PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 91 
We will now modify the results of our previous investigation to suit the case of an 
asymmetrical frequency-curve which has arisen from the superposition of two normal- 
curves having the same axis. In this case if we unite, 6 ^ = 63 = 0 , = ajli 
(= Mjyj), t ’3 = (Tg/Zi (= «oy 3 ) in Equations ( 8 ) to (13) we have (9), ( 11 ) and (13) 
identically satisfied, and ( 8 ), ( 10 ), and ( 12 ) become 
+ '^3 = 1 .( 31 )) 
equality of component group-totals and of their standard-deviations. This equality seems less likely 
than equality of means and divergence of totals and standard-deviations. Should it exist, how’ever, wc 
fall back on a sub-case of the genei'al case we have already dealt with. We need only, in Equations 
(8)-(13), put = Z. 2 , 7 i = ~. 72 ) = Ua, aiid we have 
whence 
or, 
Zi = ^2 = h "/d (1 + Wi") = A2) 7d (1 + = /'n 
“1 = 
v/CS/ia" —/n) 
Tj = (To — 
\/ 
v/(3/*2^ — / 1 . 1 ) 
1 
2 
The possibility of the solution clearly depends on 3 /( 2 ^ being greater than 
The following is an example of this special case. Air. AIerkiman gives some results for American 
target practice, on page 14 of his Text Book on Least Squares. He does not seem to have noticed that 
the resulting-curve is very far from a normal-cui-ve. I find that for these observations 
1 
1 = 
6-482 
/"I = 
0 
/''a = 
44-502 
/‘2 = 
2-486 
/‘'s = 
320-582 
/<3 = 
•104 
2405-094 
f'i = 
15-793. 
The smallness of indicates general symmetry ; assuming then that the shots were fired in two groups 
ivith equal precision, I find = C 2 and b-^ = — almost exactly. 
We have accordingly 
6 ^ = - 62 = 1-082, 
<Tj = (Tg = 1'147, 
[For the 1000 shots as a whole <t = 1'577.] 
Allowing for a uniform error of defective sighting amounting to '482, we find a compound-curve 
fitting closely Air. AIereiman’s figure, and indicating that the gun was aimed ai the centres nearly of 
divisions 5 and 7, and not at that of 6 . Six was possibly white, 5 and 7 black. Like results of course 
would arise from a change of sighting about midfiring. 
N 2 
