28 Proceedings of the Royal Society of Edinburgh. [Sess. 
twice, and consequently the number of solutions is relatively double 
the number found for the case where n is 5. 
(12) The peculiar determinant form referred to in the preceding para- 
graph, namely, the circulant of the two circulant arrays, 
a b g d 
b a d c , 
is the simplest possible case of a block circulant. The cyclically permuted 
arrays, however, need not be themselves circulant in form, but may be the 
arrays of general determinants so long as they are all of the same order, 
q say ; and the number of them may also be unrestricted, p say ; the 
resulting determinant being thus of the (pq) th order. 
(13) One of the most important properties of such determinants is that 
they are homogenetic ; in other words, that the product of two block circu- 
lants, each of p q-by-q arrays, is itself a block circulant of the same type 
and dimensions. 
As a consequence of the definition the (i, j) th element in the first of the 
p arrays — and any one of the p arrays may be made the first without 
departing from the circulant form — occurs again in the places 
(q + i,q+j), (2q + i,2q+j) , . . . 
and what we have therefore got to prove is that in the product-determinant 
the elements in the places 
(q + i,q+j), (2q + i,%q+j), ••• 
are identical. Further, if we denote the rows of the multiplicand by r v 
r 2 , ... , and those of the multiplier by p v p 2 , ... , this is the same as to 
say that we have got to show that 
r i'Pj = r Q+i'Pa+o ~ r 2Q+i' P2q+j • 
The lines on which such a proof may be effected are readily grasped by 
taking for the multiplicand the 9-line block circulant 
