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

BORDERED DETERMINANTS (HESSE, 1872)

437

HESSE, 0. (1872).
[Ein Cyclus von Determinanten-Gleichungen. Crelle's Journ.,
lxxv. pp. 1-12; or Abhandl..... Akad. d. Wiss. (Miinchen),
xi. pp. 177-192; or (in Italian) Giornale di Mat., xi. pp. 309 -317; or Werke, pp. 585-598.]
Hesse slightly modifies Salmon's notation, writing
[uaiu.1 for Ita
and, more generally, putting
[ay, /38]
to stand for the determinant obtained by bordering lu,,, vertically by
ap1, a2,.., a, and Y', 72', ' 7'#,n
and horizontally by
/1'/32) 3,., /f and 681'82'...86n
His starting point is Jacobi's theorem regarding a two-line minor
of the adjugate, as applied to the adjugate of [a7y, /3]; the result
of this is
I Uln [a-y, /38] [= a, 3] [-y, 8] - [a, 8] [7, 031 (A)
By inserting therein ~ and e for 3 and y respectively, a similar
identity is obtained; and the elimination of [a, 8] between the two
gives
jmU1f n[e, [ay, /38] - [y, /3][tAe, L 8]
[a, /3] Ly, 8] [E, ]-, /3][e, 8] [a, ~]. (B)
Next, seeing that only the cofactor of I u, I in this is affected by
changing a, /, y, 8, e, g into y, 8, 6, e, a, /3, and that the like happens
when a, 3, y, 8, e, 5 is changed* into e, a, /3, y, 8, it follows that
the three cofactors of Iu1,4 must be equal: that is to say,
[E, %][ay, /38] - [y, 3][aec, S] = [a, /3][ye, 8~] - [E, 8][Ya, 3 (C)
= [y, 8][a, S/3 - [a, Si[ey, 8/3] C
From (B) another set of three cofactors of I u1, is obtained by the
* We may note that these changes are equivalent to the performance of cyclical
substitution on a, -y, e and 3, 3, r simultaneously.