442 
Miscellanea 
Let A be the determinant of (w + 2) rows and columns, A w the minor corresponding to the 
pth column and qih. row component. Then the regression equations for x n + i and x n + 2 on the 
remaining variates of the (n + 2) group are : 
- A n + 2,n + l <Tn + l / - x » f \n + l ""n + 1 / - x\ 
*» + 1 *n + 1 — — (,■*»+ 2 *«, + 2j ^ 1 T W ^(J f 
A)l+l,?J + l 0"k + 2 1 l^n + l.n+l °"« J 
7 _ ^» + l,« + 2 On + 2 / - x q f ^,» + 2 On + 2 / - v) 
n + 2 — x n\2 — ~ T + l *n + U ° It \*J *«;f • 
^■» + 2,?i + 2 °"n + l 1 1^71 + 2, k + 2 ""J J 
and 
Now, when we put x x ...x n constant, the coefficients of x n + o — x n + i and x, l + 1 — x n + 1 and the 
partial regression coefficients of x„ + 1 on x n + 2 and x M + 2 on + t for constant 1 to 11 variates, or 
the square root of their product is the partial correlation coefficient, i.e. 
1, 2, 3 . . . nPn + 1, n + 2 = Pn + 1, n + 2 > 
say for brevity ; therefore 
_ / A w +1, n + 2 ' i 
p»+i,b+2-\/ t x - ~ /- — . — ( S1V )) 
v ^a + l,)t + l "» + 2,n + 2 ' 
^ft + l,re + 2 
+ 1, 71 + 1 An + 2, n + 2 
a well-known and familiar form*. 
Now let us look at the standard deviations of the arrays of the (>i + l)th and (ra+2)th variates 
for absolutely selected values of the n variates. 
The variability of the array of the (n + 1) variate is given by (vi), i.e. 
5=„ + 1 = o-„ + 1 —J 1 (xv), 
and of the n + 2th variate 
ov + 2 = f» + 2 x / — (xvi). 
» /l n + o, n + 2 
But R =An + 2,n + 2> 
•fl = A re + i, )l + i 
while clearly #n+i,n+i=-fl'»+2,n+2 = the second minor of A obtained by leaving out both (w+l)th 
and (?i + 2)th rows and columns. Hence we have : 
""n + 1, n + 1 = R 11 + 2, n + 2 = A„ + i, n + j, B + 2 , }l + 2 > 
and 
S(l?n+l»;n + 2) _- - - _ An + hn + 2 , 
«t — ^(l+l cr )t + 2P»j + l,n+2— — <r n + i o- )1 + 2 (xvn). 
■»n + 1, 71 + 1. n + 2, 71 + 2 
Thus finally we have from (xii) : 
y , y „ n — rr rr / A n + 1,11 + 2 ■ S / Rf.n+1 R t, 11 + 2 V \ 
*n + l *n + 2Pn + i,n + 2 — <»n + i °n + 2 It — ho I -„ pT 5 ) 
l A « + l,?i + l, 7( + 2,n. + 2 1 \"» + l,n + l ^t» + 2,» + 2 °"t / 
>Sl( 11 + 1 ^' t '' n + 2 i Rt',n + iR't,n + 2 \ *«»f 1 .. 
+ ' 5 I 5 ™ h p „-, I p tt >\ ( XV111 )) 
\-"» + l,n + 1 11 n + 2,n + 2 -"n + l.n + l -tln + 2,71 + 2/ criCi' J 
which is in complete agreement with the value found from the Gaussian hypothesis t. 
(iii) Lastly we require the correlation p U n + 1 between a selected and a non-selected variate, 
t < =n. Turning back to (ix) multiply by g t , sum and divide by JV, then : 
Rt,n + 1 °"» + l 2 \ V ( Rf, n + 1 °"« + 1 „ „ \ 
' '■• , 1 I ~ \~R ^ hSfPtv) 
n + l,n + l &t / V^n + l^t + l J 
* Pearson, Phil. Tr cms. Vol. 200, A, p. 10, Equation (xxvii) 
t See Phil Trans. Vol. 200, A, p. 17, Equation (xlvi). 
