88 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



average weights is the area of the figure between the 

 graph and the ^'-axis. 



If we call /(/) the length-weight function, 



I T J^^ '^^ ^^ equal, therefore, to the sum of the average 

 weights. Now assume that /(/) is M^ and integrate this 

 expression for the range L^ to L^. The coefficient k is 

 then easily calculated for 



_ 4 (sum of average weights) 



Obviously it is necessary to add 0'5 to the highest 

 mean length to find L^, and to subtract 0'5 from the 

 lowest mean length to find L^, the upper and lower limits 

 respectiveh'. 



Now, if such a series of average weights is found 

 and "smoothed," a curve can be drawn very 

 approximately through the points. If from the same 



series the equation g ~ ^^'-(FTp. ^^ calculated it will 



generally be found that its grapli does not agree as 

 closely as it ought witli the curve obtained by smoothing 

 the observed average weights. 



This suggests that the length-weight function 

 referred to above is not the best one. To find a better 

 one we employ the systematic " method of moments " 

 used in biometric work, and assume tlnit ili(> series of 

 average weights is represented by tlie parabola 

 g = a + hi -\- cP -\-dP + . . . Generally it is necessary 

 to find the constant (i and tlie coefficients, />, c, d , and 

 to do this successive " moments of inertia " must be 

 calculated from the rough statistics, and equated to 

 moments calculated from the theoretical equation. The 

 simultaneous equations so formed are solved to find the 

 constants. Tlie niciliod is clearly described, with 

 examples, in Palin I^lderton's " Frequency Curves and 

 Correlation," and need not be further referred to here. 



