326 
Proceedings of the Royal Society of Edinburgh. [Sess. 
on each side gives the relation 
^33 
6 *34 
+ 
C 33 
^34 
^34 
+ 
C 33 
^34 
^43 
C 44 1 
C 43 
^44 
C 43 
C 44 
6 43 
hi 
which is an example of Theorem A for the two determinants 
33 
'43 
'34 
'44 
and 
'33 
'43 
C, 
34 
'44 
If we had expanded in terms of the b’ s and their complementaries and 
taken the coefficient of any b such as b lv we would have 
@22 
a 23 
C 21 
@22 
C 23 
a 2i 
C-2.2 
a 23 
a 24 
a 32 
a 33 
C 31 
+ 
a 32 
C 33 
a 34 
+ 
C 32 
Ci 33 
@31 
@42 
a 43 
C 14 
@42 
C 43 
a 44 
C 42 
a 43 
@11 
@22 
a 23 
«£4 
@22 
@23 
@ 04 
C 22 
^23 
C 24 
a 33 
a 34 
+ 
C 32 
C 33 
C 34 
+ 
@32 
a 33 
a 34 
V 49 
C 43 
C 44 
@42 
a 43 
@41 
a 4L 
a 43 
@44 
which is another example of Theorem A for the determinants 
( a 22 a 33 a 44) an< ^ ( C 22 C 33 C 44)' 
For the proof of Theorem B it is sufficient to observe that the truth 
of the case for three determinants is seen by expanding by Laplace’s 
theorem in terms of minors formed from the as and their complementaries ; 
then the coefficient of any minor of the a’s on the one side is equal to the 
coefficient of the same minor on the other side by Theorem A. Having 
thus established the theorem for three determinants, it is extended in a 
similar manner to four, five, and so on up to any number k. 
We might use for our k determinants the mutually exclusive minors of 
order n formed from any n rows of a determinant, of order nlc. That is, 
« 
these minors would have no two columns alike. If we select the minors 
so that some of them will have columns alike, it is apparent that some of 
the terms on the right in Theorem B will have columns alike, and therefore 
disappear. 
Syracuse University, 
March 1917. 
(. Issued separately October 12, 1917.) 
