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

450 HISTORY OF THE THEORY OF DETERMINANTS

that is to say,
a lix ~ 2a19xy + a22Y2
+ au11x'2 + 2a12x'y' + a22y'2 + 2312 XYV
RUBINI, R. (1857, May).
[Applicazione cfella teorica dei determinanti. Annali di Sci.
mat. e fis., viii. pp. 179-200.]
In treating of determinants with binomial elements Rubini's
most interesting example is that in which the element in the
(r,8)th place is a,+ b,.j '-1. By substitution in his general
result he readily obtains the expansion of the determinant in
the form C+DV/-1, which is seen to alter into C-DIJ-1_on
changing the signs of the b's. The product of Ia,,,+b,,,j-1
and a - -1 is consequently expressible as the snm of
two squares. His next point is that on using the ordinary
multiplication-theorein the same product is got in the form
a11 a112 /121/ 1... L.fIn-/31fV-I
a12 +/312 a22.... a2 - 02nh/ -1
a+1 /- L-1 a2,+ 2j210J a..,,,
and that a comparison of the two forms may be fruitful of results.
When n = 2, the identity resulting from snch comparison is
(cad - be - a6~ay)2 ~ (a-by d -/3c)2
m(a2+ a2 + b2+ 32)(c2 + 72 + d2 + 2) -(ac+a7y bd~/36)2_ (ay- ac + bd - Od)2,
a result which gives the product of two snms of four squares
as a like sum.
In connection with this special example, however, note should
be taken that Hermite in a letter to Jacobi published in 1850
(see Crelle's Jotrn., xl. p. 297), had pointed ont that it followed
from the row-by-row multiplication of
a al-l a b~b -c+ y -1-by-b+ 3-1 a-al- y d~61-1 -c - -1