Encyclopaedia Britannica, 11th Edition, "Destructors" to "Diameter" / Volume 8, Slice 3

by Various

it appears that this is

= x

a,

b,

c

+ y

b,

b,

c

+ z

c,

b,

c

-

d,

b,

c

;

a′,

b′,

c′

b′,

b′,

c′

c′,

b′,

c′

d′,

b′,

c′

a″,

b″,

c″

b″,

b″,

c″

c″,

b″,

c″

d″,

b″,

c″

viz. the second and third terms each vanishing, it is

= x

a,

b,

c

-

d,

b,

c

.

a′,

b′,

c′

d′,

b′,

c′

a″,

b″,

c″

d″,

b″,

c″

But if the linear equations hold good, then the first column of the original determinant is = 0, and therefore the determinant itself is = 0; that is, the linear equations give

x

a,

b,

c

-

d,

b,

c

= 0;

a′,

b′,

c′

d′,

b′,

c′

a″,

b″,

c″

d″,

b″,

c″

which is the result obtained above.

We might in a similar way find the values of y and z, but there is a more symmetrical process. Join to the original equations the new equation

αx + βy + γz = δ;

a like process shows that, the equations being satisfied, we have