﻿908 Mr. 0. G. Darwin : A Theory of the 



law of force may be proved by taking the only other simply 

 soluble case, the inverse cube. Analytical difficulties enter 

 a little earlier here, but when the velocity is very high it 

 may be proved that the velocity curve is a common parabola, 

 whereas with the inverse square it is 1 — ns= Ar 4 /log v. The 

 difference is quite marked, and we may say that the results 

 of experiment could not be deduced from any arbitrary law 

 of force. 



§ 3. Application to Air* 



We next examine the velocity curve numerically. If the 

 range of the a particles is R we have : — 



87rNn£<r 2 R 



= Elog(l+^). ... (7) 



provided that log 10 o- 2 V 4 A 2 , the 'parameter of the curve, is 

 great enough to admit the existence of a range. We shall 

 apply (7) to the case of RaC in air, as it is with this sub- 

 stance that most of the experiments have been made. The 

 mean range is 6'S cm. and the initial velocity 2*0 X 10 9 cm. 

 per sec. The quantities N-5'44 x 10 19 , £=1*37 x 10"*, 

 E = 9'3 x 10- 10 (ES), and e/m-5'dl x 10 17 (ES) are all 

 directly known from various experiments. <7, the radius of 

 the atom, is a more difficult matter. A value can be assigned 

 from the kinetic theory of gases, but this need not be at all the 

 same as the value required here. For the kinetic theory deter- 

 mines the mean distance of closest approach of two molecules 

 in collision. Here we require the mean radius of the atom 

 at all times. It is possible to suppose the atom highly com- 

 pressible, so that the electrons are usually at a considerable 

 distance from the centre, but are driven back on it by the 

 approach of a second molecule. Or we may suppose them 

 held very close to the centre, but capable of exerting outside 

 their region large elastic forces on other atoms. In the first 

 case a will be larger, in the second smaller than the value 

 given by the kinetic theory, which moreover refers to mole- 

 cules, not atoms. If the atom of air is highly compressible 

 its radius will be say 3 X 10 ^ 8 cm. This would give for RaC 

 a parameter /e = 4'8, which is almost certainly too large. 

 Moreover, it would result from compressibility that the mean 

 radius would depend somewhat on the pressure ; but the 

 stopping of the a rays depends only on the number of atoms 

 it encounters, and not at all on their pressure. This 

 possibility may therefore be rejected, and to cover the other 



