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

354 HISTORY OF THE THEORY OF DETERMINANTS

we have
S5 S8 S7 3 2 1 a5 b5 c5
86 s7 88 = 3a 2b c. a6 b6 c6
87 88 s9 3ac2 2b2 c2 a7 b7 c7
— 21 5~bl ~ C
= 6 a 0bc2 aC5b5c1 blc
-o65b5C5 blc2 12
The other part of the proposition rests on the statement that a
similar procedure leads in that case to factors having at least one
column of zeros.
The case where the number of rows is less than the number of
different roots is not considered; but the first part of the proposition is used to obtain the modification which it is possible to
make in the relation
Sn, + aClS,+r-1 + Ca2S n+.r-2 +...+ ans} = 0
when the roots cease to be all different.
ROUCHE, E. (1858, Dec.).
[Sur les fonctions Xn de Legendre. Comptes rendus... Acad.
des Sci. (Paris), xlvii. pp. 917-921.]
Xn being the nth differential-quotient of (x2- 1) Rouche first
proves that every rational integral function of x of the nth degree
xn+alxn-l+... + a-_xC+ a,,, or V, say, which for all integral
values of k < n satisfies the equation
+1
"Vn dx = 0
-1
does not differ from X, save by a constant multiplier. To
determine Vn,-in other words, to determine its coefficients
a1, a...., an,-there is thus available the set of n equations
YVndx = 0, xV ndx =0,... -, x1lndx = 0,
-1 -1 1 -