MISCELLANEA. 
I. A Rejoinder to Professor Kapteyn. 
By KARL PEARSON, F.R.S. 
In the Recueil des Travaux botaniqiies Neerlandais, No. 3, 1905, will be found a reply to my 
recent criticism of Professor Kapteyn's theory of skew curves*. 
Professor Kapteyn's reply consists, as far as I am able to follow it, of two statements 
accompanied by a complete iguoration of the criticisms I have made on his treatment of 
skew valuation. 
His statements are 
{a) That he has arrived at a more general proof of the equation 
^=A/''(^)«-''^t^'^»-^'i^ (i) 
than Professor Edgevvorth had previously done and that I have misre^jresented his method 
of obtaining this equation. 
(Jj) That I have largely profited by his theory' and in fact adopted it as the basis of my 
own treatment. 
I wish to consider briefly the.se two points. 
(a) ydx is an elementary frequency and Kapteyn's equation can be written at once: 
= ^e-^'''^'dz 
V 77 
if z be put for F{x) — M. 
Thus whatever Kapteyn's process of deduction may be, its final result is absolutely no 
more than asserting that some quantity z obeys the normal law and x the observed variable 
is a function of this " shadow " variable z. The process is completely the same in result as 
stretching a normal curve with varying degrees of stretch parallel to its base. 
It is perfectly true that Professor Kapteyn only reaches this result after fifteen pages of 
preliminary talk, but the mathematical demonstration of (i; occupies something less than a 
page, and it involves nothing more than the assumptions made by Professor Edgevvorth 
(see Kapteyn, p. 16). The normal curve is actually assumed on p. 16, and the validity of the 
* Biometrika, Vol. iv. pp. 199—203. 
