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

DETERMINANTS IN GENERAL (SPOTTISWOODE, 1853) 83

being used for one of the two determinants, and
l' 2'... n (11) ( 12)... (1n)'
1' 2'... J or (21)' (22)'... (2n)'
(nl)' (n?2)'.. (nn)'
for the other. The reading is thus rendered tiresome, and inaccurate printing exaggerates the trouble.
In ~8 (pp. 335-337) Sylvester's proposition of the year 1850
regarding the vanishing of the minors of a non-quadrate matrix
is attempted to be proved. The matrix being, for example,
11 12 13 14 15 16 17
21 22 23 24 25 26 27 l 234567
31 32 33 34 3 6 36 7 2 3 4
41 42 43 44 45 46 47
and {rstu} being the determinant whose columns are identical
with the rtl", 8th, tth, Uth columns of the matrix, it is required to
show that if the minors {1234}, {1235}, {1236}, {1237} vanish,
the thirty-one other four-line minors of the matrix must vanish
also. In support of this Spottiswoode says truly enough that if
A, B, C, D
stand for
1_f23l 123) 123k (123l
t234J' 134J' l124J' l23J'
we know that
{1234} = 14A + 24.B + 34.C + 44.D,
{1235} 5= 1.A+ 25.B + 35.C + 45.D,
{1236} 16.A + 26.B + 36C + 46D,
{1237} = 17.A + 27.B + 37.C + 47.D;
but then to this he merely adds, "and, if these vanish, it is
obvious that by direct elimination all the others may be at once
deduced." He notes in addition that in the case of the matrix
1 2 34... n 11 12 13 14... In
l2 2 J 1 22 23 24... 2n
the vanishing of
12} {13) 12 ' ' 14
12 12 12W - 12*