0:
| a , b , c , d | | a', b', c'|
| a' , b' , c' , d'| | a", b", c"|
| a" , b" , c" , d"| | |

which, considering [delta] as standing herein for its value [alpha]x +
[beta]y + [gamma]z, is a consequence of the original equations only: we
have thus an expression for [alpha]x + [beta]y + [gamma]z, an arbitrary
linear function of the unknown quantities x, y, z; and by comparing the
coefficients of [alpha], [beta], [gamma] on the two sides respectively,
we have the values of x, y, z; in fact, these quantities, each
multiplied by

|a , b , c |,
|a', b', c'|
|a", b", c"|

are in the first instance obtained in the forms

|1 |, | 1 |, | 1 |;
|a , b , c , d | |a , b , c , d | |a , b , c , d |
|a', b', c', d'| |a', b', c', d'| |a', b', c', d'|
|a", b", c", d"| |a", b", c", d"| |a", b", c", d"|

but these are

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

or, what is the same thing,

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

respectively.

6. _Multiplication of two Determinants of the same Order._--The theorem
is obtained very easily from the last preceding definition of a
determinant. It is most simply expressed thus--

([alpha], [alpha]', [alpha]"),
([beta],[beta]',[beta]"),
([gamma],[gamma]',[gamma]")
+---------------------------------------+
(a , b , c )| " " " | =
(a', b', c')| " " " |
(a", b", c")| " " " |

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

where the expression on the left side stands for a determinant, the
terms of the first line being (a, b, c)([alpha], [alpha]', [alpha]"),
that is, a[alpha] + b[alpha]' + c[alpha]", (a, b, c)([beta], [beta]',
[beta]"), that is, a[beta] + b[beta]' + c[beta]", (a, b, c)([gamma],