[gamma]', [gamma]"), that is a[gamma] + b[gamma]' + c[gamma]"; and
similarly the terms in the second and third lines are the life functions
with (a', b', c') and (a", b", c") respectively.

There is an apparently arbitrary transposition of lines and columns; the
result would hold good if on the left-hand side we had written ([alpha],
[beta], [gamma]), ([alpha]', [beta]', [gamma]'), ([alpha]", [beta]",
[gamma]"), or what is the same thing, if on the right-hand side we had
transposed the second determinant; and either of these changes would, it
might be thought, increase the elegance of the form, but, for a reason
which need not be explained,[2] the form actually adopted is the
preferable one.

To indicate the method of proof, observe that the determinant on the
left-hand side, _qua_ linear function of its columns, may be broken up
into a sum of (3 cubed =) 27 determinants, each of which is either of some
such form as

= [alpha][beta][gamma]'|a , a , b |,
|a', a', b'|
|a", a", b"|

where the term [alpha][beta][gamma]' is not a term of the
[alpha][beta][gamma]-determinant, and its coefficient (as a determinant
with two identical columns) vanishes; or else it is of a form such as

= [alpha][beta]'[gamma]"|a , b , c |,
|a', b', c'|
|a", b", c"|

that is, every term which does not vanish contains as a factor the
abc-determinant last written down; the sum of all other factors +-
[alpha][beta]'[gamma]" is the [alpha][beta][gamma]-determinant of the
formula; and the final result then is, that the determinant on the
left-hand side is equal to the product on the right-hand side of the
formula.

7. _Decomposition of a Determinant into complementary
Determinants._--Consider, for simplicity, a determinant of the fifth
order, 5 = 2 + 3, and let the top two lines be

a , b , c , d , e
a', b', c', d', e'

then, if we consider how these elements enter into the determinant, it
is at once seen that they enter only through the determinants of the
second order |a , b |, &c., which can be formed by selecting any two
|a', b'|
columns at pleasure. Moreover, representing the remaining three lines by

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