SEA-FISHERIES LABORATORY. 175 



percentages of fish at and over each centimetre group are 

 represented on the vertical axis and the mean lengths of 

 each group on the horizontal axis. The curve is of the 

 " ogive " form, so-called by Gralton. We see at once 

 that the frequencies are concentrated round a point 

 nearer to the lower lengths than to the higher ones. 

 The modal length, that is, the length which is the 

 prevalent one, can be determined by inspection of the 

 curve, for at this point the tangent crosses the curve, that 

 is, the position of the mode is at a point of inflexion. The 



differential coefficient is maximal here, and -=-4, that 



dx 2 



is, the differential coefficient of the frequency curve from 

 which the integral curve is constructed, is at a minimum 

 at the corresponding point, that is, the tangent is parallel 

 to the #-axis, and the point is therefore at a cusp. 

 This is the only convenient way of determining the position 

 of the mode in such distributions as these, for by altering 

 the grouping of the data we alter the apparent mode, 

 while the work of determining the position of the mode by 

 interpolation, or by the calculation of a theoretical prob- 

 ability curve is very laborious. On the other hand, if 

 the integral curve described here is carefully drawn by 

 means of some mechanical aid, the position of the point 

 of inflexion can be very approximately found. A celluloid 

 set-square on which a fine straight line is scratched is laid 

 over the curve, so that the line crosses from one side of 

 the latter to the other. The straight line then appears to 

 coincide with a small segment of the curve and the two 

 points at which the line is apparently co-incident with the 

 curve are marked. The point of inflexion is then midway 

 between these points, and beneath this on the #-axis is 

 the value of the mode. 



If the frequency distribution itself is irregular, this 



