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

366 HISTORY OF THE THEORY OF DETERMINANTS

equations may be calculated, and he prints at full length the
resultants R2,2, R3,3, R4,,, that is to say, the final expansions of
a b c
a b c
p q r
p q r.,etc.,
the arrangement of the terms being such as to make evident the
fact that each resultant is unaltered by reversing the order of
the two sets of coefficients of which it is a function: for
example:-'
R2,2 = (a2r2+c2p2) - (abqr+bcpq) - 2acpr + acq2 + b2pr.
CAYLEY, A. (1856).
[A second memoir on quantics. Philos. Transac. R. Soc. (London),
cxlvi. pp. 101-126; or Collected Math. Papers, ii. pp. 250 -275.]
In addition to previously calculated discriminants Cayley gives
(No. 26) the discriminant of the binary quintic,-that is to say,
the eliminant of
ax4 + 4bx3y + 6ccx2y + 4dxy3 + ey4 = 0 t
bx4 + 4cx3y + 6dx2y2 + 4exy3 + fy4 = J
CAYLEY, A. (1856, December).
[Miemoir on the resultant of a system of two equations. Philos.
Transac. R. Soc. (London), cxlvii. pp. 703-715; or Collected
Math. Papers, ii. pp. 440-4.53.]
As the resultant, R3,2 say, of the pair of equations
ax3 + bx2y+cxy2 + dy3 = 0, px +qxy + ry2 = 0,
is homogeneous and of the 3rd degree in the coefficients of the
second equation, and at the same time homogeneous and of
the 2"n degree in the coefficients of the first equation, Cayley
* The expression for R4,4 is full of inaccuracies.
In the reprint, p. 288, the coefficient of b4cef2 should be - 1920, not + 1920.