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

112 HISTORY OF THE THEORY OF DETERMINANTS

CUNNINGHAM, A. (1874): ROBERTS, S. (1874).
[An investigation of the number of constituents, elements, -and
minors of a determinant. Quart. Journ. of Sci., (2) iv. pp.
212-228.]
[Question 4392. Educ. Times, xxvii. pp. 45, 66; or Math. from
Times, xxi. pp. 81-83.]
Cunningham devotes ~~ 8-10 (pp. 220-224) to the number of
terms in devertebrate and vertebrate axisymmetric determinants.
There is an oversight, however, in his reasoning, and his results
are correct only as far as the fifth order. He concerns himself also
(~ 14) with the number of k-line minors in an n-line axisymmetric
determinant.
Roberts' difference-equation for the number of distinct terms is
2 ~ U,-3+(n-3)U,-4+ 0,
Un - L1- (n-1) l-2 + '( -l)(n-2){ u_3+(n-3)_4 = 0,
and Cayley establishes it by taking his own, namely,
Un -- U_1 + — (n-1)(n-2)u-3 = 0
or, say, En = 0,
and showing that the other is
E,+(n-1)E,_- = 0.
The first eight values of u, he finds to be
1, 2, 5, 17, 73, 388, 2461, 18156,....
Roberts himself identifies the problem with that of finding "the
number of distinct ways in which 2n things, two of a sort, can be
made into parcels of 2."
PAINVIN, L. (1874)..[Note sur la methode d'elimination de Bezout. Nouv. Annales
de Math., (2) xiii. pp. 278-285.]
A clear exposition of the mode of obtaining the axisymmetric
"eliminant of two equations in x.