194 NEIBURGER AND CHIEN 
For usual rates of drop growth 6 does not exceed 
10-° and thus may be neglected in comparison to 
unity. The growth equation is thus 
Lé 
— eT) | exp —— 
es( E ml 
; 20 1 Tp,70° 
eRe par’ 
It is to be remembered that dr/dt occurs in 6, 
so that this is an implicit equation for the rate of 
drop growth. If the exponential terms are ex- 
panded in series and the terms higher than the 
second are neglected this equation is practically 
identical with that used by Howell. Langmuir’s 
equation for small drops is obtained by omitting 
the last expression in the square brackets; for 
large drops he also omitted the second (curvature) 
factor. Houghton’s equation is obtained by neg- 
lect of all three factors in brackets. 
To test the validity of these various approxi- 
mations, dr/dt was computed for various-sized 
drops grown from various-sized salt nuclei, using 
(1) Houghton’s equation, (2) Langmui’s large 
drop approximation, (3) Langmuir’s small drop 
approximation, (4) Howell’s equation, with the 
correct expression of the effect of solute, and (5) 
Eq. (9), and assuming constant temperature and 
relative humidity (5°C and 101%). The results 
are shown in Table 1. They show that the neglect 
of the difference in temperature of the drop from 
that of the air leads to large errors for all sizes of 
drops and nuclei. The neglect of the curvature 
term leads to errors of less than ten per cent for 
drops larger than about one micron; whereas the 
size at which the effect of hygroscopicity influ- 
ences the growth rate by less than ten per cent 
depends on the size of the nucleus; it appears to 
be roughly the size for which (ro/r)* is about 
0.0005. 
Howell’s equation gave growth rates within 
three per cent of that given by (9) for all the 
cases tested. 
The method of integration—So far only the 
growth of a single drop has been considered. The 
simultaneous growth of drops of different sizes is 
more complicated, as the distribution of sizes at 
each instant must be taken into account. 
Let ¢(r)dr be the fraction of the drops in the 
size range r to r + dr. Then at all times 
dr D f 
Siam ae 
(10) 
ie) 
[ o(r)dr = 1 (11) 
J0 
Suppose that at time f, the fraction of the drops 
in the range ra to ri + Ara is 7; , and that 
time t. the r;-sized drops grow to rj2, and the ri 
+ A ra-sized drops grow to rj. + A riz. Since the 
nature of the growth process is such that a given- 
sized drop never overtakes a larger one, at time tz 
the fraction in the range rjz to rig + A riz will 
still be ;, that is 
rratArie 
it o(r) dr 
= 
+2 
rec (12) 
= i o(r) dr = ny = constant 
Ch 
Thus 
biAri = GirAriz 
or 
g2 = dndrin/Arie (13) 
The procedure in this treatment will be to 
divide the initial nucleus distribution into size 
groups A rj, of average frequency ¢io . The 
average frequencies ¢;, at each time ¢ will be 
computed by (18). 
An approximation of the continuous size dis- 
tribution at any time is arrived at by graphical 
interpolation. It was found that the interpolation 
could be best approximated by dealing with the 
cumulative distribution, that is, the variation of 
the number larger than each given size. If No is 
the total number per em’, the number N,; larger 
than r; , called the cumulative frequency, is 
e) 2 aN 
Nine = No / d(r) dr = | a dr (14) 
rT; T? i 
where dN = Nod (r)dr is the number in the size 
range r to r + dr. 
The values of N,; were computed for each group 
at selected times and plotted against r on log—log 
graph paper. Then a smooth curve was fitted for 
each time, keeping the physical process of drop 
growth in mind in establishing the relative posi- 
tions of the various interpolated curves. The slope 
of the curves for N, were computed to obtain the 
frequency per unit size interval, dN/dr = Nod(r), 
which is called the differential frequency. 
In interpreting the results it is important to 
keep in mind the distinction between the definite 
results of the computations and the results indi- 
cated or suggested by the curves interpolated be- 
tween the computed values. 
In treating cloud development in the atmos- 
phere it is assumed that the temperature of an 
