ORGANIC VARIABILITY 



and for the hypergeometrical series 



54i 



iy. = 



y 8x 



x 



(3) 



Co + C\X + C 2 X 2 



The functions required to describe biological frequency dis- 

 tributions are usually continuous, so that finite differences must 

 be replaced by differentials and equation (3) may be written 



1 dy _ _ a + x 



y dx Co + C\X + c 2 x % 



Equation (4) is also written 



dy y(a -f- x) 



dx Co + C\X + c 2 x* 



and this enables us to give it a geometrical meaning- 



(4) 



(5) 



Fig. 3. 



The ratio dy/dx is the first differential coefficient and is, 

 geometrically, the tangent to the curve. At the ends of the 

 curve the tangent becomes indistinguishable from the #-axis 

 and therefore becomes zero, so that when y — o (at the ends 

 of the curve) dy/dx becomes zero. When — a = x (that is, 

 when the origin of the distribution is shifted from o to o 1 ) the 

 tangent becomes parallel to the #-axis (the test of a maximal 

 ordinate) and dy/dx again becomes zero. The differential 

 equation (4) thus describes the salient characters of frequency 

 distributions. It includes, as special cases, equations (2) and 

 (1), (2) when the coefficient of c 2 x 2 becomes zero, and (1) when 



