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

DETERMINANTS IN:GENERAL (BATTAGLINI, 1862) 11
the theorem suggested by Routh (1861). Thus, the sum of the
signed primary minors of j a,b~c~d4 I being known to be
1 a1 a2 a3 a4
1 b1 b2 b2 b3
1 C1 C2 C3 C4
1 d, d, d3 d4 K
we would only have to perform the operations
col2 - col3, col3- coI4, col4- col5,
row2-rOW3, rOW3-row4, row 4- row5,
to obtain
a,- a2-b,~b2 a2-a3-b2~b3 a3- a4-b3+ b4
b1- b2- c1 + c2 b2- b3-c2+c3 ' b3- b4-c3 + c4
c., - Cz2- d.,+ d,,-c - d2 d c-c- dc j
c~-c2-d1+2 C2- C3-d+d3 c3 - c4-d3+d4
which is Battaglini's form. It may also be pointed out that the
elements in the latter form are three-line minors of the former.
SMITH, H. J. S. (1862).
[Report on the theory of numbers. Part IV. Report.... British
Assoc.... xxxii. (Cambridge), pp. 503-526; or Collected
Math. Papers, i. p. 229.]
In ~ 105, p. 504, Smith denotes a determinant, whose elements
are double-suffixed, by placing between two upright lines a single
element with variable suffixes, and then appending an indication
of the extent of the variability. For example, the determinant
'ax DY2 axn ax_ Dy2 ODyb
he writes
DYa
O = U 1, 2,..,n
Dx,
Four years later Kronecker made similar use of I bi. (See
Crelle's Journ., lxviii. p. 276.)