(15) 
NOTE ON AN EXPANSION OE THE PEODUCT OF TWO 
OBLONO ARRAYS. 
By Sir Thomas Muir, LL.D. 
(1) The long-known expansion of the product of two oblong arrays takes 
the form of a sum of products of pairs of determinants. The expansion 
here brought forward takes on the other hand the form of an aggregate 
of single determinants. Of these last the first is itself a product-determinant, 
and through being bordered gives rise to all the others. Taking the case of 
two 3 -by- 5 arrays we have as an example of it 
r ^1 ••• I 
^' 7i72 ••• 73 !i 
j Pi Po P, a. Pi P. P3 a, a, 
j Qi Q3 Qs ^5 + Qi Q3 Q3 ^4 ^5 
I Rj Ro R3 C5 I Ri Ro R3 Cg 
i «5 75 - I ^4 /^i 74 • • 
»5 7:5 • • 
where jP^QoRof is the product of kinfi^c.^l and |ai/5^,y3|. Had the number 
of given columns being six instead of five, there would have been four 
additional terms in the expansion, all of them involving the elements with 
the sufiix 6, one like those already obtained of the 4^^^ order, two like those 
of the 5^'' order, and one of the 6^^ order namely, 
Pi P2 «'4 «'5 
Qi Q2 Q3 ^4 ^r, 
Ri Ro R3 c.- c,i 
a- ttg . 
/^4 A3 ^ • . . 
74 75 76 • • • • 
1 tto ... a-^ 
hi ho ... h~ 
Co ... c~ 
1 -^2 -^3 
Qi Q2 Q3 
Rj Ro R3 
-^1 -^2 -^3 " 4 
Ql Q2 Q3 ^4 
R^ Ro R3 c_|^ 
7, . 
