208 Expectation of Moments of Frequency Distributions 
we see that the terms of order \jN- in the developments of 
and E (v'^^ - I'rJX^'r., - IvJ ("'ra - 'Vj) ("'r, - 'V4) 
coincide. 
Putting Vi = r.2 = = )\ = r, we find : 
E{Vr' - Vj.y = {3 [/tor - IJ-Zf - 12r/Xr_i [/Xo^^lr+i - Mr+i/^r'] 
+ 6/'"/A>-i [j«-2r j"- + 2/x>+i - - 12r>,+i |U,'V_i ;u,o + 3?-Wi H-^"] + ■•• \ (41). 
The same formula (41) gives also the first term (of order l/N''') in the develop- 
ment of E (I'j.' — jx^y. 
(5) In the general case if we agree to denote 
{ v'n - ^n) 0''r, - 'Vo) . ■ • {v'n - t'n) by d^'^v, 
and (//,,j - iJ,r^){v'r, - fir^) . . . (v'n - iJ-n) by S^'>v, 
we have : 
+ 
Ed^''h> = Eh^'^v- 2 {v,j^-fjir,,'>E-r 
li = l ^rh-fJ-Th 
1 % (vrn^ - H-r,,) {vr,,, - H-rn) E .y 7^X77/ ;r\ + 
(42). 
We see that, in the developments of Ed'-'^v and ES^-'^v in powei's of 1/N, 
the first terms (of order 1/iV') coincide, since 
2 (/Va -l^n)E~, — 
contains no terms of order . On the contrary, in the developments of ^cZ'-^+"i/ 
and ^8 '■-'■+'> u the first terms (of order 1/N'+') are different, for 
i gPi+i) J, 
- ("rh-H-m) J'^-, 
1,-1 ^'rh~H-rh 
contains terms of order . 
Formulae (18) and (19) of Chapter II permit us to calculate Eh^'^v and 
Ed^'^v in general, to an arbitrary degi;ee of accuracy, in the same way as Ed^-'u, 
Ed^^'v, Ed^-'^v were found above. The actual expression, however, is of so un- 
wieldy a character that I shall limit myself to the calculation of the first term in 
the development of E {v^' — Vr)'-\ coinciding with the first term in the develop- 
ment of E{vr' —flr)-\ 
In the calculation of the first term of the development oi E {vj — jx^)"^ we may 
take 
