10 The Fundamental Problem of Practical Statistics 
Hence after some reductions we find 
\ 11 
Take cr,r = PQ ( 7». + 1) ( 1 + '"-^ ^ 'l 
and we may Avrite 
Slope of side of polygon P,.P,.+i 
Ordinate of midpoint of side 
^• + 1 
1 m + 1 
1+ ^ 
^ (^P(7» + l)(l +' 
Now Laplace's hypotheses amount to taking both n and m very large and 
?! relatively large as compared with m. Thus we have, if we neglect the second 
and third term in the denominator, 
Slope of P,P,+i _ --^r + i 
Ordinate o-,,^ 
The same result would be reached, if we took Q = P, i.e. took equal chances for 
each contributory event — which is really Gauss' equality of chance for errors in 
excess and defect — together with n the number of cause groups in the first 
sample very large. 
To get the Gaussian or normal curve we must then replace differences by 
differentials and we have 
1 dy _ X 
y dx a„- ' 
Laplace, however, fails to introduce the condition that neither P nor Q is to be 
small which is essential. 
Now there is a point here of considerable importance. We do not assume that 
differences may be replaced by difJerentials. What we have reached is a funda- 
mental geometrical property of the hypergeometrical histogram and what we do is 
to seek for a continuous curve, which possesses this same geometrical property. 
If our first sample be very large, i.e. we have exact knowledge of the bag or 
population contents, the last term in - will vanish and we have 
^ ^ 11 
1 dy — X 
y dx o-fl- -f |c {Q — P)x' 
where a^' = PQ { in + 1) c". 
