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88 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



average weights is the area of the figure between the 

 graph and the ^-axis. 



If we call f(l) the length-weight function, 



T J (I) dl is equal, therefore, to the sum of the average 



weights. Now assume that f(l) is kl 3 and integrate this 

 expression for the range L 1 to L 2 . The coefficient It is 

 then easily calculated for 



, 4 (sum of average weig hts) 



- — w -(£■)* 



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

 mean length to find L 2 , and to subtract 0'5 from the 

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

 respectively. 



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 



I s 

 series the equation g = It be calculated it will 



generally be found that its graph does not agree as 

 closely as it ought with 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 that the series of 

 average weights is represented by the parabola 

 g — a + bl + cl 2 + dl 3 + . . . Generally it is necessary 

 to find the constant a and the coefficients, b, 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. The method is clearly described, with 

 examples, in Palin Elderton's " Frequency Curves and 

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



