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

SKEW DETERMINANTS (BALTZER, 1857)

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(- 1)'-1 or. indeed (- 1)r-1, instead of (-1)r for the sign-factor
of (2,3,..., r-1, r —1,..., n).
The very next step taken, in accordance with the above
mentioned dictum, is to make the substitution in the right-hand
side of the equation
VJa = a1,~JA,2 + aA2 Aa3 +,,/A3.. ~ c+l?<Ajn,,,
the first term being used to decide whether (1,2,3, n) or
- (1,2,3,..., n,) has to be substituted for the left-hand side, and
the final result being
(1,2,3,...,Ln) = an(3,..., 1)~ a13(4, V 1, n,2)... +ca,,,(2,...,n- 1).
Since (3,4..., n) is the, cofactor of a,9 in (1,2,3,..., n) and
the differential-quotient of the latter with respect to a12 is the
same, it immediately follows from this that
Va 2a,-a +a,3 + 6a + a1
12 a,2, -aa,3 aLI
Baltzer, however, obtains a more general result by going back to
the corresponding more general theorem in determinants, viz.,
the theorem
A = arilA,r + a,.2A,.m +... + a A
with which he associates
0 = carA.1 + a3,2A.2 +. I b. + c a,
substituting A A f6r~ Aa; and then dividing both sides by
Va. ~~In the results,
VA = arm +... + ar,
0 C= r c~VA ___, CL
it has to be noticed that there is no term in DVaa, a,..
By comparison of the first of these with the immediately
preceding result (the recurring law of development) he dednces
the quite general identity regarding the two forms of the