1915-16.] The Theory of Circulants from 1880 to 1900. 165 
the new sets of suffixes 2,4,6; 3,6,2; 4,1,5; 5,3,1; 6,5,4: and 
so finally the complete set of terms 
- 27a 3 0 (ala 2 a B + a\a^a & + a\a 6 a 2 + cCia Y a b + c^a B a x + a%a b a 4 ) . 
The “reason of the rule ” is that 
C(0 ,1,2, 3, 4, 5,6) = C(0 ,2, 4, 6, 1,3, 5), 
= C(0,3, 6, 2, 5, 1,4), 
= C(0 ,4, 1,5, 2, 6, 3), 
= 0(0 ,5,3, 1,6, 4, 2), 
= C(0 ,6,5, 4, 3, 2,1); 
as we can show either by transposition of rows and columns, or by 
considering the circulant in its form as a product of seven linear factors 
and using the fact that, if w be one of the imaginary seventh roots of 
unity, w 2 , a) 8 , to 4 , co 5 , ft) 6 are the others. 
A caution is then given that two terms having apparently the same 
form have not necessarily the same coefficient ; and as an illustration it 
is shown that in Forsyth’s expression half of the terms with the coefficient 
35 should have the coefficient —14 and that the coefficient of his final 
term should be changed from —448 to —105, the whole expansion then 
being 
- 7 2 + a 2 a 5 + 
+ ^ 2/ ( a i a 5 + a ^ a -6 + a l a \ + a l a Q + a l a i- + a 6«2 
+ 14 2 a\[a x a 2 a x + a B a b a^j 
- 7 ^ a 0 (a\ a\ + a\a\ + a\a£\ 
- 2 1 2 al (^a\a 2 a B + + a\a b a 2 + a\ a Y a b + a\a z a Y + a\a b a l 
+ 14^ al(a\al + a\a\ + ala\ 
+ 7 ^ + a 2 api B a b + a B a b a x a± 
- 7 ^ a 0 (a\a\a\ + aja|a§) 
+ 35^ ao(^ a i a 3 a 4 a 5 + a l a Q a i a z + a\a 2 a b a Y 
- 14 ^ al(ala 2 a A a 6 + ala^a^ + a\a b a b a^j 
- \^a b ayx 2 a, B a i a b a b ; 
