1917-18.] Determinants of a 4-by-8 Array. 
219 
XVIII. — The Quadratic Relations between the Determinants of a 
4-by-8 Array. By Sir Thomas Muir, LL.D. 
(MS. received May 14, 1918. Read June 3, 1918.) 
1. The theorem which expresses the product of two determinants as a 
sum of like products is the main instrument for investigating the relations 
between the determinants of an oblong array. It is justifiably spoken of 
as “ Sylvester’s theorem of 1839 and 1851,” although simple cases of it date 
back to far earlier times ; for example, the case 
\af 2 c 3 \ \d x ejf = K Ms! \ c i e 2fs \ “ \ a i c 2 d s \ I V2/3I + bW'^l KVsl 
which was brought to light by Bezout in 1779.* 
2. The most interesting array, and the one that has come in for most 
attention, is that having n rows and 2 n columns, — in other words, the 
array formed by the juxtaposition of the arrays of two n-line determinants. 
3. Much of the difficulty experienced in dealing with the relations con- 
necting the determinants of such an array has been due to the want 
of a suitable notation for the binary products involved ; and the main 
object of this note is to draw attention to a notation that promises to be 
helpful. 
4. Taking the case where n is 4, say the case of the array 
«i b i c i A e i A 9i \ 
a 4 64 c 4 d 4 e 4 f\ g 4 \ 
we readily find that the full number of binary products 
I «A C S ' d i I I e i Mh ! > I a i h 2 C ^ I I d lf2%\ b • ■ * 
is 35, and the problem has been how to specify these 35 by a short, 
natural and suggestive notation. The solution here proposed rests on the 
fact that 35 is the sum of the four squares 1, 9, 9, 16, and that the 
double-suffix notation thus becomes available. 
* A curious fate lias attended this theorem of recent years. Although it is of funda- 
mental importance, and used to be so considered by the earlier writers of text-hooks like 
Brioschi and Baltzer, we have the unfortunate fact now facing us that no mention of it is 
made in any of the students’ manuals at present on sale by publishers throughout the 
world. For proof see Scott in English, Pascal in Italian, Kowalewski in German, am"! 
Dostor in French. 
