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

JACOBIANS (WOLSTENHOLME, 1863)

259

or, say,
P, Q, R, A, B, C,
with respect to a, b, c, x, y, z. This, because of the well-known
relation
PQ
Q R A2 +B2+C2,
must vanish; and it is thus suggested to ask for an independent
proof of the identity
a x.. - y
b y. z. -x
c z. -y x.a x. c -b. b y -c. a. c z b -a
Such a proof is not given.*
BAEHR, G. F. W. (1864).
[Note sur le changement des variables dans les int6grales multiples.
Archiv d. Math. u. Phys., xli. pp. 453-495.]
The relevant theorem regarding the Jacobian is dealt with on
pp. 474-479.
COMBESCURE, E. (1867).
[Sur les determinants fonctionnels, et.... Annales sci. de
l'Ecole Norm. Sup. (1), iv. pp. 93-131.]
* One, however, will be found in the fact that the columns of the determinant
are connected by the equation
Rcol- Qcol2+ Pcol,-Acol4-Bcol5 -Ccol6 = 0.
But it is also interesting to note that, if we express the determinant in terms
of the minors formed from the first three rows and the minors complementary
to these, we find that Q is a factor of each term of the development, and that the
cofactor vanishes because it is equal to
P Q - (A2+B2+C).
Q R