10 
MR. LEWIS F. RICHARDSON ON 
(h 2 — h x )/(t 2 —t x ). So the first order terra on the right of (3) vanishes, on taking the 
mean, if the variations of dQ/dh, in time at a fixed point, are not correlated with the 
variations in (Ji 2 —h^)l(t 2 —t^), at the same point and time. 
In cumulus cloud eddies, the variations of velocity are caused by variations of the 
potential temperature 0, so that a correlation is almost certain to exist. On the 
contrary, when the eddies are due to dynamical instability, the correlation may be 
expected to vanish. In the latter case, it is the second order term of the right of (3) 
which becomes effective, so that 
m = dje {h-Kf 
$t dh 2 '2(t 2 —t x ) 
Now suppose further that either d 2 Q/dh 2 has no variations at a fixed time and level, 
or else that its variations are not correlated with those of (h 2 —h x ) 2 . Then (9) 
simplifies to 
^e_Fe UVAfl / 10 x 
°8t dh 2 ' 2 (V-0. K ' 
Thus _ 
(h 2 —h x ) 2 /2 (t 2 —t x ) is the eddy-diffusivity Iv.(ll) 
It is seen to be identical with that derived in Part IV. above, by considering the 
diffusion of a lamina, in which the density was distributed according to the law of 
error. It is a quantity easily measured. 
Of course if t 2 — t l were sufficiently small, say second, then it would be the first 
power of h 2 —h x , which would be proportional to t 2 —t x , instead of the square. This 
suggests that t 2 —t x must be long compared with the fluctuations of the wind. On 
the other hand t 2 —t x must be short compared with the period, of say 6 hours, over 
which the averages denoted by the bar are desired to be taken. 
A similar argument can be applied to any other quantity which, like 0, does not 
change following the motion of the fluid, provided it has space-rates independent of 
the time-variations of velocity. Thus the mass-of-water-per-unit-mass-of-atmosphere 
may replace 0 in (10) with similar restrictions. 
When we consider diffusion in three dimensions there may be six coefficients of 
diffusivity corresponding to the six components of stress. 
VI.—Osborne Reynolds’ Eddy-stresses. 
But we cannot, without further investigation, apply the preceding argument to 
the diffusion of horizontal velocity in a fixed azimuth. 
Something might perhaps be deduced from the well-known theorem that, when p is 
constant, 
j^(i/>' y2 + W , + p) — ........ (12) 
where ^ is the gravity potential. 
