222 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 
necessary to find, from the weight of plaice landed, the 
approximate numbers of fish within, or above and below 
certain limits of size; and this has been done by the 
somewhat haphazard method of deducing ‘“‘ factors ’’ 
which are used to convert the weight of fish contained in 
‘“ boxes ’’ or “‘ trunks’’ into actual numbers. Given the 
length-frequency relation, and the  length-weight 
relation, the numbers of plaice within any limits of 
length (a) and (6b) could obviously be obtained from the 
total weight landed by means of the expression 
a a 
Gaff fe), $02) dx de 
G being the total weight of the catch, or series of 
catches; f(#) the length-frequency function; and ¢(2) 
the length-weight function. Both are independent 
variables, but each is dependent on a (the length). The 
actual integration need not be made—as a rule it could 
only be made approximately—but tables might be used 
similar, in a way, to those employed by actuaries. By 
such means considerable accuracy of result could be 
obtained. 
The length-frequency function can be obtained by 
analysis or empirically, with considerable accuracy, as 
shown above; but in such calculations one could not use 
the length-weight function w = Al? without serious 
inaccuracy, since in the expansion the errors would 
attain considerable dimensions. Some more approximate 
expression must be obtained. Now, in seeking for a 
function to represent the variation of weight with length 
in the case of a catch of fish such as plaice, we must find 
J, see) az 
should approximate as closely as possible to the sum of 
one such that 
