16 
ME. LEWIS F. RICHARDSON ON 
Let us now see how these two modes of disposal of smoke fit together—the one. at 
short intervals of time, governed by the eddy-stresses, the other at long intervals 
governed by the diffusivity. Let us draw the trail of smoke as it would appear to a 
distant observer looking, say, horizontally. The scatter may be represented by 
drawing two lines through the points where the smoke-density has its standard 
deviation in height above or below its mean. Let T be the time since the smoke 
emerged from its source. Suppose that the stress hh , the difiusivit-y K and the mean 
velocity v x are all constant along the path of the smoke. The time T taken to travel 
a horizontal distance x measured down the trail from the source is x/v x . Near the 
origin T = r in equation (22), and so the standard deviation in height is 
representing a pair of straight lines intersecting at the source. Further down the 
trail T = t 2 —t 1 in equation (11), and the standard deviation in height is 
± \/x • \ f ^ .( 35 ) 
v v x 
representing a parabola with horizontal axis and its apex at the source. Thus 
according to the theory, the smoke may be said to be contained within a paraboloid 
which has had its blunt end sharpened into a cone. The preceding theory gives us 
no clue as to the manner of transition from the cone to the paraboloid. 
When we can observe a sufficient length of the path traced out in space by a single 
small portion of air the eddy-stresses and the eddy-diffusivity may be deduced from 
the irregularities in the motion. With this object I have observed the motion of anti¬ 
aircraft shell bursts, and of portions of cloud, by means of an Abney level or a pocket 
sextant. With better instruments this method might yield a good deal of information 
about eddies at heights such as 2 to 5 km. One principal difficulty is that the shell- 
burst fades away after about 5 minutes, before a sufficient length of path has been 
observed to give the diffusivity. 
If the path is sufficiently high and long, the hills, trees and houses on the earth 
may be regarded as blending into a “ roughness.” Suppose this roughness to be 
uniform. Then if we had been causing smoke to issue in puffs from a fixed pipe, we 
should presumably have obtained the same scatter diagram for the puffs, within the 
limits of probable error, at whatever point of the path we had placed the pipe, or at 
whatever time we had begun to observe, within limits. If this is so, we may form 
the scatter diagram by taking its origin at every point in succession of the trajectory 
of the single particle. For instance, Captain Cave in his book on “The Structure of 
the Atrposphere,” gives several diagrams of the irregularities of height of a balloon 
observed by two theodolites, the uniform vertical motion of the balloon relative to the 
air having been eliminated. From his figure 30, of an ascent on February 19, 1909, 
the following has been deduced, by taking the origin of scattering at every available 
