86 
Proceedings of the Royal Society of Edinburgh. [Sess. 
u n 
U u . , 
• • U ln 
U 2l 
U% 2 • 
■ • *hn 
Uni 
U n 2 . 
• • u nn 
= 
u n 
^12 • ■ 
. . u ln 
tti 
u n 
U\2 • • 
• • u ln 
7i 
- 
u n 
Uu • 
• • U ln 
a i 
u n 
U Tl • ■ 
• U 2n 
a 2 
U 2l 
^22 • 1 
■ ■ U 2n 
72 
W 21 
?/ 22 • 
■ U 2n 
a 2 
u n\ 
U n 2 • • 
• u nn 
a n 
U nl 
U n‘l • ■ 
• • U nn 
In 
U n i 
U n 2 . . 
■ ■ u nn 
O-n 
tti 
°-2 - • 
■ 
0 
7i 
72 • ■ 
■ ■ In 
0 
7l 
72 . ■ 
■ ■ 7 » 
0 
ableiten, wo U eine ganze homogene Function 2ten Grades sowohl in 
Riicksicht auf die Grossen a als auf y und vom n — 2ten Grade in Riick- 
sicht auf die Grossen u ist.” It will be seen that in our extension the 
exact form of Hesse’s U is specified. 
7. It is evident that so far as the right-hand side of the identity in 
§ 5 is concerned analogous specialisation is possible when any number of 
factors are adjugate determinants. 
By making all the factors in the theorem of § 3 adjugate determinants, 
we see that the cofactor of any element in the product-determinant of the 
adjugates of the n-line determinants A 15 A 2 , A 3 , ... is equal to the 
corresponding element in the product of \ , A 2 , A 3 , . . . multiplied hy 
(A, a 2 a 3 . . .r 2 . 
8. Returning now to the point reached at the end of § 5, we take up 
the question of the three- line minors of a product-determinant, and note 
that as the principal minors of a four-line determinant are three-line 
minors, one case has already been considered, and that the result is quite in 
keeping with that represented diagrammatically at the close of § 5, the 
single point of difference being that instead of the word ‘ two-line ’ we 
have to substitute ‘ three-line ’ throughout. Taking the next case where 
three-line minors are possible, namely, when the factor-determinants are of 
the fifth order, we find the three-line minor which occupies rows 1, 2, 3 
and columns 3, 4, 5 of the products 
! ^i^2 C 3^4 e 5 I ’ \f\9<2h^'fh I > 
I ^l^2 C 3^4 e 5 I ’ I ' i m i n 2 r 3 S fb I > 
and which, as we know, are representable by 
f 3 J 4 f'i 5 
9z 9 b 
3 4 
> 0 b 
a, a R 
b K 
Ccy 
h% h 4 h 5 
k b 
h h h 
