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

ALTERNANTS (BRIOSCHI, 1854)

173

If we use column-by-column multiplication, and put s, for
ar + /3r+ yr - + r, we clearly have
4 s1 s2 1 1 a a 2
81 82 83 13 1 /3 /32 1
82 3 8 /32 1 y2
1 /3 /32 1 1 ( 62 2
I a a2 2
_ 1 y Y2
1 (3 (2
1 S 2,
and so the result is established. It will be observed that the
chosen letter 3 which occurs most conspicuously in the new form
thus obtained for ~(a, y, S) is one which the expression is quite
independent of. Further, by performing on this new form the
operations
coll-col4, col2-3 col4, col - f2 col,,
we return to the more natural form
3 a +y + a2 +y2+ 2
(a, 8) = a +y +8 a2f+y2+-32 a3-+y^3+(33
a^,2+^/,62 as +/3 + 3 a4+ 4+64
BORCHARDT, C. W. (1855, March).
[Bestimmung der symmetrischen Verbindungen vermittelst ihrer
erzeugenden Funktion. Monatsb..... Akad. d. Wiss. (Berlin), 1855, pp. 165-171; Crelle's Jomrn., liii. pp. 193-198;
Gesammelte Werke, pp. 97-1().]
The generating function in question is
-1 1 1
Et-a ti —a' t —an'
or T say, the sign of summation being meant to indicate that of
the two series of elements the one is to remain unaltered and the
other is to be permuted in every possible way. The development
of this function according to descending powers of t, t, t2,.., t,
leads to those simplest types of integral symmetric functions of