ORGANIC VARIABILITY 543 



represents the theoretical frequency curve which is to replace 

 the rough series of observations, and this curve must be such 

 that its area is equal to the sum of the areas of the rectangles 

 representing the crude observations. Let the equation to the 



frequency curve be y = f(x), then Jydx = area of the graph. 



If we now multiply each mid-ordinate by its distance from the 

 mean and then take the sum of all these products, we obtain 

 the first moment of inertia of the whole distribution about 

 its mean, that is, ... -f ^_ 2 ^_2 + y~\X-\ = y\X\ + y^x^-^- • • •> 

 since the distribution is supposed to be symmetrical. This 



gives us the equation fxydx — o. In a similar way we find 



the second moment by multiplying each mid-ordinate by 

 the square of its distance from the mean : this gives us 



x 2 ydx = C . Proceeding in this way we can make as many 



equations as there are constants in the expression to be found, 

 and by solving these equations simultaneously the numerical 

 values of the constants can be obtained and substituted in 

 equation (5). 



Equation (5) must then be integrated so as to put it in the 

 form y = f(x). There are a number of integrals all of which 

 satisfy the equation since the denominator c + c x x + c 2 # 2 

 can be written (x — a)(x — /3), and the form of the integral 

 giving the solution depends on the nature of the roots a and 

 /3 in the quadratic (x — a)(x — /3)'= o. The actual integration 

 involves mathematical technique which does not concern us 

 here, but as its results we obtain the series of Pearson Fre- 

 quency Curves, two of which, the " normal curve " (the limit 

 to the symmetrical binomials), and the Type III curve (the 

 limit to the asymmetrical binomials) we have already con- 

 sidered. The most common of these frequency curves in 

 biological work on variability is Pearson's Type IV, which is 



/* 



*+a 



2\ - m 



= y l g-t,taa-.f? 



The philosophical basis of the Pearson Generalised Proba- 

 bility Curve should be noted, for this conception will probably 

 come to be considered as perhaps the most distinctive and 

 fertile advance made by general biology during the last 

 decades of the nineteenth century. Organic variability is 



