368 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
Let « be the area of the curve between # = — a and =a», ap’, its n™ moment 
round a parallel to the axis of y through « = — a, and ap, its 1" moment round the 
centroid vertical. 
Then we have 
b 
Opn — | yx" da, 
0 
1 
-— be + ‘” ( gia +2 (1 — z)"? dz, 
“0 
= b"*'y B(m, +n+1, m+ 1), 
— pet} T(m +” +1) U(m + 1) 
rie T (m, + my +2 + 2) 

Thus, by the fundamental property of the I function, we have 
a = bn V(m, + 1) (mz + 1)/T(m, + my + 2), 
Gena) iam b(m, + 2)(m+1) 
WALT, My, + My + 2’ 2 = (m, + amy + 8) (my + m4 + 2)? 

diy b3 (m, + 3) (m, + 2) (m, + 1) 
PO (am, + my + 4) (m, + airy + 3) (My + my + 2)’ 

a, b* (m, + 4) (m, + 3) (m, + 2) (m, + 1) 
ees (m, + ig + 5) (at + Mg + 4) (a + Mg + 3) (M, + My + 2) 

From these we easily deduce by the formule connecting p and p’, if we write for 
brevity, m, +l=m), m+1=m’, andm,+m,=r: 
__ 3b* m', m’, (m', m’'s (7 — 6) + 27°) 
Ba a8 (y +1) (7 + 2) (7 + 8) 

Bm’; i's 203m’, m’, (m', — m')) 
Mee tl) MS Br +1) @ +2) 


Now, @, 49, #3, and p, are to be found by the methods indicated in Art. 4 from the 
polygon of observations, and may be supposed known quantities, when we are dealing 
with the fitting of frequency-curve to observations. 
Then, if By = py/po", and B, = ps/pm.*, € = m’',m’',, we have: 
_3@ +1) (27? + € (7 —- 6)) 

_ 4 (2 — 4e) (r +1) 8 



N= e(7 + 2) 4 alae e(7 + 2) (+ 3) 
Thus : 
B, (7 + 2) at ” eats B3 (st 2) (v7 + 3) Me 2 Bt, 
4(7 +4 1) € 3 (7 + 1) € 
whence, eliminating 7*/e, we find : 
een se Ls 
7 
3B, — 2B; + 6 
