112 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



the point of inflexion is found by bisecting the tangent 

 between the former points. Often there are two points 

 of inflexion, rarely three. The modes are found in this 

 way much more easily then by methods of interpolation 

 from the original frequency-numbers ; and the accuracy 

 is probably quite close enough when one considers the 

 other sources of error inherent in the general investi- 

 gation. 



Further graphical representation of the results of the 

 measurements by these integral curves is really the most 

 convenient method. One does not usually want to know 

 how many fish in a catch were between, say, 20 and 

 21 cms. in length, which is all that can easily be read 

 from the differential curves ; but what percentage of the 

 whole catch is over, or under certain lengths, say 8 inches, 

 and this is obtained at once from the graphs. The 

 percentage of fish caught by a six-inch trawl mesh, at a 

 certain time of the year, and on a certain fishing ground, 

 between, say, 8 inches and 10 inches, is obtained by 

 graphical interpolation, by drawing lines from the axis 

 of x, parallel to the axis of y, and then, drawing lines 

 parallel to the axis of x from the points on the curve, and 

 letting these cut the axis of y. This is usually the sort 

 of question that arises in practical fishery discussions. 



If it is desired to trace the seasonal changes taking 

 place during a period of two or three months, on a 

 particular fishing ground, with respect to the size of the 

 plaice present, the curves for these months can be drawn 

 to the same axes of coordinates. It is useful in such 

 cases to change the origin to the point of inflexion, finding 

 new coordinates by the formulae, x' = x-a, and y' = y~b, 

 (a, h) being the coordinates of the point of inflexion. 

 New scales are then drawn. All the conventions as to 

 sign then have meanings : positive values of x represent 



