206 Expectation of Moments of Frequenc]i Distributions 
- 3 [h\r.i (rs - 1) /i^3_2 + kr.,i\{i\ - 1) /a^^., ^^3_„ /io 
+ h\ Vo (r, - 1 ) /^n-i Mr2-2 /^r3+i + ^ r., (r, - 1 ) fx^^+, fio 
+ h\ (''i - 1) ^"2 /^ra-i Mrs+i i^'i " 1 ) ^'s /^r^-a ^r2+i f^rs-i Ms] 
+ 3 [f/'ir.,?-;, (r, - 1) yUri-i /^r.-i /*r3-2 Ma' + ^r.r.;, (?•., - 1) 
X ''3 Mr2-2 Mra-i f^i^ + ^ ''i (''i " 1 ) '"fl^'s Mra-i ^,-3-1 Ma'lj + • . • 
Noting that 
- {{Vri - M/'i) E {v'r^ - flr.,){v'r, - fi^r,) + {''r, " Mr.,) 
= ^ (f'n - /"n) (f'r. - Mr,) ('''ra " Mrg) + ([''i Mri - ^ ''i (''i " 1) Mri-2 M2] 
X [Mrj+rs — Mra Mrs ~ ''2 Mr,-! /^rj+i " '3 M/2+1 Mra-i + ''i^'s Mr.-i Mr3-i M2] 
+ ['2 Mr, - (r.. - 1) Mr,-2 M2] [Mr,+r3 - Mri Mrs " ''i Mn-i Mrs+i 
- Mri+l Mra-l + »'l Mri-1 Mrj-i M2] 
+ [''3 Mra - 2 »'3 (^'s - 1 ) Mr3-2 M2] [Mrj+ra " Mri Mr, " Mri-i Mra+i 
- ''2 Mri+1 Mr2-i + ^1 ''2 Mri-i Mr,-! M2]} + • • • 
we find hence the first term in the development of 
in powers of 
Putting ?"i = = ''3 = w? find : 
1 
E (v'r - f^rf = J^n {Msr " 3r /yo,.+i - 3 (r + 1 )yU2r Mr + f l)M2rMr-2M2 
- 6/- fJ,2r-i fJ-r+^ + 6r- /X.2r-j Hr-i M2 + 3?'= /X^+o M"r-l + 12r (r + 1) /ti^+i /iy flr~\ 
+ 3r (r - 1 ) /x^_„ - 9r- (r - 1 ) /x^^., /i^., 
+ (3/- + 2) /X/ - I r (r - 1 ) yti^^ 
- 9r2 (r + 1 ) ^, ^V-i M2 - mV-i M3 + (r - 1 ) ;Uo"} + . . . 
3 
E{Vr'- = E{Vr' - /J-rY + jy^ [''/^r - i'' ( '' - 1) Mr-2M2] 
X [fi.,^ - 2r fMr+i MV-i - Mr" + MV-i M2] + • • • 
= J^i fMsr - 3r /A2r+i Mr-i " ^/^2r Mr - Gr /i2y_] yu,^,^! 
+ 6?'= /isr-i Mr-i M2 + 3r= MV-1 
+ 6?- (?■ + 2) + 3?- ()• - 1 ) fMr-o 
- Qr' (r - 1) tir+, flr-i Mr-2 M2 + 
- 3r= (2r + 3) /i,^ yito - iJ?r~i Ma + 3?-^ (r - 1 ) /i^^-i /a^.o a/.-} + • • • 
\ (37), 
(38), 
v(39). 
