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

122 HISTORY OF THE THEORY OF ]DETERMINANTS

WOLSTENHOLME, J. (1878).
[MATHEMATICAL PROBLEMS... 2 nd edition. x+480 pp. London.]
Under No. 1627 (1),~ (2), Woistenholme gives
-2a a+b a+c
b+a -2b b+c 4(b+c)(c+a)(a+b),
c-Ia c+b -2c
and Ferrers' result of 1876, the latter of which includes No. 1627 (4).
No. 1630, so far as it is correct, is identical with a result, of Caldarera's
of 1871, and is included in one of Baltzer's of 1864. No. 1633
asserts that
1 ~~COS a COS (a n cos (a + +)
COS a 1. COS 3 COS (~3~y
COS (a~) +cos/3 1 cosy
COS (a~+/3 + y) co(i3+) COSy 1
and all its primary minors vanish, a fact which we may verify for
ourselves by showing that it equals
sin2 a.sin2( 3sn(+ /1~). 1
COS a 11
COS (a~) 1
COS (a~+ 3 +)11 K
SYLVESTER, J. J. (1879).
['Note on determinants and duadic disynthemes. American Journ.
of Math., ii. pp. 89-96, 214-222; or Collected Math. Papers,
iii. pp. 264-280.]
This interesting paper, which is based unconsciously on Cauchy's,expression of a substitution as a quasi-product of circular substitu-tions (Hist., i. pp. 101, 304-305) deals with the number of terms in
axisymmetric and skew determinants. In addition to previously
known results he asserts (p. 93) that if Un be the number of distinct,terms in an 'invertebrate' (i.e. zero-axial) axisymmetric determinant,,then
U,, =- (n -l)(Un-I+Un-) - 1n )n2U-,,.thus giving 0,1, 1, 6, 22, 130, 822.
-for the first seven values of un.