MR. J. S. TOWNSEND ON Till. I M FISSION OF IONS INTO GASES. l:!7 



coefficients, in the expansion of p in Equation (5), would be detenuinable by means 

 of the identity 



xMEEC/M. + C/M.+ Ac., 



and by using 7 (a) and 7 (b) we see that 



P ' - 

 ~ 



1 1. nee any function x c&n be expanded in a series of the " M" functions. 

 It can further be seen from Equations G (a) and 6 (b) that x ( r ) can be expanded in 



a series of functions <f>, where <j> is a solution of ' ; - + &*/(>') $ = ; those values 



of 6 Ixmig selected which make <f> vanish at the boundary of the cylinder to which 

 X applies.] 







G. Before determining the roots of the equation M r = = 0, it will be found useful 

 to establish the two following propositions : 



- 



1. All the coefficients -kyfLj in the expansion of R in Equation (9), are 



positive, and their sum is . 



2. All the roots of the Equation M r = . =: are positive. 



When z = 0, R must be unity, so that 



= i- 

 Also 



1,/rfr 



~ a*e\\lWi(a* - ,*)rdr " J 



This last expression is essentially positive, since r is less than a, hence none of the 

 coefficients in the series (9) can exceed ( sum of preceding coefficients). 



The second proposition is easily proved by a geometrical method, which shows 

 that when ff* is negative M r = a is a positive quantity greater than unity. 



The first few terms in M are 



M= i-^V + V 



Let us suppose that 0* is negative, and let a curve be drawn, the x axis of which 

 is r, and the y axis M. 



When x = : y = 1, dy/dx = 0, and tfyjdx* is positive. 



Hence the curve cuts the axis of y at unit distance from the origin, the tangent to 

 the curve is here parallel to the axis of x, and as x increases the tangent begins to 

 slope at a positive angle to the axis of x. 



VOL. CXCIII. A, T 



