256 Proceedings of the Royal Society of Edinburgh. [S 
ess. 
the normal curve y~ae‘ 1&1 . Let in addition the second and fourth moments 
of fix be denoted by fx 2 and ^ 4 . Then for any point x the final distribu- 
tion of the population will be given by 
i 
—(x—xf) 
y = a I fi(x')e v 2 dx . 
Let the moments of this be denoted by v 2 and v 4 : 
then 
r r r , 
v 2 I ydx = a x 2 fi(x)e w dxdx 
\J — GO J — 00 ^7 — GO 
or 
likewise 
so that 
= a 27t<t 3 J fixdx +a \/27rcrJ x 2 fi(x)dx, 
v 2 = ex 2 + y 2 ; 
v 4 = 3 a- 4 + 6ot 2 /ul 2 + / x 4 , 
v 4 _ 3o- 4 + 6o- 2 /> t 2 + //, 4 
3v 2 3o- 4 + 6o- 2 /x 2 + 3/x 2 2 ’ 
and is > or < 1 according as /x 4 > or < 3/>t 2 . 
This curve then tends to approach the normal curve of error in its 
moment relationships, but it is to be remembered that cr 2 must be small 
compared with /jl 2 , and also that if dominance exist perfect normality is 
impossible. 
Note II. 
In case of misunderstanding, it may be well to state that I have used 
the word blend in the sense that the quality resulting from the combina- 
tion of two different elements lies between that of the separate elements, 
and not in the sense that either of the elements is modified by the com- 
bination, as is sometimes done. 
( Issued separately January 13 , 1911 .) 
