8 
MR. LEWIS F. RICHARDSON ON 
The standard deviation is not then sufficient to give K. It is necessary to make 
more elaborate computations with K. Pearson’s “ incomplete normal moment 
functions.” 
These difficulties are avoided in the symmetrical case when the source of smoke is 
exactly on the ground, and the smoke does not rise or fall by its temperature. 
The results of these measurements are set out in the last column of Table IV. It 
is seen there that K. increases from 5 near the surface of land up to 10,000 at the 
height of a factory chimney. 
But before considering the results further a fuller mathematical investigation will 
now be made. 
V. General Theory of Eddy-diffusivity Deduced from Scattering. 
The foregoing theory of the diffusion of a lamina assumes that the diffusivity is 
constant throughout the space, and that the density in the lamina does not vary 
except in the smooth regular manner indicated by the “ law of error.” But it is well 
known that the wind has an intricate structure. Thus if observations of the smoke 
puffs are to yield a measure of the diffusivity from the formula 
diffusivity = 
increase in square of standard deviation 
twice corresponding increase in time 
then either the interval of time in the denominator must be long compared with the 
fluctuations of the wind in time, or the initial standard deviation must be large 
compared with the fluctuations of the wind in space, or both conditions must hold. 
The former condition is an inconvenient one in practice, because puffs are apt to fade 
before a sufficient time has passed. Dandelion parachutes, with the seeds removed, 
may be better than smoke for this purpose. 
The following theory brings to light some of the assumptions involved in the 
measurement of diffusivity by smoke puffs. It was contrived specially in order 
to avoid “ the distance through which an eddy moves before mixing with its 
surroundings,” a quantity which occurs in Taylor’s theory, but which does not lend 
itself easily to measurement, except in the case of cumulus eddies. See Section IX. 
below. 
The potential^ temperature 0 does not change at a point moving with fluid, if 
radiation and precipitation can be neglected. Now let a portion of an eddy move from 
a height h x at time t x to a height h 2 at t 2 . Then, regarding 0 as a function of h and t, 
we have 
0 {K, 0 = 0 (h 2 , t 2 ) .(1) 
* Potential temperature is the temperature which the air would acquire if compressed adiabatically 
to a standard pressure. If 0 is to be of service in dealing with cloudy air the standard pressure must be 
high enough to evaporate the cloud in all samples. 
