110 TRANSACTIONS LIVERPOOL IUOLOGTCAL SOCIETY. 



sents the frequency of the fish at lengths less than that 

 of the mode, and the other the frequency at lengths greater 

 than the mode. Now if this relation were of universal 

 application it would be of immense value, for the work 

 of fitting a catch of fish to such a curve is simple and 

 rapid. Further the equation shows that the gradient of 

 the curve is a constant one ; that is the number of plaice 

 at any length, say n + lcms., is in a constant ratio to 

 the number at n cms. Knowing, then, the numbers taken 

 at any particular length — not too near the mode — we 

 should be able to calculate what numbers were taken at 

 most other lengths. The formula would enable us to 

 estimate the value of the sample. 



But it will be seen by plotting the logarithms of the 

 numbers of fish taken in most of the catches tabulated in 

 this Report, that the equation does not generally repre- 

 sent the distribution of lengths. One cannot expect that 

 a close fit would usually be obtained, but it appears to 

 me that the deviations from the curve are often too great 

 to be due merely to errors of observation. Probably the 

 catches consisting predominantly of fish of one age-group 

 would fit the curve fairly well, but many catches evidently 

 consist of fish of various age-groups, and the presence of 

 these contrasting groups is often most distinctly indicated 

 by the double modes in the frequency-curves. The 

 equation log y = A + bz does not describe the distribution 

 in such catches. If, on the other hand, a large number 

 of plaice were caught at different times during the year, 

 and on different fishing grounds, the variations in rate of 

 growth due to different natural conditions, and the 

 capture of fish at different seasons of the year would 

 smooth down the curve and we should have a gross catch 

 of fish, apparently homogeneous, but not really so. The 

 equation, while useful in the analysis of commercial 



