186 Expectation of Moments of Frequency Distributions 
Replacing in (2) quantities of type /xj^^^n-i) by their values in terms of the 
quantities fi (cf Chapter I, formulae (15), (17) and (18)) we obtain : 
fN - IV 
V N 
r-l 2;, ''-1 1 '■ . 
M-2r + C.^,. flir-2l, (iV'^)^+' ,•?„ ^~ ^h-i+}J "^-ih^h-i+j \ 
^2/1+1 
Z'zr+i, (iV) 
i\r-i\='-+' , 
— ( 1)' ^h-i+j T2h+\,h-i+j 
■(4), 
)— 1 
J— 1 
.ii •^'•+i^"-^\-ro(^-i)''+'-- -0 
r-1 
- 2 
=11 
- (- 1)' ^h-i+jj T.2h+l,h-i+j 
> ...(5), 
and, hence, 
-o(iv^-i)'-+'^'A 
1 
2 (—1 )' ^r-i+hj ^2r+i, r-i+j 
Vr, (N) = Mr - AT [»>r " 2 ^^"^^ /^r-a Ma] 
+ 
^2, (A-) 
2^3, (iV) 
r — 1 1 
Mr 2~ ^^"^ 1^'-'' ~ ^^'^"'^ '^^ ^^'"'^ ^^j 
As iV increases the ratio tends in this manner to the limit 1. 
Mr 
(2) Putting r=2, 3, 4, 5, 6, in (2) we find without difficulty 
N-1 1 
(6). 
3 2 
M3 = Ms~]^M:i + 2y^.^M3 
2 3 3 
1^4, (A') = M4 - ^ [2m4 - 3/i2-] + [2/1, - 5fj^'] - ^3 [/X4 - Sfi.^] * 
V-o, (N) = Ms - ^ [Ms - 2/^3 M2] + ~ [fl, - 5fXi - ^3 [/is - 8/X3 /ia] 
4 
+ ]y-4[Mr.- 1^M:.M2] 
3 5 
^^6, (AO = Mh - ^ [2/ts - 5yU,4 /ts] + -p [3/i6 - 1 5/^4 yu,2 - 4/i3-^ + 9/i2'] 
- ~ [4/.« - 33yLi4 M2 - 16/i3= + 42/1/] + ~ [3/.e - 36/14 M2^ - 22/x3= + 63/i,^] 
in 
5 
- [M6 - 15m4 Ms - IOMs' + 30/1/] 
* For footnote see page 187. 
