12 
MR. LEWIS F. RICHARDSON ON 
when any symbol such as xy is the force in the cc-direction per unit area of a plane 
normal to the y axis. Tractions are reckoned positive, as usual. The above is taken 
from Osborne Reynolds’ theory, adapted and slightly generalized to suit our needs 
for a rotating atmosphere, having density diminishing with height and a molecular 
viscosity which is not neglected. 
One may form a clear mental picture of these eddy-stresses by imagining the 
scattering of smoke puffs. Let a puff emerge from a pipe at the origin of the 
co-ordinates. After a short interval of time r, let the 
puff appear at the point P on the diagram, as seen by an 
observer at a distant point on the ?/-axis. Now let the 
observation be repeated for a large number of puffs in 
succession, the time r being kept the same for each. 
We thus obtain a diagram, with a large number of points 
on it, showing the scattering of the puffs after r. Then 
the eddy-stresses are simply related to the correlations 
and standard deviations of this scatter-diagram—under 
certain conditions. For let X, Y, H now mean the 
co-ordinates of any one of the dots on the diagram reckoned from the source of 
smoke. 
Then the velocities of the corresponding puff were 
X _ H 
5 — ~ 
(17) 
provided the time r was so short that, during it, the velocity may be regarded as 
uniform and in a straight line.(18) 
Again, the velocity of the puff will be equal to that of the air which it has replaced 
provided the puff is at the same temperature as the air, and provided that the pipe 
points parallel to the Y-axis so that the impulse with which the puff leaves the pipe 
does not show in the projection on the plane XQH. 
Let us suppose that a number of puffs, n in all, are observed. In order to 
correspond with the time-mean taken over 6 hours, which was used in deriving the 
eddy-stresses from the equations of motion, these n puffs should be spread uniformly 
over a similar interval. 
From the scatter diagram we can compute first the mean velocities. For the 
mean velocities are 
v x — — 
X 
T 
n 
NX ; v Y = Y/r = 
2Y 
n 
■ ( 19 ) 
where 2 has the meaning :—take the sum of what follows it, for n puffs. 
