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

BIGRADIENTS (TRUDI, 1862)

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and that in addition we have
(aC, a,,..,..., a s, s, S) = 0,
(a0, a1,..., aC n+l, Sn,..., ) = 0,
(ac LL, * *, (,nSn+2' Snal, ' ' ', S2) a = 0.
The results arrived at are
Dr = ao'+l so 81 S2.. ' 8r-1 8
81 82 83.... 81. Sr+1
82 83 84...,+1 Sr+2
~~.......... ~r+l
= 0 when r >- 1,
' 2 S3 S4 * - 8 * 8,+1 I
=V. - ao~" 80. 81.....,._ 1
1 82 4.... s.. x
S8 8 4... 3+' 2.+12
82 3 84. r. 8+1 +.........* +1,
Here again, however, Trudi loses his opportunity from not being
acquainted with Cayley's multiplication-theorem of 1845, the use of
which enables us to transform not only D,., but the whole bigradient
array of which D. is the first determinant. In fact, it gives us for
the case under consideration another condensation-theorem. For
example, when
and we consequently have to consider the four 'simplified
remainders'
(c0,...b., b ) (b2,. ),,1), (ao....
(ao a*. ^)2 (a, a, 1,)
(b.., b4)4 (b0 * *. 6),