are less 

 than /3, it was found necessary to take /3 = 21 ; which, as we 

 have seen, must be about the correct value. 



For the purpose of getting an expression for a x in terms of 

 Xx which will agree approximately with the experimental 

 curve, we shall take seven terms in the expansion of a lm When 

 the velocity of the ion on impact is less than I 5 , the probability 

 of producing an ion is very small. The number of ions 

 formed by collisions in which the velocity is less than I x may 

 be neglected. Letting p = l we see from equation (6), 

 Section 6, that the number of impacts per centimetre in 

 which the velocity of the ion is intermediate between I 2 and 

 I 5 is 



?/3 



(e *i _ e - x^ 



Let bi be the average number of new negative ions 

 generated by a single ion in going one centimetre with a 

 velocity intermediate between I L and I 5 . Tne total number 

 of collisions will be /? in this case. Hence when the ion 

 is moving under an electromotive force X 1? the number of 

 new ions produced by collisions in which the velocity of 

 impact is intermediate between I t and I 5 is 



Hence a x can be expressed by the sum of terms : — 

 «x=Me X *-'e * l ) + b 2 (e * -e ~) 



e x,_ e xj + &c. (10) 



