Dr T. Muir on the Theory of Determinants. 
537 
B’s being in both cases first arranged in ascending order of their 
suffixes. On the other side of the identity, the use of the symbols 
is exactly similar, n-qoi the n upper elements . . . . , being 
taken for the first determinant of any term of the series, and the 
remainder for the second determinant. The number of terms in 
the series on the one side is evidently m\lq\{m - q)\ and on the 
other n\!q\(n - q)\ 
In the demonstration of the theorem greater fulness is evidently 
necessary than in the case of the previous theorem, the rule of signs 
in particular requiring attention. This Schweins’ does not give. 
He merely tells the character of the first transformation, symbolising 
the expansion obtainable, and then says that a recombination is 
possible, giving the result. 
The succeeding five pages (pp. 350-355) are devoted to evolving 
and stating special cases. This is by no means unnecessary work, 
as in the case of a theorem of so great generality it is often a matter 
of some trouble to ascertain whether a particular given result be 
really included in it or not. To students of the history of the 
subject the special cases are doubly interesting, because it is in 
them we may expect to find links of connection with the work of 
previous investigators. 
The first descent from generality is made by putting some of the 
B’s equal to A’s, the theorem then being ( l . 3) 
■^/ \*ll || ^1 

=2(-)1a; 
a'l 
^p+s 
2+1 * * ^2+^'^ 
a'p+s+l .... ^'p+s+q ....... . 
’ II Dg+fc^i . . . . A_p J . 
If in addition to this specialisation, some of the b’s be put equal to 
the a’s, the result is 
\*ll ^1 • • • ^ 1 . . . . ^p+s-h \ II ^'p+s- 
A,^, 
B 
2(-)^ 
bi .... 
Ai .... Ap_|.^B j . . . B g, 
( l . 4) 
h- 
• h+t\ 
'h+k- 
-/>+ 2^1 • • • • 
■ K ) 
‘ 2+1 
. . B 
Ai 
..aJ 
— a notable theorem, which it vmuld not be inappropriate to con- 
sider rather as a generalisation than as a special case of the theorem 
from which it is derived. Returning, however, to the preceding 
case, and putting k = 0, we obtain ( l . 5) 
