29 
1919-20.] Determinants connected with Cireulants. 
whose permuted arrays are those of 
and for the multiplier the determinant similarly formed from 
I a l/^273 I J I a 4^576 I ’ I a 7/^879 I 1 
and comparing the original forms of three of the elements of the product- 
determinant which, in accordance with our assertion, are expected to be 
identical, say the (2, 3) th , the (5, 6) th , and the (8, 9) th . The (2, 3) th being the 
product of the second row of the multiplicand by the third row of the 
multiplier, i.e. r 2 . p 3 , we may conveniently write in a form showing the 
rows in question, namely, 
7i > 72 > 73 » 74 » 75 » 7e > 77 » 7s » 7e 
this being therefore used temporarily to stand for 
^l7l + & 272+ * • • +^979* 
Now the cyclical movements which change r 2 into r 5 are exactly those 
which change p 3 into p 6 , so that in the (5, 6) th element of the product each 
y must find itself still directly under the corresponding b, the element in 
fact being 
& 7 , b 8 , & 9 , frj , , b 5 , b Q * 
77 ’ 78 > 79 ’ 7l > 72 ’ 73 > 74 J 75 ’ 76 
Similarly we may reason regarding r s and p 9 , the (8, 9) th element being 
^4 > ^5 ? bg , b 7 , b g , bg , b-^ , b 2 , bg . 
74 > 75 > 7e > 7? » 7s > 7 9 > 7i > 7 2 > 7a 
(14) Another theorem of like character is that the adjugate of a block 
circulant is itself a block circulant of the same type and dimensions. A 
proof of this may be worked up from considerations similar to those in 
the preceding paragraph. What we have got to show is that any element 
of the particular block circulant written at full length above, say the 
element b 6 , has the same complementary minor in each of its three positions, 
namely, the (2, 6) th place, the (5, 9) th , and the (8, 3) th . Now the facts are 
that the complementary of the (2, 6) th element being written, we obtain 
from it the (5, 9) th by merely advancing its last set of three rows over all 
the other rows, and then doing the like with the last three columns ; and 
that the (8, 3) th is similarly obtained by advancing its last set of six rows 
over the others, and then the last set of six columns. 
(15) The opportunity may now be taken to add a few fresh facts on 
the factorisation of ordinary cireulants. 
In the first place, it deserves to be noted that the n-line circulant has 
