516 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Baltzer, R. (1857). 
[Theorie und Anwendung der Determinanten, .... vi + 129 pp., 
Leipzig. French translation by J. Honel, xii + 235 pp., Paris, 
1861.] 
“Die Functionaldeterminante ” is the heading of Baltzer ’s thirteenth 
chapter or section (§ 13, pp. 61-72). Though the exposition is neither so 
full nor so fresh as Brioschi’s, it has the advantage in arrangement, 
concision and clearness. Jacobi’s last theorem ( De determ. fund. § 18), 
expressing the determinant as a single product, 
m /%\ m . . . /ma 
XdxJ Kdxf \dxj \dx n J ’ 
Baltzer makes his first, the proof being readily altered to suit. This 
change enables him to deal very effectively with the proposition regarding 
the vanishing of the determinant. For then he can assert that as the 
determinant vanishes, one of the factors of the said product must vanish ; 
and thence step-by-step can infer the vanishing of the succeeding factors 
including the last,— a conclusion which entails f n being expressible in terms 
of the other /’s. 
A footnote recalls the fact, which we should have noted before this, 
that Mobius had given in Crelles Journ., xii. p. 116, in the year 1834, the 
equation 
(t x u y - t y u x )(v t w u - v u w t )(x v y w - x w y v ) - 1 
where t x stands for dt/dx. 
Salmon, G. (1859). 
[Lessons Introductory to the Modern Higher Algebra. 
xii + 147 pp., Dublin.] 
Salmon gives little, and certainly nothing fresh, on the subject; but his 
unreserved adoption of Sylvester’s word “Jacobian” (§§ 53, 54; p. 37) 
doubtless helped greatly to spread the usage. 
LIST OF AUTHORS 
whose writings are herein dealt with. 
1844. 
Jacobi 
. 499 
1844. 
Hesse 
. 503 
1847. 
Cayley 
. 505 
1851. 
Bertrand 
. 505 
1851. 
Spottiswoode . 
. 510 
1856. 
Sylvester 
. 510 
1854. Donkin 510 
1854. Donkin 512 
1854. Brioschi 513 
1857. Bellavitis .... 515 
1857. Baltzer 516 
1859. Salmon 516 
(. Issued separately August 6, 1909.) 
