144 
Proceedings of Boyal Society of Edinburgh. [sess. 
The fact that axisymmetry implies equality of conjugate 
elements thus accounts at once for Kronecker’s theorem. 
(3) Proceeding in an exactly similar way we change 
1 2 3 41 
12 3 5 
12 36 
123 7 
12 3 8 

-j- 
4 5 7 8 
— 
+ 
45 6 7 
5 6 7 8; 
4 6 7 8 
4 5 6 8 
into 
123 4 
123 4 123 
5 6 7*8 ~ 
5 6 8|7 + ! 578 
123 4 
6 7 8*5 
1 2 3 
5 
123 
5 12 3 
5 
467 
* 8 + 
468 
*7 ~~ 4 7 8 
* 6 + 
5 
4 
1231 6 
4 5 7 I 8 
1 2 3 
45 8 
6 | 1 23 
7 + ' 4 7 8 
6 
5 
1231 7 1 2 7 
1 4 5 G j * S + 4 5 8 j " 6 
1231 8 
! 4 5 6 ! ' 7 
where we have now 4x5 terms, each of which is the product 
of a three-lined determinant and a simple element. On examination 
it will be seen that the simple elements consist of the ten 
4 4 4 4 ] 
8 , 7 , 6 , 5 
I 
5 5 5 
8, 7, 6 
- 
6 6 | 
7 j 
8 J 
and their conjugates 
’8 7 6 5 
4, 4, 4, 4 
8 7 6 
5, 5, 5 
\ 
8 7 
6 , 6 
8 
7 , 
and that the twenty corresponding three-lined determinants con- 
sist of the ten 
