. It is to be noticed that the
resulting equation is

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

where the expression on the right-hand side is the like function with d,
d', d" in place of a, a', a" respectively, and is of course also a
determinant. Moreover, the functions b'c" - b"c', b"c - bc", bc' - b'c
used in the process are themselves the determinants of the second order

|b', c'|, |b", c"|, |b, c |.
|b", c"| |b, c | |b', c'|

We have herein the suggestion of the rule for the derivation of the
determinants of the orders 1, 2, 3, 4, &c., each from the preceding one,
viz. we have

|a| = a,

|a, b | = a|b'| - a'|b|.
|a', b'|

|a, b, c | = a|b', c'| + a'|b", c"| + a"|b, c |,
|a', b', c'| |b", c"| |b , c | |b', c'|
|a", b", c"|

|a, b , c , d | = a|b', c', d' | - a'|b" , c" , d" | +
|a', b' , c' , d' | |b", c", d" | |b"', c"', d"'|
|a", b" , c" , d" | |b"', c"', d"'| |b , c , d |
|a"', b"', c"', d"'|

+ a"|b"', c"', d"'| - a"'|b , c, d |,
|b , c , d | |b', c', d'|
|b' , c' , d' | |b", c", d"|

and so on, the terms being all + for a determinant of an odd order, but
alternately + and - for a determinant of an even order.

2. It is easy, by induction, to arrive at the general results:--

A determinant of the order n is the sum of the 1.2.3...n products which
can be formed with n elements out of n squared elements arranged in the form of
a square, no two of the n elements being in the same line or in the same
column, and each such product having the coefficient +- unity.

The products in question may be obtained by permuting in every possible
manner the columns (or the lines) of the determinant, and then taking
for the factors the n elements in the dexter diagonal. And we thence
derive the rule for the signs, viz. considering the primitive
arrangement of the columns as positive, then an arrangement obtained
therefrom by a single interchange (inversion, or derangement) of two
columns is regarded as negative; and so in general an arrangement is
positive or negative according as it is derived from the primitive
arrangement by an even or an odd number of interchanges. [This implies
the theorem that a given arrangement can be