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

242 HISTORY OF THE THEORY OF DETERMINANTS
JANNI, V. (1878).
[Sopra una formola di Waring. Rendic.... Accad. delle Sci.
(Napoli), Anno xvii. pp. 27-31.]
The determinant form (Hist., ii. p. 211) for s,8 in terms of the
coefficients of the equation
x'"+ajxe~+a2x'2+.+ ~ a,, = 0
is taken, and expanded in terms of the elements of the first column
and their complementary minors, the result being written
_1ys, = ajD,._1-2a2D2..9 +3a3D,._, -
He then expands the D's as Trudi did in 1862; and making substitution for them deduces Waring's formula, as Trudi also did.
Another contribution is noted in chap. xx., under Janni, 1876.
GLAISHER, J. W. L. (1878).
[On a class of determinants. Messenger of Math., vii. pp. 160-165.]
The results here given by Glaisher are fairly evident consequences
of Faure's proposition of 1855, namely, that if
1+b~x~b2x2+. 1-PIx~+P2x2 _P3x3.
1+alx+a.x2+.
then Pn a. 1-b, 1.
a2-b2 aC1
a.,-b2 (62 a1, 1
I fl~
First he says that if
(1+alx-+-agx2+.)-11 -A1x+A2x2-A3x3+
it follows that A, is the same function of the a's as a, is of the A's,
and that if in addition
(1+b,x+b~x2+...)X1 I-B~x+B2x2 -B3x3 +
then we have
a,-b, I (-~ 1 I- l
a,-b2 a1 1.a.. 1132-,A R 1
a3-b3 a2 a, 1... B3-A3 B2 BA 1