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

82 HISTORY OF THE THEORY OF DETERMINANTS

FURSTENAU, E. (1879).
[Beitrage zur Theorie der Determinanten. Crelle's Journ., lxxxix.
pp. 86-88.]
The first of Fiirstenau's theorems is identical with Sylvester's
of 1839, of which an easy proof was suggested by Cayley in 1843
(Hist., i. p. 233; ii. pp. 10-11).
The second is that if the successive minor diagonals on the upper
side of and parallel to the main diagonal be multiplied by x, x2, x3,...
respectively, and those on the under side by x-, x-2, x-3,... respectively, the value of the determinant remains unaltered. In proof of
this, it is pointed out that the multiplications in question are equivalent to multiplying the element in the (r, s)t'1 place by x"-', and
therefore equivalent to multiplying any term of the determinant by
x2- ", that is, x~.
When x = -1 we have Janni's case of 1874.
CARR, G. S. (1879).
[Question 5752. Math. from Educ. Times, xxxii. pp. 54-55.]
What is here proposed to be indicated is a method for changing
a given algebraical expression into the form of a determinant. The
process, which is illustrated by the expression
abcd +bfgi+fh2+def +cghp+ahr+elpr
-fhpr - ablr- ach2 -fgh - bdf2 - efhl - cdep,
is commonplace when compared with Dodgson's of 1867.
STUDNICKA, F. J. (1879, 1880).
[Ueber eine neue Determinantentransformation. Sitzungsb...
Ges. d. Wiss. (Prag), Jahrg. 1879, pp. 487-494.]
v
[O nove pouece determinantni. Casopis pro pestovdni math. a fys.,
ix. pp. 97-103.]
The transformation referred to is the second of those spoken of
as 'condensations,' and fully dealt with earlier in the same year
by Muir (see above, pp. 80-81).