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

128 HISTORY OF THE THEORY OF DETERMINANTS

Relying on this he proceeds to determine the quotient by
considering tbe cases where (1) ab =0=cd, (2) ab =0 =ac,
(3) ab =ac = ad = be = bcl = ed = 1, his result being
~~(ab)2 (bc)2 (ca)2 {f(da)2 +(db)2 +(dc,)2} {(ab )2 + (bc)2 + (ca)2}
-+ (cab)2 (bc)2 + ( bc)2 (ca)2 + (ca)2 (ab)2i
~ 21](ab)2(bc)2(cd)2(da)2(ac2
where there are four expressions under the first f and six under
the second; or
- (da)4 + (db)4 ~ (dc)4 +(b)+(c)}
E(ab)2 (b c)2 (ca)2 - (da)2 ~ (ctb)2 + (dc)2} {(ab )2 (bC2+ a2
L+ (da)2 (db)2 + (db )2 (dc)2 + (dc)2 (da)2
+ (ab)2 (bC)2 ~ (b c)2 (ca)2 ~ (ca)2 (ab )2j
where there are four expressions under the f, one corresponding
to each face.
Although there is no explicit mention here of determinants, it
being unnecessary, it has now to be noted that Sylvester had the
determinant-form of W before him throughout: he even says
that he had tried to express the quotient as a determinant, but
had been unsuccessful. Without further restriction, then, as
to form, his proposition is If F, G, II, K be the complementar'-y mninors of the elements in the places 11, 22, 33, 44 of
the determinant
(ab)2 (ac)2 (ad)2 1
(ab)2. (bc)2 (bd )2 1
(ac)2 (bc0)2. (cd)2 1 or 2W say,
(ad)2 (bd )2 (cd)2. 1
then the resu~lt of rationalising
its divisible'by W. This is not all, however; for Sylvester
having noted the analogous case connected with the relation