1890-91.] Dr T Muir on some unproved Theorems. 77 
third row entails like consequences. But the first three elements of 
the first three rows constitute likewise the first three elements of 
the first three columns ; and the elements of a column are related 
to each other exactly as the elements of a row are; consequently the 
vanishing of these nine elements entails the vanishing of all the 
other elements of the first three columns. Finally, viewing these 
last elements as constituting the first three elements of the 4 th and 
remaining rows, we see that all the 225 minors will vanish if the 
nine minors common to the first three rows and first three columns 
vanish. 
Had the original determinant been of the n th order and the 
minors formed from it been of the m th , the compound determinant 
would have been of the order C n> m> and all the elements of its first 
row could have been shown to vanish if 
0 = |1,2,3, . . . , m- 1, m| = |l,2,3, . . . m- 1, m+ 1| = . . . , =|1,2,3, . . . m- 1, n\, 
that is to say, if n — m + 1 of them vanished. The general theorem 
we have thus proved is — All the minors of the m th order formed 
from a determinant of the n th order will vanish if (n-m + 1) 2 of 
them vanish. 
If the original determinant be axisymmetric, the compound deter- 
minant is also axisymmetric, and therefore the said (n-m + 1) 2 
minors are not in this case all different. In fact, instead of there 
being n-m + 1 different minors to be counted in each row, there is 
1 less in the second row, 2 in the third, and so on, the total thus 
being only 
(n - m + 1) + (n - m) + (n - m - 1) + • • • + 1 . 
i.e. -m+l)(n-m + 2) . 
This result was enunciated without proof by Sylvester in the 
Philosophical Magazine for September 1850. It appears from the 
foregoing to be an easy deduction from Cayley’s theorem published 
seven years before. 
Cayley’s next theorem is bound up with a certain notation intro- 
duced by him, and forms indeed the fundamental justification for 
the use of the said notation. To indicate that all the 15 determi- 
nants of the 4th order formed from the array 
