SEA-FISHERIES LABORATORY. 919 
a rule, as to how it ought to go. Values of y for each 
value of 2 are now read off on the curve, and if the 
latter is described carefully on a rather large scale, there 
is no difficulty in obtaining a series of figures which 
should sum up to very approximately the same value as 
the original (integrated) series. The original series is 
replaced by this graduated series, and then the process 
of integrating by summation is reversed. The figures in 
column 4 of the example are those on the same line in 
column 3a,* minus the figures of the line next below. 
The figures in column 4 can then be plotted: they do 
not form a perfectly smooth series, but there is never 
any doubt as to how the curve ought to be drawn. 
This is probably the best method of smoothing such 
series as are here dealt with when an analytical 
expression for the function cannot be obtained. There 
are many plaice-measurement series which are obviously 
bi-modal: one sees this from inspection of the figures. 
Others cannot easily be so identified, but when they are 
integrated there is never any doubt as to the presence 
of two or more modes, and unless the integral curve 
shows more than one point of inflexion, one ought not to 
regard the series as bi-modal. Of course a curve might 
be drawn smoothly through the original figures plotted 
as points on a graph, but there would always be some 
bias displayed in so drawing the line. 
Fig. 5 represents another example treated in this 
manner, and the dotted curve in fig. 1 represents the 
curve drawn by the same manner. In this last example 
it is easy to see that the frequency curve obtained by the 
method of moments differs considerably from that 
obtained by smoothing by graphical integration. The 
latter curve is not very accurately obtained, but it is 
* The smoothed integrated series not given in the table. 
