9254 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 
for a series of rough values. One could not have done 
this in any other way without introducing the 
undesirable tendency to make the curve go the way he 
wanted it. To arrive at the figures in column (6) the 
2\m 
function vol iV « ) has to be integrated, and the definite 
integrals representing the limits January-February, 
January-March, and so on, have to be evaluated. But 
this is so difficult as to be quite impracticable, and I 
have approximated to the integral curve by calculating 
ordinates for the beginning, middle, and end of each 
month, and then finding the area of the curve between 
the ordinates at the beginning and end of the month by 
means of Simpson’s quadrature formula 
1 
J va i : (y, + dy, +%) 
and hope this may be accurate enough for the purpose. 
The areas have then been summed, giving the figures 
of column (6). 
Now, although the above is the most obvious way 
of obtaining regularity from very rough data, it is likely 
that it is not the best way. Professor D’Arcy W. 
Thompson has pointed out to me that, since the growth 
of a fish is to be represented by a cyclic curve—since it 
grows from year to year—a sine curve ought to be used. 
That this is true is indicated by the fact that constants 
representing Pearson’s ‘‘Type IL’ frequency curve 
were. obtained from the calculation, though obviously 
Type VII ’’—the normal curve of error—should have 
applied, the range being infinite. 
