274 
Transactions of the Boyal Society of South Africa. 
for the case where the involved elements are those of the special deter- 
minant whose matrix is the sum of the two matrices 
(do «I 
-a^X^ 
-asX ) 
-a^X 
-aeX 
! — a8/\7 ... — a^X^ — a.X"" — a^X 
and yet the latter identity is not included in the former. 
2. Instead, however, of attempting a generalization of the new identity 
by making the elements of its left-hand member perfectly general and 
then seeking for the corresponding change in the right-hand member, we 
may adopt a reverse procedure, namely, view the elements on the right- 
hand as being changed from special to general, and then inquire what 
alteration is thereby necessitated on the left. Thus, instead of Mr. 
Eoseveare's expression 
«i a^ a. - a^X' 
a^ aj—a^X^ a^ — a^X^ 
— a^X"^ a^ — a^X^ a^ — a^X'' 
-a,X^ 
aj ^2 a^ — ayX^ 
a^ ai—a^X^ a^ — a^X^ 
- a^X^ — a^X^ a^ — a^X^ 
- a()^'^ — a.^- — a^X^ 
— (X,\5 
a^ 
a. 
-%x," 
a. 
— a^X^ 
a. 
a^X^ 
- aeX^ 
a. 
-a^X 
— a^X^ 
— aJ^ 
^3 
-a^X^ 
a, 
~aeX 
-aeX^ 
a^ 
(Xi 
a^ 
-a^Xi 
aeX^ 
«o 
-a^X' 
— a^X \ 
which is a function of 7ii)ie variables, let us pass to 
u 
/^I 1^4 
7i 72 73 74 
^1 ^2 h ^4 
^3 
73 
75 
X + 
/3i 1^3 
7i 72 73 
tti 
l/3x /34 l^s 
'71 72 74 75 I 
which is a function of tioenty-one variables, and ask if there be a con- 
densed expression equal to the latter. 
A little examination of the determinants involved in it shows that all 
of them are 4-line minors of the array 
/3x 
7i 72 73 74 75 
