BesoJvability of the Minors of a Compound Determinant. 
99 
(5) The distinguishing feature of the equality established in § 3 is the 
prominence held in it by the collocation of elements a^h.^ ; and we see that 
to every such collocation, thirty-six in all, there corresponds a like equality 
having on its left an irresolvable minor. In this connection the next point 
is to ascertain whether there be not other irresolvable minors in which a^),, 
plays the same part. Observing the positions occupied in 
^2 Ctg 0-4 
\ 63 63 64 
^\ ^2 ^2> 
d^ cZg c?3 d^_ 
by the minors involved in the equality (I) we see that while retaining a^ho 
unchanged in all its positions we may make (a) interchange of the third row 
with the fourth, (/5) interchange of the third column with the fourth, and 
(y) both of these interchanges simultaneously. Doing this we obtain 
(a) j |(XjZ>ol \ci-^d.^\ I63C4I j = 't^i^2^3l la^W^i + aibolaih^d^c^] 
1125' 
!l34i ~ I^A^sl \^ibod_^\ — a-fic,\a-J)oC.^d^\. 
(13) j la^hol la^c^i Ih^d^,] j = I^^A^g] \aibc^d_^\ + a^hla^hcyC^d^i 
I125I ~ '^iVsl l^hMd ~ a^h^\afi.^c^d^. 
(y) I l^i^ol \a^d,^\ \bc,H\ — l^iMs' l^iVJ + a^'bM^hc>d^c.^\ 
134 
^" ^" 134 ~ l«i^2^4.' l^-iMsI + aYh,,\a^.yC.^d^\. 
There are thus four different irresolvable minors marked out by the 
collocation a^h.^, the set of four being most suitably arranged thus : 
125 
134' 
1251 '^1^2^41 i^iMsl + a.Ma^h^^d,^\ = j-^g^ 
125 
134 
134 
125 
As a consequence we know of the existence of 144 irresolvables among the 
400 three-line minors of the second compound of \aj}.,c.^d,^\ ; also that the 
144 may be arranged in pairs the members of which are equal. 
(6) In turning to resolvables it is convenient to deal first with an 
equality immediately derivable from (I), namely, 
\(iibo\ \ciibs\ l^^sl 
\a^cJi \ct^c^\ [aoC.^l 
M hd,\ \h,c,\ 
l^i^^^al ki^2^3l 
7§ 
