Miscellanea 
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(iv) That the fundamental physical constants are not ascertainable from Kapteyn's 
constants, and this alone seems to me sufficient to deprive his method of all practical 
significance. 
(v) That his assumptions would involve the existence of a number of organic variables, 
the distribution of which followed a truncated normal curve ; no such variables have bceti 
observed in the very wide biometrical experience we have had. 
(vi) Further that if they did exist, we ought to discover a number of perfectly correlated 
organic characters. Hundreds of correlations between organic characters have now been 
investigated, but no case of perfect correlation has yet been discovered. 
Professor Kapteyn instead of replying to my criticisms (i) to (vi) states that he has 
reached a result more general than Edgewortli's. This I fail entirely to agree with and 
I believe no mathematical logician would agree with it either. He next asserts that I have 
in some way purloined his result (i) under a form (iii). My rejjly is that (i) or (iii) are of no 
importance at all until we come to select forms for the arbitrary functions involved, and 
that if they were of importance, I am not indebted to Professor Kapteyn for form (iii), for I 
used it for years and published it some months befoi'e his paper appeared. 
I am quite ready to leave the result even of testing the practical value of the two series 
of curves as empirical descriptions of frequency to the computator ; and this for the simple 
reason that Kaptej'n's curves have been tested by a trained computator and fail to fit at all in 
certain cases where mine do fit. The source of this failure is shown in my paper ; Kapteyn 
has not got general skewness and general kurtosis with his formula. But of this more on 
another occasion. Kapteyn promises us a general method of determining the analytical form 
of his F{x). I shall look forward to his paper with the greatest interest, for it involves 
indirectly no less than a revolution in physics. It amounts to the determination of the 
arbitrary analytical function which expresses the relation between two jjliysical quantities, 
from a graph of their observed relationship. Clearly if we can find F{x) in (i), it is identical 
with the discovery of <^ (.*'), the functional form of the relation between two physical characters 
f Xi 
X and y. The solution will be of the greater value because every observed class z= j ydx is 
J 
subject to the probable error "67449 v^z (1 - z/iV) where N is the total frequency, .so that the 
form of Fix) has to be determined analytically, not from exact knowledge, but from a 
knowledge that y lies with a defiuite amount of probability within a certain belt of varying 
breadth. The gain in power to the poor physicist who is too apt to select y = S{c,^x"') to 
describe his observation curves will be enormous. 
this is done the question to be answered is: What is its probable error? Every constant used in 
my frequency theory is uni(iuely and absolutely given as soon as the moment coefficients have been 
ascertained and its probable error can then be found. It is accordingly an absolutely significant con- 
stant for the frequency distribution quite apart from its relation to any special form of curve. And 
it may be compared from one distribution to a second, without any assumption as to the goodness of 
fit of curves. For example, just as we can test whether differs significantly for two distributions, 
so we can also test whether any function, 
differs significantly, and this will be one test of true differentiation in the distributions. Thus we may 
test if 
7 = 2/i.,/^.j and ^; - ifi.^jix^- - 1 
are significantly different for two distributions. This is perfectly legitimate whether we take 7 and p 
constants of my curve (Type III) or not; they are unique functions of the /x's. But when Professor 
Kapteyn expresses his frequency in terms of a constant which may have values in the same case from 
0 to 00 , it must be obvious that he has at once destroyed the fundamental purpose of frequency 
investigations, which lies in testing by the theory of probable errors the difference of random samples 
of two populations. 
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