146 Proceedings of Royal Society of Edinburgh. [skss. 
and the other is got from it by substituting for each element 
its conjugate. The full set of binomial factors is thus 
n _ '2n n _2n-\ 
2n n 1 2n - 1 n * 
n + 1 2 n 
2 n n + 1 ’ 
As for the determinant which is to be the cofactor of any of these 
differences, its two lines of indices must contain exactly all the 
indices not found in the said difference, the first line being always 
1, 2, 3, . . . . , n — 1 , 
and the second being therefore got from 
n, n + 1, n 4- 2, . . . . , 2 n 
by dropping out the two indices which appear in the annexed 
difference. This being grasped, there then only remains to be 
determined the law of signs of the terms. Looking again to the 
triangle of elements we at once observe that in each of the left- 
to-right lines of the triangle the signs are alternately positive and 
negative, and that so also are the first signs of the various rows 
taken in order. If in addition to this we only note the fact 
that as we thus move from place to place in the triangle, there 
is a corresponding alteration in the sum of the indices of the 
binomial factors, we see that the determination of the sign 
of any term can be made dependent on the difference between 
the sum of the indices of its binomial factor and the sum of 
the indices of the first binomial factor of all. 
Our enunciation of the general theorem will thus take the 
following form : — 
If /x and v be any integers , y being the less i taken from the series 
n, n+ 1, n + 2, . . . , 2n ; 
and a, /3, y, . . . , a> be what the series becomes when g is removed , 
and a, /?, y, . . . , i p what it becomes when both are removed ; then 
1 2 3 ... 2n 
in connection with any even-ordered determinant 
1 2 3 
2n 
we have 
