of Edinburgh , Session 1885-86. 
835 
and thence expanding in terms of the elements of the first column 
and their complementary minors, and observing the corresponding 
result to (B 2 ) for two rows of m - 1 elements each, viz. : — 
<Cl\ 
h m ~\ , . . . b{ 
. . . bf~ 
m— 2 
2 
-3 A Vm - 2 
3 • • y »i-l 
— j (X 1 5 2 I j #1^3 
l '«i— 2^>w— 1 I J 
we have the identity 
I a ih\\ a AW a A\ \a m -M 
= ■ • • K | 0 2 6 3 1 1 «2 & 4 | • V | I 
- a 2 m - 1 6 1 6 3 . . . I II a A | ... | | 
+ 
_|_ ( l) m | II I * * • I a m-d ) m-l | • • • (C 2 ) 
Using the Law of Complementaries, we should then obtain the 
conjugate theorem (involving elements formed from m letters 
a, b, c, ... k, l ). 
&3C4 . . . h m 1 1 ^2^4 * * * n I • • • | b-^2 • • • h m _2 | . | (X-fyCz • • • I'm I 
= A^Bi I b^cgd^ . . . l m | m 1 — A 2 B 2 I b^Cyd^ . . . l m \ m 1 
"h A3B3 | \cfl± . . . l m \ m 1 — . 
+ ( - l) m_1 . A m B m | . . . Z m _! I 5 "” 1 , 
where the coefficient of | af) 2 c z . . . l m | m_1 in the left-hand side 
consists of \m{rn - 1) factors, formed by taking the m suffixes m - 2 
together; and in which coefficient A 15 A 2 , A 3 , . . . A m denote the 
products of those factors in which the suffixes 1, 2, 3 ... m respec- 
tively occur, and so have each J(m-l) (m-2) factors; while 
B p B 2 , B 3 , . . . B w denote the products of those factors in the 
expression 
I «A C 4 * • • K || «A C 4 * • • K I | «A C 3 * • ‘ /r m-l | 
in which the suffixes 1, 2, 3, ... m respectively occur, and so have 
each m- 1 factors. 
