1895.] Contributions to the Theory of Evolution. 259 



The differential equation to the generalised frequency curve is shown 

 to be of the form 



1 dz x 



Z dx 



If we put y3 3 = we have the curve corresponding to the skew- 

 binomial ; if we put /3 2 = /3 3 = we have the normal curve. In the 

 most general case we are led to two principal types of curves 



The second of these curves is marked by a limited range and skew- 

 ness. Its theory method of fitting to actual statistics and its 

 geometrical properties are discussed, and the curve is shown to 

 involve in fitting only the use of a table of r -functions a table which 

 already exists. 



The first of these curves has skewness but no limit to range. This 

 unlimitedness of range is not, however, necessarily significant. There 

 is a limit to the height of adult males, or at any rate to the ratio of 

 their sitting to standing height, but we do not hesitate to express the 

 results in terms of the normal curve. The fact is that both normal 

 curve and generalised curve are only close approximations to series 

 point-binomial and point-hypergeometrical series which can them- 

 selves give a limited range, and we ought to fit these series rather 

 than the curves to our observations.* 



The criterion to distinguish between the application to any special 

 case of curves (i) or (ii) is the negative or positive value of 



which we have seen vanishes for the curve corresponding to the skew 

 point-binomial. 



The complete treatment of curves of the first kind is shown to 

 depend on a certain integral called a Gr-function. This G-function 

 has been discussed in a recent paper by Dr. Forsyth, to whom the 

 author had referred for information with regard to it. It is built up 

 of F-functions with imaginary arguments. The function has not yet 

 been tabulated, but various formulae are given for its evaluation, and 

 it is hoped that its values may shortly be calculated for the range 



* The fitting of the first series is discussed in this memoir ; the fitting of hyper- 

 geometrical series is reserved as the memoir is already of considerable length. 



