26 
Proceedings of the Royal Society of Edinburgh. [Sess. 
the elements of a term of the given determinant. Further, it is seen 
that the 2 n terms thus connected with a solution cannot be altered by 
transposition of rows or by transposition of columns ; and that therefore 
two solutions that do not involve in their construction the same set of 
2 n terms are essentially different solutions. 
(8) When n is prime a solution is always got by taking the array 
of elements whose array of letters is identical with C(a, b, c, . . . ) n and 
whose array of suffixes becomes identical with C(l, 2, 3, ... , n) after 
the last n— 1 rows of the latter have been reversed in order. For 
example, when n is 5, a solution is 
b<2 Cg d ^ 6 b 
e 2 a 3 & 4 c 5 d 1 
d 3 e 4 ct b b 4 c 2 
C 4 d b 6^ 0L 2 ?^g 
b b Cl d 2 e 3 a 4 . 
In other words, we take for our first row the diagonal of the given 
determinant, and thence get the other rows by cyclical forward move- 
ment of the letters and cyclical backward movement of the suffixes. 
(9) Other solutions are got by starting with a first row whose 
elements belong to an as yet unused term of the given determinant 
and permuting the letters and the suffixes separately and in different 
ways as before. In the case where n is 5 there are twelve solutions, 
which, by reason of the nature of the law of formation, we can specify 
sufficiently by giving merely the first row of suffixes, namely, 
1 2 3 4 5 
1 2 3 5 4 
1 2 4 3 5 
1 2 4 5 3 
1 2 5 3 4 
1 2 5 4 3 
1 3 2 4 5 
1 3 2 5 4 
1 3 4 2 5 
1 3 4 5 2 
1 4 2 3 5 
1 4 3 2 5, 
it being understood that the first row of letters is always 
a b c d e . 
That these solutions are essentially different from one another is evident 
from the fact that in their construction not a single term of the given 
determinant appears twice, every additional solution using up, as it 
were, ten new terms, and the number of solutions thus being 
