Miscellanea 
175 
where a=ic ^4 (1 +pN) (1 +qN) - {N- 2nf 
.(XX). 
'jA{\-\-pN){\-^qN)-{N-'2.nf 
«i = ^(iV + 2) 
If on the other hand p lies outside the above limits, then the sum of the series is given by the 
areas of the curve 
where and a-i are the roots of the quadratic equation 
(u + 1) (iV-n+l) {l+qN) (1 +pN') c N{N-<2.n){p-q) ^ 
(iV+2)2 2(.V+2) ■« + ^'—y 
and v = (^+2)/(ai-ff2) J 
Thus, (xix) which falls under my skew curve Type IV, and (xxii) which is included in my 
skew curves of Type I, complete the full solution of the problem of the random sample. 
The partial integrals of (xix) and (xxi), which can be fairly easily found grai^hically, fall 
under the incomplete ^'-function*, and the incomplete /3-function. 
The incomplete r- and /3-functions can be determined by aid of tables which have just been 
calculated and will shortly be published. 
Thus we see that the skew curves (vi), (xix) and (xxii) directly arise in the course of our 
investigations when we come to deal in full generality with the problem of random sampling. 
But what we know so far of cell-division and detcrminantal theories of inheritance suggests 
forcibly that the character of any sub-class of a population is fixed by a random sample of a 
number of determinants, the size of the sample being commensurable with the number of deter- 
minants. In all such cases the distribution of frequency will approximate to the curves we have 
here discussed. They thus cease to be approximations in any other sense than the Gaussian or 
normal curve is an approximation when the probability integral is used to determine the 
probable error of a random sample. 
It is true, indeed, that they contain a good deal more than the general theory of random 
samples. Thus the general frequency curve must be of the form 
\ dy x + a 
y dx~ fix)' 
If we take f{^x) = c,,-\-ni- + c.A~) +...+cA-) +..., 
then I have given the finite difference equation which determines the successive c's in terms of 
the moments and shown that the convcrgency ratio of the successive constants is a factor (less 
than unity like in general the skewness and kurtosis) which vanishes for the normal curve. It 
will, I think, be obvious that to give the general rule for finding as many terms as we please, give 
their degree of convcrgency, and then retain three because they are found to fulfil all jiractical 
requirements is a process more legitimate than to assume every function must be of the form 
F{x) = {x+kY, 
and give no measure at all of the deviation from this form, and no statistical illustration (such 
as that of random sampling) in which such a function habitually and necessarily arises. Yet 
such is the course recently adopted by Professor Kapteyn and considered by him " rational " as 
compared with mine. 
* I term the complete G-function, G (r, v)= j\m'' 96+"^ de. This has been tabled by Dr A. Lee, 
B. A. Report, Dover, 1899. The incomplete G-functiou is G (r, v,e) = \ sin'' gc"^ d^, and has not yet 
been dealt with. 
