P. HISTORY OF ' AATHEMATICAL - School of Mathematics

100 A HISTORY OF MATHEMATICAL NOTATIONS
Salmon' ordinarily uses Cayley's two vertical bars, but often
"for brevity" writes (aI, b 2 , ca .••• ) where aI, b 2 , ca, .•.. , are elements
along the principal diagonal, which resembles designations
used by Bezout and Jacobi.
E. H. Moore- makes the remark of fundamental import that a
determinant of order t is uniquely defined by the unique definition of
its t 2 elements in the form au" where the suffixes uv run independently
over any (the same) set of t distinct marks of any description whatever.
Accordingly, in determinants of special forms, it is convenient to
introduce in place of the ordinary 1, 2, .... , t some other set of t
marks. Thus, if one uses the set of t bipartite marks
~ : 1 , ,m)
gj ( J -1, , n
and denotes au. by ajihk, the determinant A = lajihkl of order mn,
where throughout
_ b(i)
(h)
(
~ ' h: 1, , m) ,
ajihk - jh • Ci k z, k-l, , n
is the product of the n determinants B(iJ of order m and the m
determinants C(h) of the order n:
A ... s» .... B(n) • C(l) .... C(m) , where B(i) = Ib }Q I' c» = I c~Z) I.
Kronecker introduced a symbol in his development of determinants
which has become known as "Kronecker's symbol," viz., 5 i k,
i were h k = 1,2,
1 .2
.... , m
,an
d. 0 h ·>k· 1 Th al
Uik = W en z< , Ukk =. e usu
, J •••• J n
notation is now 5;; Murnaghan writes [;]. A generalization of the
ordinary "Kronecker symbol" was written by Murnaghan in 1924 4
in the form {I, ..... , r m (m:::; n, in space of n dimensions), and in
S1, ••••• , Sm
1925 6 in the form [r i r2 •••. rm], and appears in the outer multipli­
8182 •••• 8 m
cation of tensors.
1 George Salmon, Modern Higher Algebra (Dublin, 1859, 3d. ed., Dublin, 1876),
p. 1.
• E. H. Moore, Annals of Mathematics (2d ser.), Vol. I (1900), p. 179, 180.
• Leopold Kroneeker, Vorlesungen tiber die Theone der Determinanten, bearbeitet
von Kurt Hensel, Vol. I (Leipzig, 1903), p. 316, 328, 349.
4 F. D. Murnaghan, International Mathematical Congress (Toronto, 1924);
Abatracts, p. 7.
'F. D. Murnaghan, American Math. Monthly, Vol. XXXII (1925), p. 234.