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

254 HISTORY OF THE THEORY OF DETERMINANTS

and a comparison of this with the definition of zi, makes it reasonable
that the z's be termed the adjunct functions of the y's. Less
expected is the identity
W(Y1'. y. Yn)W(z.+l,... Z ) = W(y1,...., ),
which is obtained by simply using the multiplication-theorem, and
which, when k = 0, becomes
W(yl,.. y. W(zl,,.. ) 1,
or, in words, the Wronskian of a set of functions and the Wronskian
of the adjunct functions are algebraical reciprocals.
The subjects of ~~ 4, 5 (pp. 252-255) are the quotients P(y, z,),
P(y), etc., and in the results, as might be expected, reciprocity is
still a feature.
The paper concludes (~ 6) with an application to differential
equations.
TRANSON, A. (1874).
[Reflexions sur 1'evenement scientifique d'une formule publiee par
Wronski en 1812, et demontree par M. Cayley en 1873. Nouv.
Annales de Math., xiii. pp. 161-174.]
[Loi des series de Wronski: sa phoronomie. Nouv. Annales de
Math., (2) xiii. pp. 305-318.]
The ' formula' referred to is Wronski's "loi supreme," which
Cayley had dealt with two years before. Although in telling
sympathetically the story of Wronski's poorly appreciated labours
Transon draws particular attention to the use of determinants,
he merely restates (~ 4, pp. 313-315) the fundamental property of
them enunciated in 1815 (Hist., i. pp. 475-476), and gives the
simplest illustration possible.
CASORATI, F. (1874).
[Sui determinanti di funzioni. Merm.... Istituto Lombardo, xiii.
pp. 181-187.]
When dealing with other forms of determinants whose elements
are differential-coefficients, Casorati formulates two theorems
regarding Wronskians. The first, however, is that more than once