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

COMPOUND DETERMINANTS (CAYLEY, 1865)

181

Again, by going through three similar operations, the sets of columns
used being 156, 426, 453, we obtain on addition
( —k4156-X1+ 2156'X2+ 3156-X3)
+( 1426X1 —k-5246-X2+ 3426-\3)
+( 1453-X,+ 2453*X2 —k6453'X3) = 0,
which, with the help of our second result, can be cleared of k and
becomes
(1426+1453)X1 + (2156+2453)X2 + (3156+3426)X3 = 0.
The desired resultant equation thus is, as in Sylvester's paper of
1863,
4123 1426+1453 1456
5123 2453 +2156 2456 = 0.
6123 3156+3426 3456
For the next higher case of the problem the corresponding equation
is also given.
REISS, M. (1867).
[Beitrage zur Theorie der Determinanten. viii+113 pp. Leipzig.]
The second section (pp. 25-54) of this important memoir is devoted
to compound determinants whose elements are equigrade minors
having for each row one invariable set of row-numbers, and for each
column one invariable set of column-numbers. To this special
class belong almost all the compound determinants hitherto met
with. From the definition it is evident that every member of the
class can without risk of mistake be specified by giving its diagonal
term: for example, we can safely put
13 13 13
24 46 62
131 35 51 1 or 35 35 35
24 46 62 24 46 62
24 46 62
and
234 134 124 123 orthe djugate of j 1234
234 134 124 123 frteajgeof1234