The theory of determinants in the historical order of development, by Sir Thomas Muir.

LINEAR EQUATIONS (TRUDI, 1862)

85

first deduction. A proof is also given of the converse of one of
the theorems regarding the eliminant, namely, that if
a11 a 112 ~ a,,ln U1
a21 a 2. a2 U2 - n
n+l,1 n-i+1,2 * * n+l,n 1 ln+l
then any one, say the last, of the equations
r=l
is a consequence of the others; the procedure being to perform on
the given vanishing determinant the operation,
col0+ - (xIol + x2col2+... + X2 coln),
with the result
{ ul,+ - (a,,+1,1 + a=+1,2X2 +.. + an+1.x-) } aln =0,
after which we have only to assume I al = 0.
Lastly, attention is drawn to the simple fact that if we have
n linear equations
alx + a2y + a3z +a4 = 0
bx+ b2y + b z+ b4 = 0
c1x+ c2y + c3z+ c4 = 0
determining n unknowns x, y, z, then any other linear, function of
the same unknowns, D1x+D2y+D3z+D4, is readily expressible in
terms of the coefficients, namely, is equal to
I a1b2c3D4 IJ. a3b2c3I
BALTZER R. (1864).
[Theorie und Anwendungen der Determinanten. 2te verm.
Aufl. vi+224 pp. Leipzig.]
Baltzer now notes (~ 8. 2) how the solution of n non-homogeneous
equations in n unknowns is deducible from the solution of n homogeneous equations in n+1 unknowns, without, however, referring
to the possibility of the reverse procedure. He also gives the
second half of Kronecker's contribution already mentioned (p. 14),
namely (~ 8. 3, iii.), that if the minor a a... akk I of I a,,... a,,, |