688 Proceedings of the Royal Society of Edinburgh. [Sess. 
but is not afterwards employed. The same author’s term “ minor ” 
(minor e) is adopted, this being represented in the French translation 
by “ mineur,” and in the German by “ Unterdeterminante.” “ Complete ” 
is used as opposed to “ minor,” and “ principal minor ” for what nowadays 
we call “coaxial.” 
Conspicuously frequent use is made of differentiation in the specifica- 
tion of minors ; and it is well to note that, though the work in this 
way becomes cumbrous, there is a certain effectiveness attained by 
the usage. Thus, A standing for 2(±a n <x 22 . . . a nn ), Brioschi, like 
Jacobi, obtains 
0a dv 02 A 
da rl 
9_A 
0dLo 
0 + a s2 
= a sV ^ - + 
da r2 da sl 
da rl da s 2 
0 + 
+ a sr 
+ a s 
8 2 A 
da rl da sr 
S' 2 A 
n da r2 dct s 
0 A 
ca rr 
= a s 
02 A + a + 
L da rn da sl °'da rn da s 
and then, by using the multipliers a rl , a 
+ 0 
, a rn and adding, finds 
A = 
a r i a r2 
0 2 A 
+ 
«rt 1 02 A ! 
. . + 
Ct r l Cl rn 
a sl a s2 
da r ^a s2 
Cl si « s3 1 ba rl da S 3 
a sl a sn 
0 2 A 
which is Laplace’s expansion-theorem for the case where the minors of one 
set are of the second order. The remaining cases, he says, can be established 
in the same way. 
Again, having proved the multiplication-theorem (row-by-row) 
PQ 
R 
where 
P = i *hi ^‘22 • * • ®nn) j Q — ^( L ^22 * * * ^nn) •> R = k( + ^2 
he obtains by differentiation with respect to elements of P 
• ^ nn ) ? 
0P Q = 0R 
da, 
dc r 
0R 7 
0C r2 
= y y 
da rs da p(T ^ 
by<r bys 
0R 7 
dc rn 
(i) 
02R 
5 
(2) 
bc r ydc px 
an element 
of P and an 
element of Q 
0P dQ _ 0P 0Q 
da rs db p( j da rcr db ps 
ZZ 
ttx<J Q'xs 
b y<T bys 
0 2 R 
3<V* c p» 
(3) 
