MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 377 
Thus, if we know @ and p14, we can find the successive p”s. Now 
ar /2 
cos’6 e~ 8 dé, 
2 
— 7/2 
C= yor | 
1 a . 
= yen | sin’6 e*" dé, 
0 
and depends on the integral [sind e* "a6, which I propose to write G (7, v). The 
: “0 
us 
result above for »’, shows us that the more general integral f cos?6 sin’@ e*” d# can 
0 
always be expressed in terms of G-functions. Further: 
77/2 
CL yo | _ cos’ ‘Osin de" dé, 
— 7/2 
yoaey (7 i av 
=| cos’6 e~° dd = — —a. 
eee 2 
Thus we find by the formula of reduction above : 
ees Ste eae A 2 lsat nae RORY wis Bryn dG 5) 
Roma eraay + vr’), h3= Caney 2 -+ v°), 
, at 
aa EER CEI ES a a 

Referring to centroid vertical, we have : 
[Ly 7° (7 — 1) (7 SP v’), (pb) — rr — 1) (7 — 2)? 
_ 3a! (+) {7 + 6) (2 +) — 879} 
ba = yt (r bss 1) (7 ne 2) (r a 3) 

These may be rewritten, if z= r? 4+ v’, 
PS Cae ad 4032 .4/(% — 7”) 
bee ei(pae ES ee 7 (r — 1) (r — 2) 
_ _dsate{(r + 6) 2 — 87°} 
Pa = 14 (p — 1) (7 — 2) (7 — 8) 


As before, putting B; = p,7/p.? and By = py/po”, we have 
(r= 2)? _ 


22 G2 Gee 
ne) = eae te ae 
Gee aE pe Me ome oe 
MDCCCXCV.— A. a 
