238 Novel Projyerties of Partial and Multiple Correlation 
we can throw (xviii) into the form 
' " s-i " V (1 — R^s.l-(s">-n) (1 — R^s".l-is')-n) 
^ . sV'Pl-(s)-n ~ ss"Pl-(6')-n X s?'Pl-(s")-« fxviU^ 
Here on the right we have only partial correlations of the order n — 3, but the 
expression involves a multiple of the order n — 1 as well as those of order n — 2. 
Of course in the second radical s" and s' may be interchanged owing to the existence 
of (xi). On the whole it appears best to confine our attention to (xv) and (xvi) 
as the fittest representatives of (vi) and (vii) expressed in correlation forms, (xv) 
has long been in use either to determine ihe partial correlation coefficients of 
higher orders by repetitional processes, or better to verify results given by (iii). 
The calculation of the higher multiple correlation coefficients by (ii) can be verified 
by the aid of (xvi) if the partials have also been found. But if once the value of 
A„ has been determined the continuous product formula may often be advan- 
tageously used. Looked at from the determinantal standpoint this may be rea'hed 
as follows : - ■ 
A = ^ Xss's' As«.v'/s"v" A, , to n - 1 terms 
^s-s ^sss's' Aj^s'sY's" ^sxs's's"s"s"'s"' 1 
= (1 - R's.l-n) (1 - (1 - R\':x-i.s',~n) (1 - R\'".l-issW',-n)-{l-r''Pq), 
if the pth and i/th variates be the last to be excluded. 
Hence* 
1 - R's.i-n = (r:r7?v.i_,,-n) (i - -RV. i- J "(i - ^VM-(.sV)-n) (I - r'pq) 
(xix). 
The trouble of this method is that A„ has to be calculated at each stage, if 
we deduce by a repetitional process. If we use it merely as a verification 
process for (ii) we shall not verify A„ unless it is worked out twice. Of course 
A„, if wheat all large, is more troublesome to calculate than the 3x3 determinants 
of formula (xvi). 
However the primary object of the present paper is not so much to provide 
verification formulae as to show by direct determinantal analysis certain relations 
known and unknown between the higher multiple and partial correlation coefficients. 
* This must not be confused with the formula 
1 ~ ss'P"l2- 
"(1 - s,''P\-U')-n) (1 - ss"'P'\-U-s")-n) (1 -ss""P\-(sVV")-.i) ••• i^-r^PQ) 
see Yule, R S. Proc. A, Vol. lxxix. p. 189, and Pearson, R. S. Proc. A, Vol. xci. p 491'. 
