372 Mathematical Contributions to the Theory of Evolution. 



I propose to consider these points in reference to the skew frequency 

 distributions discussed in a memoir in the 'Phil. Trans.' for 18y5 (A, 

 vol. 186, et seq.) in another place. The present memoir, however, 

 shows that these skew distributions give results immensely more pro- 

 bable than the Gaussian curve, and thus confirms in the case of errors 

 of observation the results already reached in the case of organic 

 variation. 



Mathematical Contributions to the Theory of Evolution. — X. 

 Supplement to a Memoir on Skew Variation." By Kael 

 Pearson, P.Pi.S., University College, London. Eeceived May 

 22— Bead June 20, 1901. 



(Abstract.) 



In the second memoir of this series a system of curves suitable for 

 describing skew distributions of frequency was deduced from the solu- 

 tions of the differential equation 



jL d U = ^0 + hx £x 

 y dx aQ + aiX + a 2 x 2 



These solutions were found to cover satisfactorily a very wide range 

 of frequency distributions of all degrees of skewness. Two forms of 

 solution of this differential equation, depending upon certain relations 

 among its constants, had, however, escaped observation, for the simple 

 reason that all the distributions of actual frequency I had at that time 

 met with fell into one or other of the four types dealt with in that 

 memoir. A little later the investigation of frequency in various cases 

 •of botanical variation showed that none of the four types were suit- 

 able, and led me to the discovery that I had not found all the possible 

 solutions of the differential equation above given. Two new types 

 were found to exist — 



Type V: y = y^e-il* (ii) ; 



with a range from x = to x = oo,and 



Type YI : y = y Q (x- af^x (iii), 



with a range from x = a to x = oo . 



These curves were found to be exactly those required in the cases 

 which my co-workers and I in England, and one or two biologists in 

 America, had discovered led in the earlier Types I and IV to impossible 

 results, i.e., to imaginary values of the constants. 



In the present memoir the six types are arranged in their natural 

 order, and a criterion given for distinguishing between them. They 

 are illustrated by three examples : (a) age of bride on marriage for a 



