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

HESSIANS (BALTZER, 1857)

399

BALTZER, R. (1857).
[THEORIE UND ANWENDUNGEN DER DETERMINANTEN, mit.....
vi+129 pp. Leipzig, 1857.]
In Hesse's converse theorem of 1851 (March) Baltzer (~ 13, 3)
wisely substitutes for A =0 the condition
Cl1l + C22 +... + Cnn = 0
(which by a property of Jacobians implies A =0), his proof being
that the substitution
Xk = bk1y1 + b22y2 + + bk,n-lIn-l + Ck1y
will then give Dv/Dy, = 0, for
au ata axi auu ax2 3au axn
ae, ~axy aY, ax2 ayL, ax,, aye,
= UC1 + U2C2 + +~Ab.
In the second place, from the same n +1 equations, namely
- (~-1)" 1 1X1 +.ll U +... + ~,x = ~
- (rm-1)f1 + z11x1 +.. +,? = 0
- ( - )(m 1) + ULX +.'. + x, = 0,
he obtains (~ 14, 4)
- (m-l) ' X'. x. ' x: = A: Vy: V... V,
= V1: V21:.: V2nl
= wh: V1h: c he corr ondg. s in
where the V's are the cofactors of the corresponding u's in the
resultant of the set of equations. The first and (r+l)th lines
imply respectively
-(nm-1):. = A V,,
and - (m-1): X = V,. V:,,,
the former of which gives
V,. - - IA, (a)