DISCUSSION 
acterized by an increasing number of modes be- 
cause of the change in shape. But I do agree, as 
in the case of Dr. Mason and his colleagues in 
their paper in the Transactions of the Faraday 
Society, that we have to go to some representation 
of the spectrum other than the complete details. 
[had thought of the possibility of using some sort 
of other representation in the computation we 
did, to see changes in the results these had with 
time. 
Dr. Claes Rooth—The change in time of the 
spectrum shape is given by the relative changes 
in magnitude of the moments of the distribution. 
The indication here is that any such changes 
‘xaused by condensation alone would be very 
slight. The formation of bimodal or more compli- 
‘ated distributions has to be attributed to mixing 
and differential settling, and to the partition of 
the nucleus spectrum into an actively growing 
part and a population of stable nucleus droplets. 
Such a partition will always occur at cloud base, 
but the nucleus droplets would normally be very 
difficult to observe because of their small size, 
and most observed cases of bimodality do not fall 
into that category. As to the number of moments 
that have to be considered, it would obviously 
have to be quite large, if an accurate description 
of something like the shape of a bimodal distribu- 
tion is desired. But the situation might be differ- 
ent if we want to apply our model to the develop- 
ment of a physical property of the spectrum other 
than shape, since a property like average mass or 
scattering power is defined by one or a few of the 
moments of relatively low order. 
Dr. W. Hitschfeld—I would like to comment 
on the particular method you chose for represent- 
ing the distribution. You chose an average over 
number. I think this may be mathematically at- 
tractive, but physically such averages are not 
very attractive because the distributions we know 
and measure are always incomplete, notably at 
the small-radius end. Any ignorance there would 
lead to serious difficulties in evaluating a ‘mean 
radius.’ Mean radii, weighted by particle mass, 
surface area, or scattering power, or in fact any 
power of the radius higher than the first are 
largely free of this defect. 
Dr. Rooth—Vhe set of zero-centered moments of 
the distribution function, as defined here, is equiv- 
alent to the set of mean radii derived with an ar- 
bitrary power of the radius as a weighting func- 
tion (see Eq. (2)). If one wants to study the time 
variation of a property like the scattering power, 
he has to make use of the proper moments of the 
distribution. If the physical property with which 
he is concerned involves the radius taken to a 
power equal to or higher than that involved in 
the basic averaging process, then he is all right, 
otherwise he is certainly worse off than if he had 
included the moments of lower order. Take, for 
instance, the supersaturation. It is seen to be in- 
versely proportional to the linear mean radius. 
Now this fact is a consequence of the physical 
model used, and not of the particular method of 
mathematical analysis applied to it. If we have 
available a measured droplet spectrum, together 
with data on the cooling rate and other pertinent 
factors, then our best estimate of the ambient 
supersaturation in the cloud would obviously be 
founded on the observed mean radius, in spite of 
the fact that this is less well defined than any of 
the weighted averages suggested. So I do not go 
along with the direct implications of your com- 
ment, but if I may interpret it as a warning 
against the pitfalls of uncritical combination of 
mathematical method and physical reasoning, 
then I would support it wholeheartedly. 
