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

JACOBIANS (BERTRAND, 1851)2

243

may be held to give implicitly the f's as functions of the x's, and
therefore by a previous result
Dx1Dx2 __2 _ n_
-i.f-12 =nk2~1 ~1 22.
Dfi Df2 ' *fn
*' cA '"
_ __0n D01 _b 1
Jaf aDf2 afn
which is the relation desired. If q does not involve f,, if 02 does
not involve f, or /2, if 03 does not involve f, or g2 or f3, and so
on,* the determinant in the denominator takes the value (-1),
and the relation becomes one of equality.
The last section deals with the theorem regarding the change
of variables in multiple integrals,-a theorem which in the ten
years from Jacobi's memoir to Bertrand's had been discussed by
Boolet and Dienger.+
SPOTTISWOODE, W. (1851, 1853).
[ELEMENTARY THEOREMS RELATING TO DETERMINANTS,.....
viii + 63 pp., London. Second edition, as an article in
Grelle's Journ., li. pp. 209-271, 328-381.]
Spottiswoode has a special chapter (~ x. pp. 51-57) headed
"On Functional Determinants," its contents being a selection of
Jacobi's theorems in unimproved form and a reprint of the first
three paragraphs of Cayley's paper of 1847. In the second
edition (~ ix. pp. 338-343) there is no change, save that the
extract from Cayley is left out.
* Or if f, be not involved in 0n,, and neither f,, nor fnl involved in 0^-1, and
so on.
1 Cambridge lfath. Journ., iv. (1843), pp. 20-28.
+ Archiv d. Math. u. Phys., x. (1847), pp. 417-421.