Mobility of the Positive Ion at Low Pressures. 163 



pass away from them the only causes which can ultimately 

 lead to divergence are : (a) the eccentricity may become too 

 large ; or (b) there may be too close approach to the dis- 

 turbing mass, so that the mean motion and therefore the 

 semi-major axis must bo restricted. If we consider three 

 Lindstedt series representing the complete solution for one 

 integration, we get p x , p 2 , p$ in this form. Eliminate the 

 eccentricity and the major axis. Then we get a series which 

 does not involve these quantities, and therefore shows no 

 tendency to diverge under any circumstances. Hence this 

 is a true integral. 



It may be added that such an integral would probably lead 

 to interesting results in astronomical work. In particular, 

 the fact that <J> involves q s might enable us to examine the 

 motion of the line of apsides in a new way. 



4. Summary. 



The restricted problem of three bodies in three dimensions 

 probably admits of a hitherto unknown integral, which may 

 be capable of expression as a convergent trigonometric 

 series. 



Bristol, 

 Oct. 29, 1912. 



XVIII. Further Experiments on the Mobility of the Positive 

 Ion at Low Pressures. By George W. Todd, M.Sc. 

 (Birm.), B.A. (Cantab.), late 1851 Exhibition Research 

 Scholar of the University of Birmingham* . 



IN a previous paper t the author has shown that the 

 mobility of the positive ion in gases at low pressures 

 departs from the inverse pressure law, but at pressures much 

 lower than is the case with negative ions. For different 

 gases the pressures at which the mobilities begin to show 

 abnormality appear to be approximately proportional to the 

 normal mobilities in the oases, so that the denser the gas the 

 lower the pressure is at which the mobility begins to increase 

 beyond that given by the inverse pressure law. In the case 

 of the negative ions also J, for the same apparatus, and when 

 certain conditions are kept constant, a similar relation is 

 found to hold between the " critical " pressure and the 

 normal mobility as holds for the positive ion. The curves 



* Communicated by the Author. 



t Phil. Mag. Nov. 1911, p. 791. 



% Proc. Camb. Phil. Soc. xvi. p. 653. 



M2 



