e""

it is further seen that the factor which multiplies the determinant
formed with any two columns of the first set is the determinant of the
third order formed with the complementary three columns of the second
set; and it thus appears that the determinant of the fifth order is a
sum of all the products of the form

= |a , b | |c" , d" , e" |,
|a', b"| |c"', d"', e"'|
|c"", d"", e""|

the sign +- being in each case such that the sign of the term +-
ab'c"d'"e"" obtained from the diagonal elements of the component
determinants may be the actual sign of this term in the determinant of
the fifth order; for the product written down the sign is obviously +.

Observe that for a determinant of the n-th order, taking the
decomposition to be 1 + (n - 1), we fall back upon the equations given
at the commencement, in order to show the genesis of a determinant.

8. Any determinant |a , b | formed out of the elements of the original
|a', b'|
determinant, by selecting the lines and columns at pleasure, is termed a
_minor_ of the original determinant; and when the number of lines and
columns, or order of the determinant, is n-1, then such determinant is
called a _first minor_; the number of the first minors is = n squared, the
first minors, in fact, corresponding to the several elements of the
determinant--that is, the coefficient therein of any term whatever is
the corresponding first minor. The first minors, each divided by the
determinant itself, form a system of elements _inverse_ to the elements
of the determinant.

A determinant is _symmetrical_ when every two elements symmetrically
situated in regard to the dexter diagonal are equal to each other; if
they are equal and opposite (that is, if the sum of the two elements be
= 0), this relation not extending to the diagonal elements themselves,
which remain arbitrary, then the determinant is _skew_; but if the
relation does extend to the diagonal terms (that is, if these are each =
0), then the determinant is _skew symmetrical_; thus the determinants

|a, h, g|; | a , [nu], - [mu]|; | 0, [nu], - [mu]|
|h, b, f| |- [nu], b,[lambda]| |- [nu], 0,[lambda]|
|g, f, c| | [mu],-[lambda], c | | [mu],- [lambda], 0|

are respectively symmetrical, skew and skew symmetrical:

The theory admits of very extensive algebraic developments, and
ap