1904 - 5 .] 
Dr Muir on General Determinants. 
931 
made the subject of statement without an enormous periphrasis, 
and could never have been made the object of distinct contempla- 
tion or proof.” 
The main object of his paper is then taken up, but the subject 
of the notation is twice recurred to, — once to say that it is “ very 
similar to that of Vandermonde ” as he had learned from “ Mr. 
Spottiswoode’s valuable treatise,” and the second time to point out 
that on second thoughts it is better to unite two umbral elements 
in the form 
a 
a 
rather than in the form 
aa , 
because then the analogy upon which the extension of the notation 
from simple to compound determinants is grounded will be better 
apprehended. 
The theorem used above to illustrate the power of the £ umbral ’ 
notation should not be lightly passed over, being far more deserving 
of the epithet ‘ new 5 than the notation employed in formulating it. 
Although at a later date it came to be of less moment because of 
its inclusion as merely one of a class of theorems, viz., the class 
known as £ Extensionals,’ there can be no doubt that at the time of 
its discovery it was, as Sylvester himself styled it, ££ a remarkable 
theorem.” Taking, for the sake of illustration, the case of it where 
r = 3 and s = 4, viz., 
I a-^b 2 c 2 d^ | 
i a A C 3 6 4 I 
I a ~ft c 2pzf 4 | 
I a 1 b 2 c 3 g i | 
I «A C 3^5 I 
I I 
I a i^2 C sf5 I 
I a i h 2 C *9$ I 
I a i^2 C S d 6 I 
! a A C 3 e 6 I 
! «A C 3 I 
I «A C 3?6 i 
I a A C 3^7 I 
| a Y b 2 c^ | 
I a A c 3 fi I 
I «A c 30V i 
= I aA c 3 ! 3 - I a A c A e bf(i9i I* 
we see that if we delete a^b 2 c z everywhere on both sides we are 
left with 
d b d Q dyj 
e 4 e 5 e 6 e 7 
f 4 J 5 f Q D 
9± 9$ 9 6 97 
I d ^f&97 1; 
