Dr T. Muir on the Theory of Determinants. 
533 
He then combines the terms which contain as a factor, the 
terms which contain as a factor, and so forth, the result 
being by the law of formation, 
+ \aAei\¥Afe9l\ - 
The identity of this aggregate with the similar original aggregate 
constitutes the theorem. 
The only point left in want of explanation in connection with it 
is the construction of the aggregate of products presented at the 
outset, it being, of course, impossible that any aggregate taken at 
will can be so transformable. A moment’s examination suffices to 
show that when once the first product of all 
V'bfe9'l\ 
is chosen, the others are derivable from it in accordance with a 
simple law, — the requirements being (1) no change of suffixes, (2) 
the last letter of the first factor to be replaced by the letters of the 
second factor in succession, (3) the signs of the products to be + 
and - alternately. As for the first product of all, it is not difficult 
to see that the orders of the determinants composing it are quite 
immaterial Instead of taking determinants of the and orders, 
and producing by transformation an aggregate of products of deter- 
minants of the 3'''^ and 4*^ orders, we might have taken determinants 
of the + and orders, applied the transformation, and 
obtained an aggregate of products of determinants of the and 
(m 4 - orders. This is the essence of Schweins’ first generalisa- 
tion. His own statement and proof of it leave little to be desired, 
and are worthy of examination in order that his firm grasp of the 
subject and his command of the notation may be known. He says 
(p. 345)- 
“ Die Keihe, welche in eine andere libertragen werden soil, sei 
Q = 
114.. 
^n+l\ . 1 
. . . A„ y 1 
1 h. 
|Ei. 
\ 
• • ^a:-l ^*+1 • • • ) 
wo 
x—1, 2, . , . 
m + 1 . 
Der erste Factor 
wird nach 515 
in niedere Summon aufgelost 
n-y+l 1 
^ / \ 
II "*1 • 
. . a.y_-^ + 
Ai 
a„b, ) 
- ) 1 
IIa,, 
wo 
2/=l, 2, . 
. . 
} + 1 
