6 
MR. LEWIS F. RICHARDSON ON 
approximately according to the ordinary “ law of error.” Their scatter in height is 
measured by the “ standard deviation,” computed by the familiar methods.* In 
order to find the diffusivity K the observations are compared with an appropriate 
integral of the approximate equation 
Such a one is 
3 <-\2 
°X _ iz o X 
dt a h 2 
(i) 
A -(h-A*)* 
v _ -^-1 + 
\Zu+a 2 
where A x , A 2 , A 3 are constants. 
This integral represents a horizontal lamina in which the density x is distributed 
about a mean height A 3 according to the law of error. The square of the standard 
deviation of the mass in the lamina can be shown to be 
(4* + A,)5 
(3) 
So if the scatter of the same set of particles be observed at tbe beginning and at the 
end of an interval T of time, it follows that 
K = 
2T 
(increase during T of square of standard deviation). . . (4) 
But the increase of the square of the standard deviation is equal to the square of the 
standard deviation of the change of height. Accordingly 
K = 
2T 
(square of standard deviation of change of height during T). . (5) 
In this last transformation we have assumed that K is sensibly independent of 
height. This is permissible because the range of scatter can usually be made small. 
For the same reason the density of the air may be taken as independent of the 
height, so that we may obtain from K, the constant £ which we require when 
pressure is taken as independent variable in place of height, in accordance with (l) 
above. This procedure is not perfectly satisfactory but it is very convenient. It 
gives £ = g 2 p 2 K and “ eddy-conductivity ” = pK. 
There is no need for the changes in height to be simultaneous for all the portions 
of air, and in practice it is much more convenient to let them be successive. 
Varieties of particles. —I have observed the scattering of smoke from a 
smouldering wick, from burning weeds, from factory chimneys and from ship’s 
funnels : also the scattering of portions of cloud near the horizon and of puffs of 
ammonium chloride from a special apparatus. Chimney smoke is not to be 
recommended, as it rises through the air. Clouds and steam may mislead one by 
* Vide ‘ Computer’s Handbook,’ M.O. 223, Section V. 
