437 
1888 - 89 .] Dr T. Muir on the Theory of Determinants. 
By taking the identities 
O-aAj + a'Aj + ....+ a<»> A^, 
0 = «A. + OjA; + . . . . + al A ( J, 
E = “*A* + • • • • + a< l >A 'l’ 
0 — a A, + 
a A, 
+ a 
(n) * («) ; 
using the multipliers 
A 
v' 
O,*' » 
A *’• 
> 
a M' 
* ’ > ‘ 
and adding, there is obtained 
i,i' 
n, V » 
4. 
R. A 
A^A^ - A ( ;?A ( p, 
fC tC rC fC J 
— a result at once recognisable as a case of the theorem regarding a 
minor of the adjugate. Next by starting with Bezout’s identity 
connecting any eight quantities, the particular eight taken being 
A*>, A®, A®, A®, 
A« i®, 
A« 
A (0 
and making six substitutions of the kind 
A?A<£> - A«A« - B.A#, 
just seen to be valid, there arises the identity 
A*’* A -i- A 
^kjc'^k"k'" + A 
Ic,k" k"',k' 
+ Kx*Kr = °- (xxm. 11) 
This clearly belongs to the class of vanishing aggregates of products 
of pairs of determinants ; but in order that its true character may 
be seen, and comparison made possible between it and others of 
the same class already obtained, a more lengthy notation is necess- 
ary. Taking for shortness the case where the primitive determin- 
ant is of the 8th order, but writing it in the form 
and making 
i, i = 3, 6 and A, A:", k"' = 5, 6, 7, 8, 
