232 Nocel Properties of Partial and Multiple Correlation 
of lower orders. Thus Iis.i-{s")-{s"')-n or simply ■Ss.i-.(s"s"')-k would signify 
the multiple correlation coefficient of the sth characteristic for w — 3 other charac- 
teristics, s" , s'" and of course s being excluded. Similarly ss'Pi_,s'v"si>)-n would 
be the partial correlation of the sth and s'th characteristics for w — 5 other 
characteristics supposed constant, i.e. the n variates with the exception of 
s, s', s", s'" and s'^'. 
("2) Preliminary Pro-positions. To express First Minors in Terms of 
Second Mitiors. 
While values of the second minors in terms of the first have long been known, 
i.e. the equations of the form 
A A A2 , 
— ^ (iv), 
A — A ss ^s's" ^s.i' ^ss " j \ 
'-^sss's" \ ' V^)y 
the reversed results, i.e. the expressions for first minors in terms of second, seem 
to me less familiar, at any rate I have not come across them in the literature with 
which I am acquainted. 
The forms to be demonstrated are 
(A,,,vA,,,sV' - A\„v') (vi), 
' Hft'fi" 
.(vii), 
where 
V ' " = 
' ss s 
A.V, 
A.S" 
(viii), 
As's, 
As's" 
A,"„ 
A,'.', 
As"s" 
and 
V2,,'s" = A3 
A.',',",", 
.(ix). 
As"s"ss' ' ^s's's's 
A(j"s"ss' ) ^S"S"SS 5 A.55.,'5" 
A <;'s'f;"s , Agjs'j" , Ajjj's' 
To prove these results I start from (iv) and (v) to express the value of 
A,s-'s's"s" Agss's" ~1~ Aj's'jj" Aj"j^."5s' . 
We find 
. A 4- A A _ (A,','A,»,- - AV ,-) (A ,,A,-,» - A,3-A,,-0 
^^s's'6"!;"^sss's" "T ilAs's'ss"^.s"s"ss' — A " A 
(A,.,-A,»,- A,v-A., ) (A,"3"A,,' - A,", A.",') 
= ^ (A,,A,v A.V' - A,,A2,,^. - A,vAVs - A.-v A^, + 2A,v'A,-',A„-) 
A . " V ■ " 
s • ^ ss s 
A^ 
which gives us at once (vii) 
