Notes on Semicirculants. 163 



character. For example, returning to circulants of the 4th 

 order, we may form a determinant by taking any two rows from 

 C (a, b, c, d) and any two from C (d, c, b, a). In every such case 

 it is clear that a + b + c + d, a—b+c—d would be factors as before ; 

 but it is not so readily seen what the co-factor would be. Investi- 

 gation shows that it takes one of five forms, viz., 0, (a — c) 2 , (a — c) 

 (b — d), (b — d) 2 , (a + b — c — d)(a — b — c + d): so that we have the 

 general theorem : Every determinant formed by taking two rows 

 from G (a, b, c, d) and two rows from C (d, c, b, a) is resolvable into 

 linear factors or vanishes, (X.) 



Similarly we find that Every determinant formed by taking any 

 three consecutive rows of C (a, b, c, d, e) and any two consecutive roivs 

 of C (e, d, c, b, a) is resolvable into three factors, one linear and two 

 quadratic. (XL) 



