212 
Miscellanea 
Again there are n{n-\) terms in S{x\^x.i), 
. S {4,S {x^^ Xj)} ^ 4 {n -1)2 (w^.,.r, a-'i^^-a) 
_ 4 (w — 1) 2 . . mean value of X2) 
~ WTn^ ■ 
But the mean vahie of associated in the sample with Xi will be ^^^.t'l or since (Tx. = crx^ it 
is rxi , 
. 2 {4;S' (xi^x-i) } ^ 4 (?i - 1) 2 (Wx. • Xi'i . r) 
JV. 71* " N. 11^ 
i(n-l)r ..... 
2 {6^(^iW )}_ 3(7^-1) 2(?ix,.r,-.t^i^.r2^) 
^^^^ ~ Ji3 • ^ 
■ 3 (7t - 1) 2 (Tlx, . x-^ . mean value of x<^) 
[Now the mean value of x^ is equal to the square of the s.D. of the x-^ array of ;!:'2's, {jU2 ( 1 — 
added to the square of the mean value of ^•2, {f^x-^)'\ 
- SWx,{?-^^'i '' + ^iV2(l-0} 
■n? N 
= ^^5^^{'-V4 + (1-^-^)m/} (iv). 
. . 2 {12^(.r i%2-^3)} _ 6 - 1 ) - 2) 2(mx...-,:.3-^i^-^2-%) 
_ 6(w-l)(?i- 2) 2 • ■yi^A-2 ■ mean value of x^) 
The mean value of x^ for values Xi and x-i of the other two variables is given by the equation 
-^33 1 c 
where the Ks are the minors of the determinant 
1, r, r 
r, 1, r 
r, r, 1 
Substituting we get 
nix, = {xi + x.^) . ^ = {xi + x.i) . — 
2{l2S{x,^XoX3) } ^ e{n-l){n-2 ) r 2 (w^.x, (V.g2 + -V-^2^)) 
By (iii) and (iv) ^ 6{n-l)(n-2) _^ {,^^ + ,2^^ + (i _,2)^^2| 
= ^^^^^^r^-'-V4 + (l-r)M2^} (V). 
j^^g^j . 2{24SJsCiX2X3Xi)} ^ (?l-I)(w-2)(?t-3) 2(?*a;ig,y3 XjX2 ^3 •»4) 
_ (?i - l)(?i — 2) (m- 3) 2 (wa;, x2a;3-.3^iA"2'^-'3-mean value of ^4) 
