146 Proceedings of the Royal Society of Edinburgh. [Sess. 
and then squaring, etc., but which is better investigated by seeking the 
cofactors of 
0° or 0 6 , O'orO 7 , 0 2 or 6 8 , 0* or 6» , 0 4 or 0 10 , 6 5 
in the result of the multiplication. These are found to be 
( 
a, 
c, 
- d , 
c, 
rO 
e 
XX 
rO 
1 
c,d, e,f) 
i.e. 
a 2 - 2 bf+ 2 ce - d 2 , 
( 
-b, 
a , 
- f, 
6, 
-d, 
e i 
) 
i.e. 
o, 
( 
c, 
~b, 
a , 
-/» 
e, 
) 
i.e. 
2 ac-b 2 - 2 df+e 2 , 
( 
— d, 
c>- 
~b, 
a, 
-/» 
e 
) 
i.e. 
o, 
( 
e, 
- d, 
c, 
~b, 
a, 
-fi 
) 
i.e. 
2ae - 2bd + c 2 —f 2 , 
( 
e,- 
- d, 
c , 
~b } 
a § 
) 
i.e. 
o, 
bringing 
clearly 
out 
their law 
of formation. 
The product itself is 
evidently 
(a 2 - 2 bf+ 2ce - d 2 ) + 6 2 (2ac -b 2 - 2df + e 2 ) + 6\2ae - 2 bd + c 2 -/ 2 ), 
where 0 2 is one of the third roots of 1, and we are consequently entitled to 
conclude that 
C (a ,b,c,d,e ,/) = C (a 2 - d 2 - 2 bf+ 2 ce , - b 2 4 - e 2 + 2 ca - 2 df, c 2 -f 2 - 2 db 4 - 2ea ). 
Glaisher actually uses this to compute the final development of the six- 
line circulant. After correcting three misprints * and effecting condensa- 
tion by employing a symbol for “ alternating cyclic sums ” we find the 
said development to be 
2 ( ± « 6 ) - 3 2 { ± “W+ ce + rf2 )} 
+ 2^ ± a 3 (c 3 + 3 & 2 e + + 6 bed + 6tZe/)) 
+ 92 ,( ± <* W ) + 9 ^ { ± « 2 (& 2 / 2 “ 2 c 2 V )} . 
where, for example, 2 { - a 2 (b 2 f 2 — 2c 2 c£/) j stands for 
a 2 (b 2 / 2 - 2 c 2 df) - b 2 (c 2 a 2 - 2 d 2 ea) + c 2 (d 2 b 2 - 2 e 2 fb) - 
o 
but where, nevertheless, ±^ 2 c 2 e 2 ) does not stand for 
a 2 c 2 e 2 — b 2 d 2 / 2 + r 2 e 2 a 2 — d 2 f 2 b 2 4- e 2 a 2 e 2 - f 2 b 2 d 2 
but merely for a 2 c 2 e 2 — b 2 d 2 f 2 .\ 
The rest of the paper (§§ 10-18) does not contain anything fresh so far 
as circulants are concerned. 
* 2 e 3 a 3 , 12 c 3 bcd , 12 a?def should be 2c% 3 , 12 a?bcd } 12 c 3 def. 
