The theory of determinants in the historical order of development, by Sir Thomas Muir.

144 HISTORY OF THE THEORY OF DETERMINANTS

NAEGELSBACH, H. (1871).
[Ueber eine Classe symmetrischen Functionen. Sch.-Prog. 42 pp.
Zweibricken.]
This is an elaborate investigation of the quotient of I a a... a I
by the difference-product of the a's, the contents of its forty-two
well-filled pages being divided into twenty-two sections with seventyone numbered results. These last, however, are of very unequal
importance, most of them in the early sections being already known
although their authors are not named, a considerable sprinkling
being trivial, and many, especially in the later sections, belonging
strictly to other branches of algebra than determinants.*
The most important theorem is dealt with in ~~ 6, 7 (pp. 8-11).
Beginning with Bellavitis' result of 1857, namely,
0 1 ~h-1 h+l
aLo (ll a ~t al a CI
0 1 h- hA-i
6ao aC1.. a. a.. a ayb
where C_7, is a function of the a's defined by the identical equation
(x-aj)(x-a))... (x-an) - x-CX' -+C2 —...
and 52 denotes the difference-product of the a's, he continues the
use of Bellavitis' procedure. That is to say, by annexing an additional row and column he forms the determinant
0. -1 T 7h+l. 1 t 1
a 1 a1 ' 1 '... 6
o 1 h-1 h —i a n+l
2...2 2 2 a
a a... at1 a,... a a'+
30 0h-1 h+] U L+1
X0 X'.. - X1. X,
which from what precedes is known to be
= C0-hla(l, a2,... a ) *(a, a2,..., a, )
The main results are clearly stated on pp. 149-152 of his paper "Studien zu
Fiirstenau's neuer Methode..." in the Archiv d. Math. u. Phys., lix.
pp. 147-192.