400 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
“ur 
Oy [ yx? (b — x)" dx 


1 NY n.mv—1 a \2 
= He = | l+s—p 
Yo ee Ve i te. fa 
Hence, if the range b be large and x be small, this series converges very rapidly, 
and we may often take with sufficient approximation even only its first term. Thus 



pe hep) 
[io 1 = 2—p 
Pet en ernie 
Bla a oes | 
b nearly. 
7 = pst 
[ho 3 — 4A—»p 
ee Yoo" el -» 
~_ 1l—p 5) 
Now «@ is given by the statistics, and we note that if p has been determined to a 
first approximation by the method of moments, we can now improve the values of the 
moments of the areas near the asymptotic ordinate by the use of the above 
expressions. 
For example, if p = ‘5 as a first approximation, we have 
and as the area up to a short distance from the asymptotic ordinate is generally a 
considerable proportion of the total area, the above values very considerably medify 
the calculated moments. 
In the case of the curve 
Yy = yye Pe", 
we have the result 
1 ya ih yn } 
l+s—p 2+4+s8s—p 1.2.3+s—>p) ; 

op’, = yg’ +1? 1 
Hence, as before, if y and # be small, 
