37 
1918-19.] Determinant of Minors of a Set of Arrays. 
reason than the consequent overcoming of this difficulty of notation, that 
the determinant is of a type which is subject to condensational trans- 
formation, being changeable into a determinant of the 3 rd order in 
which the elements are minors of the 4 th order. Applying this con- 
densation-theorem (. Messenger of Math., xxxv, pp. 118-121), and noting 
that the rows of the new four-line minors all belong to the group of 
twelve rows which we started with and which we may specify by their 
ordinal numbers 
1, 2, 3, 4, 5, 6, 7, 8-, 9, e, t, 
we obtain for our determinant 0 the form 
26^1 
26ffi 
26(59 
37el 
37e5 
37e9 
48rl 
48r5 
48r9 
or, if we indicate each of the twelve rows by the variables appearing in 
it, the form 
a 
b 
c 
“ b 
c 
d~ 
~b 
e 
d 
b 
c 
d 
e 
f 
9 
f 
9 
h 
f 
9 
h 
f 
9 
h 
i 
j 
k 
_i 
k 
l _ 
k 
l _ 
—j 
k 
l_ 
a 
b 
e ~ 
— c 
d 
c 
d 
e ~ 
e 
d 
e 
e 
f 
• 9 
1 
9 
h 
i 
9 
h 
i 
9 
h 
i 
J 
k 
_k 
l 
a _ 
-l* 
l 
a_ j 
1 
-h 
l 
a _ 
~ a 
b 
e ~ 
~ d 
e 
f~ 
~ d 
e 
r 
d 
e 
f 
e 
f 
9 
h 
i 
j 
h 
i 
j 
h 
i 
j 
i 
j 
k 
_l 
a 
b _ 
_ l 
a 
b _ 
a 
b _ 
where, for example, the element in the (1, l) th place is 
1 a + b + c ab + ac + be abc 
1 b + c + d be + bd + ed bed 
1 f+g + h fg+fh + gh fgh 
1 j + k + l jk +jl + kl jkl . 
(5) A knowledge of certain properties of this latter type of 4-line 
minor is thus a necessary preliminary to a knowledge of 0. Those 
requisite for our purpose are : If any two rows have two variables in 
common, the difference of the remaining two variables is a factor : if in 
addition one of the two repeated variables occurs in a third row, the 
determinant is expressible as a product of six differences : and if two 
