352 
two z's in the one, and their corresponding x's in the 
other, while the abscissae are values not of x itself, but 
of z — F(x); or, in other words, a normal curve obtained 
by substituting for every x the corresponding value of a 
z — F(x). If we succeed, the form of F{x) will be known, 
1 
F'(x 
tant for our knowledge of the physiology of growth. 
50 far no special supposition was made as to the type 
of F(x. Kapteyn now introduces a type by which a 
considerable number of observed curves were found to 
be satisfactorily represented; namely 
EDGE Er 
It implies, that the effect of normally distributed causes 
has been proportional to 
1 
F’(x) 
or simply to (x + k)l—a, q being a constant. 
The equation of the normal curve, being in its general form 
h — h°(z — M} 
Te .e 
and consequently also the ,reaction factor“ , impor- 
Re 
q 
UÈE= 
now becomes 
re . Nr h°[(e 6) MIE 
and that of the theoretical skew curve derived from it 
hq — h?[(x + kja — M} 
Re de 
YEN “x 0e e 
To obtain the four parameters M, h, k, q, we choose 
on the line of abscissae four equidistant points x1...x4. 
Let © be the probability integral from the starting point 
of the curve to an x in general, then we get four equations 
by putting the ©’'s of the theoretical curve in the points 
X1....X, equal to those of the empirical one. If the 
solution succeeds, we may speak of a warranted fourfold 
