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

CONTINUANTS (PAINVIN, 1858) 433
will suffice. Increasing each element of the first row by the
corresponding elements of the other rows,-an operation which
as before we may symbolise by
rowl + row + row......,
-he removes the factor r+ 3a -3 and finds left the cofactor
1 1 1 1
3(a-1) r-1 2a
2(a-1) r-2 3a. c~a-1 ' —3 ~
On this are performed the operations
col-col2, col2-col3, col3-col,.....
the result being a determinant of the next lower order
3a-r-2 r-2a-1 2a
- 2-2ca 2a-r r-3a-2
1- a a-r+2
Finally, after changing the signs of all the elements here, the
operations
row1 + rows + rows +...,
row2 + row3+.,
row3+.....
are performed, the result
r -a a
2(a-1) r-a-1 2a
a-1 r-ca-2
being a determinant exactly similar in form to the original, but
with r-a instead of r. This, therefore, in turn may be transformed into
(r+a-2) - 2a
a-1 r-2a-i,
and so on.
The value thus obtained for the above-written determinant of
the (n +l )th order is
(r + a-n)(r+na-n- 2a+ 1)(r+n a-n- 4a+ 2)... (rv-na),
M.D. II. 2 E