OF PLANETAKY ATMOSPFIERES. 
O 
») 
3. Afi'plication of Maxwell’s Law. 
The results tabulated in my British Association paper of 1893 were based on the 
assumption that the speeds of the molecules were grouped according to Maxwell’s 
Law, so that out of any N molecules at any point the number whose speed lies 
between c and c + dc is 
\/ TT 
e 
.(1). 
where 2I^{ttK) is the mean speed, and 3/2A the mean scpiare of the speeds of the 
molecules (Watson, ‘ Kinetic Theory of Gases,’ p. 6). 
If V be the least velocity required to carry oflP a molecule to infinity, and given by 
the relation = p,/a, then the number of molecules having sufficient speed to fly off 
from the atmosphere will he 
VTr J v 
which on putting hc^ = d gives 
4N 
\/ • V V/i 
f’ 
Ywh 
To calculate the integral, we have on integration by parts 
[ e ^t^dt = ^te + -g [ e 
' t ' h 
= i ■ 
where Erfc t denotes the complement of the error function of t. 
The values of '2rr~^ Erf t are tabulated for small values of t in Woolsey Johnson’s 
‘Method of Least Squares,’ p. 153, but in the present problem the values of t are 
necessarily large. Employing the series for Erfc t given in Woolsey Johnson, p. 46, 
Ex. 11, we deduce that the number of molecules moving with speed greater than the 
critical speed is 
+ A _ A + lA _ 1^6 , 
v/tt 1 2(2 2V ^ 2V 2V ~ 
where t = v^h. 
For very large values of t, such as occur in many of the calculations, the first term 
of the series need alone be taken into account for jjurposes of rough approximation. 
From this formula were calculated the results given in Table I. of my Nottingham 
paper, and wliich it may be convenient to reproduce here. This table shows the 
values of N when the number of molecules given by the above formula is taken to be 
unity ; in other words, the number of molecules of oxygen or hydrogen at difierent 
temperatures out of which there is, on an average, one molecule moving wdh sufficient 
B 2 
