Miscellanea 
441 
This establishes the first proposition* of my Phil. Trans, memoir, namely : that selection 
about the means with any variabilities gives the same mean value for a non-selected but correlated 
variate as if all the selected variates had been taken at their mean selected values. 
We have clearly : 
V n + i — X n+ i — Tjn + i -t3 I — ) V 1X />- 
Now if we are dealing with N manifolds of variates : 
S (>7 2 ft + i)//V = a standard deviation indicated by + 
Sittt 2 )^ = standard deviation of selected tth organ =s ( 2 , 
S{y n + it;t) =0 because the Zth variate is not selected in reference to the (?i+l)th variate. 
Hence if we square (ix) and call 2„ + j the resulting variability of %„ + x due to the selection 
of the n- variates, we have 
2 11 + 1 = 0" ? t+ l \~2 HA ™ 2 + 2 ' S _ R2 2 Pit' 
L(T + 1 1 \n- n + l,», + l / V ^w + l.m + l 
But as we have already seen ?;„ + 1 is not correlated with g t . Hence we shall find the value of 
S 2 )i + i °y putting all the s ( 's zero, or by concentrating the selection at a single value of a manifold. 
It is therefore the value of 2 2 „+i for an array of x n + i for definite values of z\, x% ... x n , i.e. by 
(vi) <r n +i equals a n + i a/ ./'''" " - , where + is the determinant of n + 1 rows and columns. 
V a n + 1, n + 1 
Thus finally : 
„ 2 2 f %±jj I 5/ ffi.tt + 1 S t 2 \ . Q y ( &t,n + l R t ', n + l S t Sf \\ ,v 
2^+1=0-^+1 1 o +'Mt?2 ^j +2 ' b l^?2 ~2pw r — ( x) - 
[-">» + 1,»+1 1 \" )i+ 1,7' + 1 a t / \ + 1.H + 1 <T« /) 
This is in complete agreement with the value given as Equation (xlv) of my Phil. Trans. 
memoir t, and deduced there on the assumption of a Gaussian frequency distribution. 
(ii) I now turn to the second of my problems the correlation between the (« + l)th and 
(«. + 2)th variables. In this work R' n + 2 ,n + -2 denotes the determinant of n rows and columns 
bordered by the (n + 2)th variate correlations, those of the (?t+l)th being omitted. Clearly 
as in (ix) 
" ( R' t , <r \ 
1 \ /l '7i+2 ) ti+2 a t J 
Multiply (ix) and (xi) sum and divide by the number of the manifolds, N ; then if p„ + + ■_, 
be the correlation after selection of the (?t+i)th and (?i + 2)th variates, we have : 
v ~ _ §_ (jh + lVn + 2) , / c. Rt, n + 1 1 "t, n + 2 St 1 
^11 + 1 + 2 p n + l,ii+ 2— if +C«, + lO"ii, + 2 I' 5 " 
J7 ' WtI Kts ] A3 pi 9 
Rt,n + \ Rfn + 2 s t s t' . \ 1 v/ Rf,n + 1 R't,n + 2 s t s t' 
+<5) Iff 7? ^"'j + b 75"^ — 7?^ — P"' r ( xu ^ 
As before vln+iVn+z) b e given by the mean partial product moment of the (»+l)th 
and (n+2)th variates for constant values of the n variates concentrated at their selected means. 
This can be found without appeal to the Gaussian frequency surface by extending the formula 
(vii) to + I variates. 
* Vol. 200, A, p. 13. 
f Phil. Trans. Vol. 200, A, p. 17. 
Biometrika viii 56 
