496 
Proceedings of Royal Soeiety of Edinhurgli. 
From each array, by taking every set of n columns,- form deter- 
minants, arranging them in any order, provided it be the same for 
all the arrays. Add together all the 1st determinants formed from 
the first s arrays, and multiply the sum by the corresponding sum 
for the second s arrays ; obtain the like product involving all the 
2nd determinants, the like product involving all the 3rd determinants, 
and so on. Then, the sum of these products is equal to the sum 
of the products obtained by multiplying each array of the first set 
by each array of the second set. 
Or we may put it alternatively as a formal proposition, thus : — 
If s rectangular arrays he taken, each luith m elements in the row 
and n elements in the column, m being greater than n, viz. 
Xp X^, , X, 
and other s rectangular arrays of the same kind, viz.. 
UJ ^2’ 
and if the minor determinants of the order formed from X^, X 2 , 
then 
s 
^11 
^12 • • 
. Xic 
fii 
^12 • • 
’ ^1,C 
^'21 
^22 • • 
, . i^a.c 
^21 
^22 • • 
• ^2,C 
Xgi 
Xg2 • • 
4. 
i . . 
• ^s,G 
+ X21 + . . . + (^11 + ^21 + • 
4 - {x-^2 + ^22 + • • • + if 12 + ^22 d" • 
+ 
• +^«i) 
• +^^ 2 ) 
+ {X^^Q + ^2C + • • • + ^s,c) (^l.C + ^2C • • • + ls,c) 
= (X 1 + X 2 + . . . -f X,) (E 1 + H 2 + . . . +H,) 
where C stands for ^.e., m{m - 1 . . ... {m-n-¥ 1)/1.2.3 . ... n. 
Now, counting the terms here as we did in the case of Cauchy’s 
theorem, we see that on the left-hand side there are C multiplica- 
tions to be performed, each giving rise to 5 x s terms, and that there- 
fore the full number of terms in the development of this side is 
s 2 C; 
also that on the right-hand side the number is 
