162 
Proceedings of the Royal Society of Edinburgh. [Sess. 
which is the factor desired. In like manner from the existence of the 
form C(X, Y, Z) there is obtained the factor 
C(aj + a 6 + a n , a 2 + a 7 + a 12 , . . . , a b + a 10 + « 15 ) . 
The same result is established by operating directly on the original 
circulant; and this procedure has the advantage of giving at the same 
time a convenient expression for the cofactor. Thus by performing on 
C(n 1 , a 2 , . . . , a 15 ) the operations 
rowj + row 4 + row 7 4- row 10 + row 15 
row 2 + row 5 + . . . 
row 3 + row 6 + . . . , 
and writing 
£ , rj , £ for «! + a 4 + . . . , a 2 + a h + . . . , a 3 + a 6 + ... , 
we obtain 
i 
yj 
{ 
i 
V 
£ 
£ 
V 
£ 
£ 
7) 
£ 
£ 
V 
£ 
£ 
£ 
V 
t 
£ 
V 
£ 
£ 
V 
£ 
£ 
7] 
£ 
£ 
V 
V 
c 
i 
V 
£ 
£ 
7] 
£ 
£ 
V 
£ 
£ 
7] 
£ 
£ 
a i3 
a u 
a i5 
a l 
a 2 
a s 
a 4 
a 5 
a 6 
a? 
a s 
9 
a 10 
a n 
a i2 
«2 
a 3 
a 4 
a 5 
a 6 
a 7 
a s 
a 9 
a 10 
a n 
a !2 
«13 
a u 
a l5 
a i 
By column-subtraction twelve corresponding elements in the first three 
rows can then be made 0, and the determinant resolved into two ; for 
example, if we choose the last twelve elements there is obtained 
C «,*,£). 
a i 
a 13 
a 2 
~ a i4 
a 3 a i5 
a 4 - 
a i 
«12 
-s 
a i5 
- a 12 
a i 
~ a !3 
a 2 - a 14 
a 3 - 
a !5 
. . . 
«11 
- a 8 
a u 
“ a n 
«15 
~ a i2 
a i ~ «13 
a 2 - 
a !4 
a io 
— a 7 
a 5 
-a 2 
a 6 
-a 3 
a 7 -a 4 
a 8 
-Cl 5 
• • * 
a 4 
- a 13 
Since 3 is prime to 5, the pair of circulant factors can themselves have 
no factor in common save a x + a 2 + . . . + a 16 , and therefore 
(a 4 + a 2 + . . * + ^i5)C(a 1 , d 2 > • • • » 5) 
must be divisible by the product of the said pair. As an expression for 
the resulting quotient, which must be of the eighth degree in the original 
elements, Torelli obtains a determinant of the eighth order. He also 
extends the reasoning to a circulant having its order-number resolvable 
into more than two mutually prime integers : and finally makes application 
(§§ 8, 9) of his results to generalise Stern’s cofactor theorem of 1871 and 
a theorem of Rummer’s of 1851 on complex numbers. 
It will suffice to add that throughout his paper Torelli makes due 
