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

128 HISTORY OF THE THEORY OF DETERMINANTS

ALBEGGIANI, M. (1875).
[Dimostrazione d'una formula d'analisi di F. Lucas. Giornale di
Mat., xiii. pp. 107-112.]
With considerable fullness of explanation the determinant is
expanded in a series arranged according to descending powers of x,
the result being
X-) + C1_l,-1 (-a- C ()2 n-2-... - (-I )-1( a)-
i.e. x(x-ac:))-.
Specialization is then needlessly used to obtain the otherwise selfevident result
x a a.... a a
a x a.... a a
a a x.... a a
(x-a),
a a a.... x a
1 1 1.... 1 1 A,
and Sylvester's evaluation of the circulant
C(X, 1, I,... *, I),.
JOHNSON, W. W. (1877).
[Problem 156. Analyst, iv. pp. 90-91.]
As the interchange of any two variables of the determinant
b c....
a 1 c...
a b 1....
may be effected by an even number of interchanges of lines, it
follows that the determinant must be a symmetrical function of
the variables. In the next place it may be shown that the determinant must be of the form
1-xab + /yabc — zabcd +...
and lastly that
x, y,z.. 1, 2, 3,....