K. Pearson 
179 
, 1 clY 1 /, 6-, X 
whence YdX = -xV^.>^^^. 
and F= F„^.e~2^7' i) (vi). 
Edgeworth himself has made other suggestions as to suitable values for f{X) 
and accordingly of F(X). Now it is quite clear tliat assuming the character to 
be a definite function of another character which really obeys the normal law, 
there is no more reason for assuming one form oi f{X) than another, because 
we are in absolute ignorance of the nature of this function. Kapteyn's, or 
Edgevvorth's, or Galton's ai'e equally valid, and the only test of their relative 
suitability lies in the extent to which the resulting curves fit actual data. 
Clearly to assume a-=f{X) is to assume the actual frequency distribution to 
follow any law whatever. It is only screening the generality of the assumption 
where <j) is unknown, by an appeal to the supposed universality of the Gaussian 
curve and by a perfectly arbitrary selection of the subsidiary function f. 
But there is another manner of looking at this proposal. Returning to the 
equation 
ldY__ X 
YdX~ a,?F{Xy 
and writing &- = ar„- F{X), we see that it becomes identical in form with the normal 
equation, i.e. 
YdX~ ct"'- 
In other words the distribution of any frequency may be looked upon as given in 
the neighbourhood of any point by a normal curve of standard deviation cr^\/ F {x). 
Hence the conception arises that if the causes which produced variation in the 
immediate neighbourhood of any value a,, of the character, were constants for the 
whole range of variation, we should have a normal curve of standaixl deviation 
(T^\/ F {x^)*. In reality there is a continuous and gradual change of the tendency 
to variability as we pass from one value of the character to a second f. Analytically 
* This method of looking at the matter throws light ou another point. If a curve be of limited 
range, it signifies that <r = 0 at certain points, or the curve stops because we have reached the limits of 
local variation. In a curve of unlimited range it is not the capacity for local variation but the absence 
of individuals to vary, which is the special feature. 
t The matter is of such importance relative to some of Ranke's criticisms that I give another proof 
of equation (iv) here, based on the conception of an infinite number of infinitely small cause groups 
which Ranke considers can only lead to the normal curve. Let be the () + l)th term of a binomial, 
skew or symmetrical, say for simplicity the latter, i.e. (i + l)". Then 
.'/ < 4-l ~ '.Ir _ w + 1 - 2r 
4 (;/,■+! +2/r) ~ i(n+l)' 
Now let (■,. be the distance between (/,. and ij used in plotting these ordinates to obtain a curve, and let 
it be related to some small value c„ by the relation c,. = c'q x function of r = i:„ x ip (r). Let A',, be measured 
23^2 
