236 Novel Properties of Partial and Multiple Correlation 
we have 
1 _ (1 - R\. i_„) (1 - R\:. (1 - i^V. 1 
1 - ^ 'P«-2 (1 - R\'.^-n) (1 - i?V.l-n) 
(l-^Vl-.')-n)(l-^Vl-CO-«) ~ " /'^-<.^'-")- 
Hence 
n p2 \2 (-^ ~ -^^s. l-( s')-n) ~ -^^g. 1-(.s" )-k) Pn-2 /vtr;\ 
^■^ s.l-«^ — \ ^ V^Vlj. 
^ ~ sVP"l-(£)-m 
This is an expression for a multiple coefficient of the {n — l)th order in terms of 
multiple coefficients of the (w— 2)th order and partial coefficients of the {n — 2)th 
and (n — 3)th orders. 
Equation (xvi) just obtained may be put into another form by aid of the 
relations* 
^ ^ s.l-n 
ss'P 1-n 
s"P 1 
1 — 7?2 
leading to 
n — Wl — -,n2 1— ^ s.l-nl 
{l-R\.l-U^-n){^-R\.,-is'l-n) 
by (xvi). 
1 - 
Hence 1 - sW'P\-i,-n -~ (1 ~ ' w'/_ ,,,2 ) (^^i)'''^- 
\^ ss'f \^ ss P 1-n) 
Thus (xvi)i'»s is the reverse of (xv), giving a partial correlation of the (n — 3)th 
order in terms of those of the {n — 2)th order. For example, if there be three 
variates, 1, 2, 3, 
1 _ J.2 ^ 1 32^1^ ~ isPi^ ~ 2iP3^ + 232P1 • 13P2 • 21P 3 
(1 - 12^3^) (1 - ,3P2^) 
leading to = _32Pi + 12P3 • 13P2 
which can be easily verified by substitution of the values of the partial correla- 
tions, or be seen at once from the polar triangle. 
For the particular case of three variates we may use (xvi) instead of (xvi)^'^, 
writing it 
n_7?2 V2_ (1 - ^'12) (1 - As ) h - 12P3, - 13P2 •••(xvii), 
\ 1 ■ 23/ 12 
^ ~ 23 - i2P3> 1' - 23/>l 
~ I3P2; ^ 23P1; 1 
which will be found on substitution of the values of the partial coefficients on the 
right and the multiple on the left to reduce to the familiar result that the square 
of a 3 by 3 determinant is equal to the determinant formed by its minors. In this 
case of three variates, if rjs, r^i, r^^ be taken as the cosines of the sides of a spherical 
* R. S. Proc. A, Vol. xci. p. 490. 
