1904 - 5 .] 
If 
Vanishing Aggregates of Determinant Minors. 861 
(2p + q\p + r\q\ 
\ a l @2/ 
(2p + q | p + r | q\ 
\ a i /V 
is axisymmetric, the aggregates in the product P< vanish. It will 
he observed that this is entirely independent of the elements 
denoted by X. 
If in addition to ?i = 0 we put h= 1 , then we have the theorem 
of arts. 29 and 31 of Muir’s paper. As stated by him in its most 
general form (art. 3 1 ) it is as follows : 
11 If two arrays B and B' be taken, B containing q + 1 rows and 
p + q - r columns and B' containing p + q — 1 rows and q columns , 
and a new determinant , /\ p , of the (p + q)* 71 order be formed having 
for its first p rows all the rows of B except the p th , each preceded by 
r zeros, and for the first column of the remaining space the last q 
elements of the v th row of B , and for the other columns the p + q - 1 
rows of B', then 
p=p + i 
2Ap(-r‘ 
p = i 
vanishes if the last coaxial minor of the of h order in A be 
axisymmetric” 
In this statement of the theorem, as pointed out by Muir, it is 
evident that we might substitute for “ the last q elements ” the 
words “ any q elements,” with the proper change in the statement 
of the minor which is to he axisymmetric. 
If we put h — 0 and Jc = q-r= 1 we have the case given in 
art. 29. 
Syracuse University, 
April 1905 . 
( Issued separately August 29 , 1905 .) 
