98 Froceedings of Boyal Society of Edinburgh, [jan. 6, 
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Continuing this last process, viz., completing the 5*’" row of the 
third determinant, by inserting the elements X, Y, Z, and at the 
same time inserting one of these new elements in the place (5,3) of 
the remaining determinants, and so on, we finally arrive at the 
identity 
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involving 5^ + 2 *4 + 3 + 2+ 1 elements, or in general +1) 
(3/»-2). 
3. The connection of this with Kronecker’s theorem is easily 
made apparent. Take a determinant with umbral elements, and 
for the sake of variety, let it now be of the 4^^ order, viz. — • 
1.1 1,2 1,3 1,4 
2.1 2,2 2,3 2,4 
3.1 3,2 3,3 3,4 
4.1 4,2 4,3 4,4 , ' 
and let the new elements inserted during the formation of the 
identity be 
