1919-20.] Determinants connected with Circulants. 27 
(10) The law of formation from the first row shows that the signs 
of the terms used in connection with any solution must be all alike, 
so that we have six solutions involving positive terms and six involving 
negative terms. 
(11) When n is composite no solution is obtained in this way, the 
fault being that it leads to half of the elements being used more than 
once and the others not at all. For example, when n is 4 it gives 
in which <x 2 , a 4 , . . . do not appear. 
In this case, however, there actually can be found a proportionately 
greater number of solutions than in the case where n is 5. In the 
first place, the array of letters is in every solution 
abed 
bade 
c d a b 
d e b a ; 
that is to say, the circulant array of a pair of two-line circulant arrays. 
In the second place, the suffixes of the first row may be any one of the 
six sets 
1 2 3 4 1 2 4 3 
1 3 4 2 1 4 3 2 
1423, 1324; 
and if the set taken be 
fX V p (T 
then the set for the third row is 
1/ /x or p , 
and the sets for the second and fourth rows the reverses of these. For 
example, 
a-^ bg c ^ c?2 
bi y a ^ dg Cj 
Cg dj ^4 
d 4 c 2 b± a 3 . 
In the case of each solution belonging to the first triad the involved 
terms of the given determinant are all positive, each pair of solutions 
having four terms in common : in the case of the other triad the terms 
are all negative. In the full set of solutions each term is thus involved 
