1899-1900.] Dr Muir on Determinant Minors. 
147 
2 <-> 
1 2 3 ... n - 1, [x 
a /3 y to 
2 <-> 
3n-(jx-fv) 
1 23. ..n 
a /?7 $ 
1 1^--^ 
ib\\v fl) 
(5) From this, of course, a variant enunciation of Kronecker’s 
theorem at once follows, viz., 
If jx be any integer taken from the series n, n+ 1, n + 2, . . ., 2n 
and a, /?, y, . . . , to be what the series then becomes , then in 
1 2 3 ... 2n 
connection with any even-ordered determinant 
whose coaxial minor 
we have 
n, n + 1 , n + 2, . 
2n 
n, n+ 1, n + 2, . . ., 2n 
1 2 3 ... 2n 
is axisymmetric , 
Z(-) 
n-ju. 
= 0. 
1 2 3 ... n — 1, fi 
a f3 y to 
The advantage of this form of enunciation lies in the fact 
that it localises the axisymmetry which is necessary for the 
validity of one of Kronecker’s identities, and thus by implication 
indicates the number of such identities which hold in the case 
where the axisymmetry of the parent determinant is complete. 
This number is clearly the number of coaxial minors of the (n + l) th 
order contained in a determinant of the (2 w) th order, i.e., C 2 n,n-i> 
The same, of course, is also evident from the fact that instead of 
taking 1, 2, 3, . . . , w — 1 for constant indices in the first line, 
we might with equal reason select any other n - 1 indices from 
the 2 n available. 
(6) With the general theorem now in our possession, other 
special cases of it similar to Kronecker’s can easily be obtained. 
Perhaps the most important of these is that where the coaxial 
minor of the parent determinant is skew. To get this we have 
only to substitute - — for v - in the general enunciation, the result 
v /x 
being : — 
1 1 2 3 ... 2n 
Di connection ivith any even-ordered determinant 
whose coaxial minor 
n, n + l,n + 2, . . ., 2n 
n, n + 1, n + 2, . . . , 2n 
1 2 3 ... 2n 
is skew , we have 
2 <-> 
n-/x 
1 2 3 . . . n- 1, n 
a f3 y , w 
= 2 Z(-) 
3a-(/x+i/) 
H,v 
1 2 3 ... n - 1 
a Py 
