138 MI;. .1. s. nnvxsKND ON THE DIFFUSION OF IONS INTO GASES. 



/f-fj tit/ 



From the differential equation of the curve x* y^ + x = 6 s (a? x*) xy, we 

 can easily trace qualitatively the form that the curve takes as r is increased from 



to a. 



We have seen that initially, when x is small, y, dy/dx, and cPyjdy? are positive 

 quantities. Let us suppose that it is possible for dy/dx to be negative for any value 

 of 7- less than a, the curve taking the form of the dotted line in the figure. 



Before dy/dx, which starts with being positive, can become negative, it must pass 

 through a zero value at x = b. 



The differential equation then gives 



6 2 ^ = - 0* (a 2 - b 2 ) 6 2 Y. 

 tar 



Hence d'y/dx* is positive, therefore as we go along the axis of x in the positive 

 direction from b, the tangent to the curve again begins to make a positive angle with 

 the axis of x, so that y begins to increase. This shows that dy/dx cannot be negative 

 at any point between x = and x = a. Hence the curve must have a form some- 

 what similar to the continuous line in the figure, the value of y when x = a being 

 greater than the value of y at the origin. Hence the function M r = a cannot vanish 

 for any negative value of (P. 



7. When r is made equal to a in M, the expression becomes a function of ^a 4 , with 

 numerical coefficients. The two smallest roots of the equation M r = = are 

 6la* = 7 '313, and %* = 44'56, which were found by expanding the function M r = <l in 

 ascending powers of (Pa*. For the determination of these roots, eight terms in the 

 expansion were found ; the larger roots cannot conveniently be found by this method, 

 but for the purposes of this investigation their determination is not necessary, as the 

 terms which they introduce into R are smaller than the experimental errors. 



The other numbers which are required are 



IPf '=-1321, RT " = "0302, 



0f s L dr J ffy? [_ rfr _ 



= '0926, 



= '0279. 



