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

316 HISTORY OF THE THEORYV OF DETERMINANTS

where A 0 a, Al a,,
A2- ao a1 A 3 a, a2,
a, a2, a2 a3~
A4 ao a, a2 A 5 a, a2 a3,.
a, a2 a3 a2 a. a4
a2 a3. a,4 a3 a, a,,
which is what Hankel would have been led to had he gone a little
farther.
SYLVESTER, J. J. (1862).
[Sur une classe nonvelle d'e6quations diffhrentielles et d'e'quations
aux differences finies d'nne forme integrable. Comptes Rendus...Acad. des Sci. (Paris), liv. pp. 129-1.32; or Collected Math.
Papers, ii. pp. 308-312.]
The existence of the equation
UX PiUX+i +P2UX~2 P3UX~3 +UXT4 = 0
where P.' ~P2, p3 are constants, implies that
WUX UX>+1 Ux+2 Ux+3
UX+ Ux1+2 Ux43 Ux+4
Uw+ UX+3 UX+4 U+
ux+3, UX+4 UX~5 uX+G
is independent of x; for if we perform the operation
row, - p1row2 + p2 row3 - p3 row4
and make use of the said equation, the determinant is simply
changed into
Ux+4 UX+5 ~t+ U
UX+4 Ux+2 UX+ ~3 +
Ux+2 Ux+3 Ux+4 UX+S
U 43 X+4A UX+ UX
-in other words, we learn that the given determinant is not
altered by substituting xr+1 for x. From this Sylvester concludes
that the difference-equation