39 
1918-19.] Determinant of Minors of a Set of Arrays. 
~ d 
e 
f~ 
a 
b 
c — 
~ b 
c 
d~ 
e 
h 
3 
c 
d 
e 
e 
f 
d 
c 
a 
b 
d 
h 
e 
f 
d 
h 
_ e 
j 
k_ 
_k 
c 
a _ 
mj 
k 
c 
which by mere interchange of rows becomes 
h j e 
~~ b a c~ 
e f d 
j e k 
a c k 
f d h 
e d f 
c d e 
deb 
a b c __ 
d h e _ 
j k c _ 
and therefore is equal to 
(k-h)(d-j)(/-j)(a -e)(b - e)(c-e) 
. (k - b)(d - a)(e - a)(d — c)(h - c)(e — c) 
. (h- e)(c - d)(k-d)(c-d). 
(7) As regards the actual vanishing of 0 little can be learned from 
the original 4-line form, save in the unimportant cases where special 
values are assignable to as many as three of the variables : namely, such 
values (for example, a, b, c = e, fi g) as will make all the elements of a 
row vanish, such values (for example, a = e = i = 0) as will make some 
other sufficiency of elements vanish, and such values (for example, 
a, e, i = d, h , V) as will make two rows identical. 
There is one case of this kind, however, for which we must turn for 
help to the derived 3-line form of 0, namely, where j , k,l — a, b, c. When 
this substitution is made it will be found that each element of the first 
column vanishes from having two rows identical. The same happens 
when a, b, c is put equal to d, e, /, and for the same reason : also a similar 
result when d, e,f is put equal to g, h , i, or when g, h , i is put equal to 
j, K i- 
(8) The cases in which special values are given to two variables may 
be summed up in the following pair of propositions : — 
(a) If any one of the variables be fixed on, and the two variables 
in front of it in the series be put equal in order to the two behind 
it, the determinant vanishes: for example, if c, d = /, g, or j, k = a, b, 
then 0 = 0. 
(/3) If three variables be equated which are so situated in the series 
that the first and second are separated by two places, and the second and 
third by either three or four places, the determinant vanishes : for example, 
if a = d = h or i, then 0 = 0. 
For the establishment of these results little, if anything, more is 
