MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 349 


XL, - > 
Here 4, yg, Us, +--+ y, are the frequencies of deviations falling within the ranges 
a +4c, x, +4¢, x, +4c...2%,+4¢..., and the tops of the ordinates are joined 
to form a frequency-curve in the usual manner. 
Let M’, be the nth moment of the system of trapezia about the Jine Oy, then 
M., — S 2y; ( a 
Vie 
n we 1) 2," — 203 + uv (a ae 1) “ 2) (a el 3) GUST G2 + eee )t. 


+ 
ww) 
In particular, if we take Oy in the position O'y’ at distance ¢ from y,, we have 
x, = re, and accordingly, 
n(n — 1) (m7 — 2) (wv — 3) 
uv (n—1) _, “ 
12 Noo t 360 Ni 
n(n —1)(n — 2) (n — 3) (1 — 4) (1 — 5) LL, 
1 20160 N 

Miee—t62 (W’, ar 

n—6 ar ete.) 2 
where N’, = S (y,7°). 
In particular, 
, y 
/ as p) y 
7 CONIC. 
M 
M 
M, = (N, + ¢N), 
M 
M 
I 
sO: (N’, ote aN); 
4=e (N’, + Nj T5N%), 
M’, = o® (N’, + EN’, + 4N)). 

When we put M’,/M’, = p’,, and N’,/N’, = v’,, these reduce to 
by aa cv, 
jig =O (vg + GM0)s 
Bh, = (v's + 3r); 
Big = CF (4 +g + T5Y0)s 
Bs =e (V's + 3y'5 + 3r))- | 
Now let pw, be the value of the mth moment of the trapezia system about the 
vertical through its centroid divided by its area. 
