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

82 HISTORY OF THE THEORY OF DETERMINANTS

essentially consists in expanding the right-hand determinant in
terms of minors formed from the first three rows and minors
formed from the last two rows. His other fresh proof is
dependent on the connection between determinants and simultaneous linear equations. Taking the two sets of equations
ax a/ + ay + z = u aul - bu - c. - 71
aX+ y +(yX+ am= 12 |1 1 +b 2 + C =3 = V1
3X + f3Y + Z= %2 (t1 + b'm2 + Cl3 = V 2
yx + y'y + a" + b"2 + c"3 3= 3
and substituting for u1, u21, u3, in the second set and solving,
there is obtained for x an expression whose denominator is
known to be
a b c a aa a
a' b ' C. dc ' /3"
a" b " c y y y"
In the second place, by substituting for uin only there is obtained
aax + aa'y +aa"z + bu2 +cu3 = v1
a'ax + a'a'y + a'a"z + b'Vu2 + c' t3 = v
a"aax + a"a'fay + a"a"z + b"ug2 + C"n c3 = V
3xe+ /3'+ "z- U2 = 0
yx+ 7'y+ 7y" - 3 O J,
whence comes for x an expression whose denominator is
aa aa' aa" b c
a'a a'a' a'a" b' c'
a(a a a a"a" b" c"
3 d' 3" -1
y V y. -1.
A comparison of the two denominators is supposed to establish
the desired result; but, although the dropping of the two
negative units in the five-line determinant is quite justifiable, no
allusion is made to it. It may be added that Sylvester's umbral
notation is used throughout in dealing with the subjects just
referred to,
l 2... v) (11) (12)... (n)
[1 2... or (21) (22)... (2n)
(no) (n2)... (nn)