202 Expectation of Moments of Frequennj DistriMitions 
III 
(1) Denoting by w' the difference X (js-) — vi^ and by d/u,/, the difference 
fx,/ — fjif, we find 
v; = i p/ - A',,v)]'- = t Pj' i (- 1)" c/ - m,y-^ 
i = l i = l h=K\ 
= /a/ - C'r'w' ft'r-i + ^V" /^'r-i " '^V' + • • • I (27). 
= /Ay + — Cj.'co' /i-r-i — Cfwdfj! + G/co'' ix^_., ! 
+ C/w ''d\i r--. — C/c}'"' fif_i — G/oj'-^dfJb'r-s + ■ • • j 
W. F. Sheppard, in his well-known investigation " On the Application of the 
Theory of Error to Cases of Normal Distribution and Normal Correlation*," 
terminates the development at the third term, taking 
Vj.' = /"■r + dfMj.' — ro}'/j.r-i (28). 
Hence : 
Z'y' — IV = I'r — f^r — f^/^/ "~ '"Mr-i 
E {I'j! — Vr)" = E (r/ — /J^r)' = (df^r )" — ^''P-r-i Ew'd/Jbj.' + ''VV-i Eco'" 
A^ 
AT' 
We thus obtain, with full accuracy (cf Chapter III (13)), the first term of the 
development of E {v^' — v^f in powers of This is explained by the fact that 
the terms rejected by Sheppard in the formula (27) do not yield terms of order — in 
E (Vj! — V^y. Owing to the same circumstance, we also get accurately the first 
term in the development of E {v\^ — Vrj){v'r^ — I'r.), starting with (28) : 
= E [dfl'r^ - 7\ /ly^_, O)'] [d/J.'r., - v., «'] 
,(29). 
* Phil. Trans. A, Vol. 192. 
