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

LESS COMMON SPECIAL FORMS (CAYLEY, 1854) 453
equations involving it must of course lead us back to some
form or other of the equation with which we started. Thus,
multiplying K by a and using the first of the four derived
equations we obtain by subtraction
0 = (a/ + 8)(y'- ')(a'- ')
- (ay + )(' - ')(a' -y') + (ad+^,3)(' -y')(a'- '),
whence we deduce in the same manner as before that the
expression * on the right when changed in sign is equal to 4;
and using any pair of the remaining equations we reach either
the form of 4t previously obtained or one of the two forms
derivable from it by means of the simultaneous circular
substitutions
fy, 8 = y, 8, 3,
p3, y', ' = y', 8', 3'.
CAYLEY, A. (1858, February).
[A fifth memoir on quantics. Philos. Transac. R. Soc. (London)
cxlviii. pp. 429-460; or Collected Math. Papers, ii. pp. 527 -557.]
The second part (~~ 96-114) of the memoir deals with two or
more quadrics, and forming part of it is a digression (~~ 105-114)
on involution and the anharmonic relation. The determinant 4r
thus again makes its appearance, and associated with it is the
determinant
1 a +a' aa'
1 13+t' A 3' or Y say,
1 3+/' y',
for the reason that, when 8 = a' and ' = a, 4t is readily shown to
be equal to
(a'-a)Y.
* This second form of ' may be got directly from the determinant by expanding
in terms of the two-line minors formable from the first and third columns, and
the minors complementary to these. Of course we also have
v = (a'' + y'a') (a -) (y - d) - (a'y'+/P'')(a -)( - 8) + (a'' + 'y') (a- ) (Q -y).