14 
MR. LEWIS F. RICHARDSON ON 
These are based on the assumption that the number of particles is not less than 
say 20, and that the scatter is “normal,’’ so that the logarithm of the density of 
particles would be a quadratic function of x, y,h\i a very large number of particles 
were observed. Reference may be made to the fundamental papers on probable 
errors: Filon and Pearson, ‘Phil. Trans.,’ A, vol. 191; W. F. Sheppard 
‘ Phil Trans.,’ A, vol. 192. Equation (26) follows from the probable error of the quantity 
called by K. Pearson the “ product moment coefficient taken about the mean.” 
Equation (25) follows from (26) on putting x = h, or may be deduced independently 
from the probable error of a standard deviation taken about the mean. 
Corrections for the Motion of the Parachute Relative to the Air. 
When observing eddy-stresses by the aid of the parachutes of -plant seeds it is 
desirable to allow for the velocity of the parachute in still air. For large specimens 
of the parachute of Taraxacum officinale, after cutting off the seed, the velocity in 
still air was found to have a mean of 12 cm. sec. -1 with a standard deviation of 
2 cm. sec. -1 . It may be shown from the equation of motion that this limiting velocity 
is acquired in a negligibly short interval of time. Thus call the limiting velocity 
c downwards, the instantaneous velocity downwards u. Then if the friction is 
proportional to the velocity 
(7 (mass) = c x F,.(27) 
where F is a constant. But, when accelerating, 
(mass) 
du 
dt 
g (mass)— uF, 
(28) 
Eliminate the mass between these two equations and there results 
l dJ ^ = -(“-<)•.< 29 > 
So that the discrepancy between the actual velocity u and the terminal velocity c 
sinks to e -1 of itself in a time equal to cjg, which for the taraxacum parachute having 
c = 12 cm. sec. -1 would be only a hundredth of a second. 
Less negligible is the variation of the velocity c from one parachute to another. 
What we actually observe is not p H the upward velocity of the air, but v H —c. Now 
write V for the mean velocity of the parachute in still air, and c for the deviation from 
the mean. Then in finding the direct stress hh we must perforce work out first the 
“raw” moment {(p h —c) 2 } = [{(r H -c) + (p' H — c')} 2 ]. O n expanding and remembering 
that a bar put over the product of a dashed and a barred symbol, causes the result to 
vanish, and also that p' H — c' — 0, it is found that 
{( ph - c ) 2 } = fu-ef + vfvf + c'c' 
