1904-5.] Prof. Metzler on Axisymmetric Determinants. 721 
while on the other side the group c is constant as columns and 
the group b is divided up between the rows and columns. 
Similarly for the other aggregates. 
If s + t>r , then D = 0 and A and B could not exist, as minors 
of order r could not have more than r rows constant. If 
s' + 1 — r , , then s = s and t = t' and A and B both reduce to the 
single term 
(2r\t' + s\ 
(2r\r\ 
l «J. 
V «J 
or 
(2r\t' + s\ 
/2r 1 r \ 
V «J 
V “i ) 
Similarly for t'>r and for s>r. 
Other forms may be obtained from these aggregates by applying 
the law of complementaries and the law of extensible minors to them. 
4. If now we turn to Professor Hanson’s paper and observe 
that in his relation 
K = M = k K\ - !,My 
or M + ^M A = A K A ' 
the expression M + ^M^ is a sum of the form of the conjugate of 
D, and A K A ' is of the form of A, then it will be seen that his 
theorem is, except for the lack of a sign factor, the same as one 
of the relations in III. 
As examples of III. we have 
( 1 ) 
1234 
5678 
6781 
4523 
= -2 
1236 
4578 
where ^ 
1234 
5678 
1234 
5678 
1235 
4678 
and 2, 
6781 
4523 
6781 
4523 
6782 
4513 
6783 
4512 
1234 
V > 
1456 
1237 
(2) 2 
5678 
-2 
2378 
= -2 
4568 
( 3 ) 
2 
12345 
6789r 
--2 
45678 
1239t 
= -2 
12349 
5678t 
Syracuse University, 
Syracuse, N.Y., February 1905. 
( Issued separately May 20, 1905). 
PROC. ROY. SOC. EDIN. — YOL. XXY. 
46 
