858 
Proceedings of Royal Society of Edinburgh. [sess. 
to some of those in 
2 n | k 
a 
, then some of the terms on the left 
vanish, and we have the special case given by Muir,* which, as 
stated by him, is as follows : — 
“ If to n - 2 - a columns of n elements each there be appended 
the cf, c 2 th , . ... , c a th and r th columns of a given determinant 
of the n order , and from the resulting array the r ih row be deleted , 
there being thus produced a square array of the (n - 1)^ order, the 
determinant of which is D r , then the aggregate 
will vanish when the given determinant is axisymmetric. ,> 
As an example, take the case (n , k , h , g) = 5 , 4 , 2 , 0) 
12345 | _ 
12345 
12346 
12356 
12456 
78916 “ 
78916 
~ 
78915 + 
789l4 
— 
78913 
13456 
78912 
23456 
7 89 1 1 
12 
78 
2 
1345 
916 
- 
13 
78 
2 
245 
916 
+ 
14 
78 
2 
15 
78 
2 
234 
916 
+ 
16 
78 
2 
234 
9l5 
+ 
123 
78 
2 
25 
78 
2 
134 
916 
- 
26 
78 
2 
134 
915 
+ 
34 
78 
2 
36 
78 
2 
124 
915 
- 
45 
78 
2 
123 
916 
- 
46 
78 
2 
235 
916 
145 I 
916 
125 
916 
123 
915 
+ 
24 
78 
35 
78 
56 
78 
135 
916 
2 
2 
124 
916 
123 
914 
and put 9 , 1 = 1,2 respectively, then the last two terms on the 
left will vanish, having identical columns, and the relation 
becomes : 
12345 
12 
345 
13 
245 
78126 
— 
78 
2 
126 
“ 1 78 
2 
126 
56 | ^ 
123 
+ 
78 1 2 
124 
+ L.c . , paper I. arts. 17 to 28. 
