The Product of Two Ohlong Arrays. 
17 
the expression of a combinatorial as a sum of products of pairs of 
combinatorial, namely, 
(4) The single determinant used at the end of §1 as the equivalent of a 
product of two oblong- arrays is historically interesting. Its counterpart, the 
similar expression for two square arrays, was first used by Spottiswoode in 
1853 in bringing forward* Sylvester's theorem of the year before in regard 
to the multiplicity of form of the product of two determinants ; and as we 
have pointed out elsewheref an extreme case of it must have been used by 
Sylvester himself, namely, the case 
i ai 0-0 
^5 
H 
(Xo . . 
• «5 
«1 
^2 
a-^ 
\hh ■ 
■ h , 
ft • 
■ ^5 
h 
hi .. 
h 
! ^1 ^2 
71 
72 • 
• «5 
^1 
.. 
H ft 
7i 
-1 
72 
-1 .. 
fh y, -1 
With this before us it is important to note that this expression for the 
product of two oblong arrays and the Binet-Cauchy expression of 1812 are 
the two extremes of a series of such expressions, and that the one used by 
us at the end of §1 is an example of the third of the series. + 
Cape Town, S.A. ; March 28, 1917. 
* Crelle's Journ., li, pp., 238-248. 
t Hist, of Dets., ii, pp., 199-200. 
X other related papers are : 
MuiR, T., "On a determinant formed by bordering the product of two deter- 
minants/' Messenger of Math., xi (1882), pp. 161-165. 
Nanson, E. J., "On partial compounds," Messenger of Math., xxvii (1898), 
pp. 17-19. 
