1921-22.] 
A Theorem of Frobenius’ 
347 
aggregate on the left cancel each other ; and this can without much diffi- 
culty be done. 
The theorem we have thus reached is of the type already touched on 
by Le Paige in 1880, by Deruyts in 1881-2, and by myself in 1888, the 
common feature of the members of the type being the summation of an 
aggregate of determinants,"^ 
(9) When the zeros of the original aggregates are confined to the 
diagonals (§§ 6, 7), the substitution - theorems thence derived take a 
different form in the right-hand member. The result reached may be put 
shortly thus. If the diagonal elements in the original determmant or 
Pfajfan he 
'1 ? ^'2 ’ '"3 J 
and the quantities substituted in the aggregates o/ §§ 6, 7 for zeros occupy- 
ing the places (1, 1), (2, 2), (3, 3), . . . he 
■X, y, z, . . . 
then the right-hand member becomes 
2/(^3 
For example, 
^2 
«3 
- 2 
X 
^2 
^3 ; + 2 
X 
^2 
«3 
- 
X 
«2 
h 
i 
h \ 
Y 
63 
Y h. 
^1 
^2 
^'3 
«-2 
Cg 1 
^1 
^2 
^3 
e-2 ^ 
and 
= (a,-X(5,-Y)(r3-Z); 
- 2, ^2 
h h 
X 
h, K 
X 
- a. 
hi 
= K-X)(^4-Y)(.3-Z). 
Kondebosch, S.A., 
24^/i May 1922. 
* Mem. 
Soc. E. cles Sci. (Liege), t. ix, x ; Proc. Roy. Soc. Edm., xv, pp. 96-105. 
{Issued separately October 16, 1922.) 
