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

152 HISTORY OF THE THEORY OF DETERMINANTS
in accordance with Naegelsbach's theorem. Of much less note is
the last of the series of deductions, exemplified by
1 -x.1 - =. -x =1, -X,x 2, -X3, X1.. 1 — x I..., 1 — X
Thenceforward (pp. 436-448). Gordan occupies himself with the
main subject of his paper, showing first how Sylvester's bigradient
may be transformed into Euler's product of differences (Hist., ii.
pp. 369-370, 374-375) and passing on to the analogous trigradient
arising from three given equations in x.
WEIHRAUCH, K. (1874).
[Zur Determinantenlehre. Zeitschrift f. Math. u. Phys., xix.
pp. 354-360.]
The second section (pp. 359-360) establishes the determinant
form for the difference-product by a gradational proof, (n-l1)t
order to n.th
MALET, J. C. (1874).
[On certain symmetric functions of the roots of an algebraic equation.
Transac. R. Irish Acad. (Dublin), xxv. pp. 337-342.]
Malet's result is essentially the same as Naegelsbach's of 1871,
and is obtained in the same way. He evades the difficulty of the
sign-factor.
GARBIERI, G. (1874).
[I DETERMINANTI, con.... xiii+267 pp. Bologna.]
Garbieri establishes the already known (1864) identity
2 n-1
w1 a. a,.... al
2 2 - 1
w2 a2 a..... a. a
2 n-i
Won an as. an
(_ 1)-1 2 al" W2 +W +
2 a" I~ alf'(a,) f(a anf(an)
I Nf Wrz~~~~

where

f(x) = (x-al)(x-a,... (x-a,).