152 Proceedings of Royal Society of Edinburgh. [: 
+ 
45 
( l 1 
45 
( l + 
45 
45 
\6 1/ 
46 
\5 ~ 2/ + 
56 
46 
/2 5\ 
46 
/2 5\ 
46 
45 
\6 _ 1/ + 
4 6 
\5 2/ 
56 
56 
/3 4\ 
5 6 
( 3 4 U 
5 6 
45 
oT 
i 
t—* 
i 
46 
\5 2/ + 
5 6 
each line of which may again be condensed by substituting for it 
a determinant of the third order, so that we shall have finally 
451 
45 6 
456 
45 1 
426 
456 
456 
426 
356 
456 
45 6 
356 
4 4 4 
4 5 6 
5 5 5 
4 5 6 
16 16 16 
4 ~ 3 5~2 6 "" 1 
4 4 4 
4 5 6 
2 5 2 5 2 5 
4 ~ 3 5 2 6 ~~ 1 
6 6 6 
4 5 6 
3 4 3 4 3 4 
4 _ 3 5 ~ 2 6 ~ 1 
5 5 5 
4 5 6 
6 6 6 
4 5 6 
16 16 
When ^ = 5 = 2’ * * •> — that to sa Y> when the elements 
of 
123 
456 
are in order identical with those of 
654 
32 1 
— the 
right-hand side vanishes, and the theorem degenerates into the 
simpler one which suggested it. 
(10) The corresponding theorem in connection with 
12345678 
12345678 ! 
is readily seen to he 
5 6 7 1 
5 6 2 8 
153 78 
4 6 7 8 
5 6 7 8 
+ 
5 6 7 8 
+ 
|5 6 7 8 
+ 
5 6 7 8 
15 6 78 
5 6 7 8 
5 6 7 8 
5 6 7 8 
15 6 7 1 
5 6 2 8 
5 3 7 8 
4 6 7 8 
