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

222 HISTORY OF THE THEORY OF DETERMINANTS

t, u, VI. w de'Signant n variables, Si le cU'terminant du
s~yste'me de n quantities
t... w
dt du.. dw
d~t d2u... d2W
est O'at a' zervo, onl a necessairernent I'6equationb
at +bu + cv +.~gw 0
oA& a, b, c,.., g sont des constantes." The theorem is spoken of
as known, but no reference is given.
TISSOT, A. (1852).
[Sur un determinant d'int~grales deffinies. Journ. (de Liouville)
de Math., xvii. pp. 177-185.]
Tissot incidentally asserts that if YI Y1 I.2 9, all satisfy a,
linear differential equation similar to that dealt with by Malmsten
above, then
Y W,- yPdx
where y/ is independent of x. It is further stated that Liouville
proved this in vol. x. of his journal, but such is not the case, the
subject being not even referred to in that volume.
PROUHET, E. (18-52)..[Me~moire sur queiques formules generales d'analyse. fou~rn.
(de Liouville) de, Math. (2), i. pp. 321-344.]
The third and last section Q~ 40-45) of Prouhet's memoir is
headed "The'ore'mes sur quelques determinants de fonctions."
The first theorem proved is that just mentioned as having been
used by Malnmsten. He then takes the set of rn +1 equations,
x,.dO(u~p0) + xl-d1(up0) + -... + xm.-dCP"(u50') - '1+1u0
x0.dlu0(1$) + x1.d1(u~p1) +... + x.-dm(u~pl) = dm+l(uP1)
x0.dlopmqlr) + x1.dl(aom) +..+ x,,dm(umI'7) -dm+l(u~pm),