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

LESS COMMON SPECIAL FORMS (SYLVESTER, 1855) 457
and then multiply together the elements of the diagonal, rejecting every term such as abba, abb,.... in which the letters form
a cycle. Two examples are given, but no justification of the
"rule" is vouchsafed. The examples area+ab+ac -ab -a = abc +(ca, +cb)ab+(ab+ac)bc+(bc+ba)ca.
-ba b +b+b, -b +a(baca + baCb+ Cabc)
-Ca -Cb C + Ca+ Cb + b(bab + Cb + abCa)
+ c (acbc + aeba + bcab),
a+ ab+ +ad - ba - Ca - da
-ab b+be bd + ba C - -db
- aC - be c +Cd a + Cb - dc
-ad - bd -Cd d+da+db+dc
= abed + (d db+ + a(b(Cdda+....)+ (bcdda+....)
The arrangement of the two developments almost raises doubts
as to whether the "rule" had been utilised, suggesting indeed
that in the latter instance, for example, the cofactor of ab was
first obtained in the form
Cd+Ca+Cb -de
-Cd da+db+dc
and the cofactor of a in the form of a similar determinant of the
third order. The " rule," however, is noted by Cayley in Crelle's
Journal, lii. (1855), p. 279.
The number of terms is (n+ 1)n-1, n being the order-number of
the determinant. This Sylvester obtains by putting a, a,, ac,...
all equal to 1. It will be observed that from the form of the
development we thus have
1 + 32 + 3-3 = 42
1 + 4-3 + 6-8 + 4-16 = 53
I + 5-4 + 10-15 + 10-50 + 5-125 = 64