390 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
We have: 
D(@=N) 45 7 m (m — 1) (m = 2) j 
[hp = [in = TOD i [Bien ie ae it Poe 1.23 1? baa + ete. 

Thus we find: 
by = 0, 
Py = (vg — vy + i65), 
Hts = CO ('5 — By, v'y + 2n'), 
Py = Cy (v'g — Avy v's + 6 Pv'g — Bv44 + {rn — v'? + H5}), 
Bs = 8 (v’, — Sy’) vy + 10? v's — 10n'? vg + 40’? + f3 0’, — dv, v’, + HP 7). 
Comparing these results with those given in the ‘ Phil. Trans., vol. 185, p. 79, 
Eq. (4), we see that treating the curve as built-up of trapezia instead of loaded 
ordinates introduces the parts into the values of the p’s enclosed in curled brackets. 
These additions are small, but in many cases quite sensible. Since the series of 
trapezia gives in general a closer approach than the series of loaded ordinates to the 
frequency curve, and, further, since the calculation of these additional terms is not 
very laborious, it will be better for the future to calculate the moments of any 
frequency curve from the above modified formule. 
(5.) Returning now to the point-binomial, we have : 
Yale eG ; 
vg = 1+ 8ng + n(n — 1) @?, 
v's = 1+ 7ng + 6n (n — 1) + n(n — 1) (a — 2), 
v= 1+ ling + 25n (n — 1) q? + 10n (mn — 1) (n — 2) 4° 
+ n(n — 1} (n — 2) (n — 8) q¢. 
Thus : 
Hy = & (npg + ¥), 
— npg (¢— P), 
by = C8 (q's + npg (2 + 3 (n — 2) pq)). 
= 
oo 
I 
If, instead of taking trapezia, we had taken a series of rectangles, but not, as in § 2, 
concentrated their areas along their axes, we should have found the following - 
system : 
Ho = © (npy + 1), 
Hs = — npg (q — P)s 
Hy = (so + npg (3 + 3 (1 — 2) pg). 
