MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 351 
Hence if we write : 
o* (npg + «),* 
Hs = — pq (g — Pp); 
py = ct (G + npg (e+ 8 (n — 2) p9)), 
= 
w 
I 
we have: 
For trapezia : — Lo 
| 
pac 
fay) 
g 
I 
{ 
5S 
| 
l— 
21 
& 
| 
— 
ol 
For rectangles : G25 3 
For loaded ordinates: . =0, «|=0, 6 
I 
and the above general system may be applied to all cases. 
Writing 
9 


ae mets 
210. B= : 
we have by elimination the cubic for z: 
2° (6 + 38, — 28.) + 2 (2e3 — 3 + 9Bie, — 48 €)) 
+ 2 (Ze, + 9B Ee” — 2B,€,") + 38,6? = 0 
The remaining constants of the binomial are : 
ops a5 = (el sae GIP) 
= a/-#+ 
ete 
(6.) Let us illustrate these results by a numerical example. Plate 8 gives 
Dr. VENn’s curve for 4857 barometric heights. Along the horizontal, 1 cm. equals *1” 
of height of barometer, and the scale of frequency is 1 sq. em. = 28°304 observations. 
The centroid vertical and the second, third, and fourth moments about it were found 
for me{ by the graphical process described, ‘ Phil. Trans.,’ vol. 185, p. 79. We have 
the following results :— 
and 

fey 
| 
# This result seems of considerable importance, and I do uot believe it has yet been noticed. It gives 
the mean square error for any binomial distribution, and we see that for most practical purposes it is 
identical with the value “pq, hitherto deduced as an approximate result, by assuming the binomial to 
be approximately a normal curve. 
+ If we take z+ «4 =x the fundamental cubic reduces to 
(6 + 36; — 262) x3 — 2-3) x? + ox — as = 0; 
a form in which the coefficients are easily calculated and the nature of the roots discriminated. 
t By Mr. G. U. Youre, who has given me very great assistance in the laborious calculations required 
in the reduction of frequency curves. We have used, with much economy of time, the “ Brunsviga” 
calculator. é 
