SEA-FISHERIES LABORATORY. 223 
the average weights. If we integrate kl? for the limits 
of length representated, we will seldom find that such 
an approximation can be obtained. 
The example given in the table on p. 224 illustrates 
this. The catch of fish has been sorted into 1 cm. groups 
and the averages of the weights of the fish in each group 
have been calculated and summed—it is 4,789 grams. 
The co-efficient k deduced as above explained is 1°01 and 
37 
if 0:0101 13 dl—4565. 
16 
so that the difference is quite in the wrong direction. 
Some more approximate function must be obtained. 
If we plot the logarithms of the lengths and average 
weights, a ‘‘mean straight line’’ can very easily and 
accurately be drawn among the points. We thus get the 
linear analogue of the parabola ai”. 
| log w = log a + n log l 
and from this the index n and the co-efficient a can be 
obtained graphically ;* or the average weights may be 
fitted to the equation 
w=a+ be4+ ca? + dzx3+ 
by Pearson’s method.t 
However obvious it may appear that the weight 
should vary as the cube of the length, this is by no means 
necessarily the case. In the case of the catch represented 
by the table, the index of J is 3°227 and the co-efficient 
a (or k) is 0:005. The integral 
: 0°005 13-227 dl 
16 
is 5168. Further, it will be seen from the table that the 
average weights obtained by the equation w=0°005 [3-227 
* The method is described very clearly by R. Howard Duncan in 
‘¢ Practical Curve Tracing,” Longmans, 1910. 
t Biometrika, Vol. II. Palin Elderton, op. cit., p. 30. 
