348 - MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
(3.) To find the nth moment of a trapezium ABCD about a line parallel to its 
parallel sides, y, and y, being the lengths of the parallel sides, «,, x, their distances 
from the moment-axis, and x, — #, = ¢. 
A 


Y) 



WN 
VRNALAAAYAV AL 

iN 


AN 

a 
As 
A 
ANAAAARRAAYER 




PAA 

Sy 
8 
OF SS) f 
‘ 2, 1 
Let M, be the mth moment. Then 
M, = | ya" de 





Yo — I xint? —7 n+2 ai Ifa — YoLy poittl —7. a+1 
fy — 2, n+2 Ly — 2; n+1 
[29/26 Di ees DQ p26 9 OG) (@=2) 5 an 
Sy ee EL 
WE |e [4 [5 / 
Bie nN Pp ie Dee INCOR) is \ 
= Se Ob i a XL, ; PM ee 
ule type + ee ee 
(4.) Now consider a curve of observations made up of a series of trapezia on equal 
bases, as in the accompanying figure : 
2 
(1—B,-3(8—:)) Mate Hs" (1—§ B\—3 (3—Bo) 
aaa IPr“BB-BM) x 0 FE x 0 Gags IE A—3 BAe) 


ane MX O— Toa GPIB B-P)-38- Ba —9 BP 2-8) BD) ye ote, 
where By = ptz?/uo° and Bg = py! p19”. 
This appears to be the more general form of a result given by Professor Epcrworra, ‘Roy. Soc. 
Proc.,’ vol. 56, p. 271. 
For the normal curve “3; = 0, “4, = 3,7; hence, if p does not differ much from q, A, and B, — 3 will 
be small, and we may neglect their products with x//n. Thus approximately 
2? 5 ne 
Y = YoC” my, By, ( ina) 
This agrees with Professor Epcrworvn’s special case if we expand the second exponential. His 
“negative frequency ’’ is accounted for by the fact that he has only taken the first terms of a long 
series, 7.e., 


¥ —22/2n5 J Bay #3 
PSOne ll = 25” (2 os 3) ; 
I have not considered this form of the skew-curve at length, because it is only a first approximation to 
the more general forms considered in this paper, and further, because it is only applicable in practice 
within extremely narrow limits. 
