25 
1919-20. J Determinants connected with Circnlants. 
in the said equality and removing Q x from both sides we obtain 
V=(F a -G & )(2«+20^; 
in other words, the factors of V are the same as those of A 5 , save that in 
Y the linear F a — G& takes the place of the quadratic Q r 
Similarly, or by mere change of e into e _1 , we have 
u=(g 0 -f 6 )(2,«+2 & H. 
(5) By taking the sum of all the rows in any one of the determinants 
A 5 , U, V we not only see that 2<x + 26 is a factor, but we can remove the 
factor and have the co-factor left expressed as a determinant. There is 
thus suggested for the future the problem of removing in some analogous 
fashion the linear factors F a — G & , G a — F 6 from V and U respectively; 
or, better, from V/(2<x + 2&) and U /fZa + 'Eb) respectively, and so obtain a 
determinant form for Q 2 , and therefore for the other quadratic factor also. 
In this connection we may note that by merely performing the 
multiplications and condensing we obtain 
Qi = ( 2 ■ a i - 2 v) + ( 2 a A 
that therefore 
q,-(2v - 2v) +(2 °a 
2 6 A)( e + e “) + (2>s - 2 & A)( <2 + eS ) ; 
2 ft A)(‘ 2 +‘ 3 ) + (2 a i a s -2 ft A)( t+ {4 ) . 
and that, again by mere multiplication and condensation, there results 
QiQa , i. e. A 6 /( 2 a + 2 ^ l ) = “ B2 ~ ° 2 ~ AB " AC + 3BC > 
where 
A stands for 2 fl i 2 B2^ 2 ’ 
O O 
b „ 2“a-2 6 a 
o o 
^ ” 2 a i a 3-2 & l 6 3- 
(6) This association of two circulants that are symmetric with respect 
to different diagonals provides the solution, when n is prime, of a very 
interesting problem of arrangement; namely, the problem of placing 
the n 2 elements of | o q& 2 c 3 ... in an n-line square array so as to 
have each row containing ' every letter and each column every suffix. 
The problem was first proposed twenty-five years ago by J. D. Loriga 
in the Intermediate des Math., i, pp. 146-147, and no solution has as 
yet appeared. 
(7) At the outset it is clear that in the desired square array — or 
“ solution ” as we may call it — each row and each column must contain 
