172 Proceedings of the Boyal Society of Edinburgh. [Sess. 
Further, it is shown that such circulants being centrosymmetric can 
be resolved into factors in quite a different way, the repeated factor 
now appearing not as a pfaflian but as a determinant ; so that by the 
equatement of the two resolutions we obtain equality between the 
two determinants 
CLg CIq ^4 ^3 ^4 
a x - a 3 a 2 - a 4 a 3 - a 5 
& 2 ^4 ^2 ^3 
&2 ^4 <^5 $2 ^4 
<N 
e 
i 
CO 
e 
i 
(N 
e 
s 
1 
eo 
e 
5 
a B ~ a 5 a 2~~ a 4: a \ ~ a S 
and the above-given pfaffians respectively. 
Bickmore, C. E. (1898). 
[Question 13748. Educ. Times , li, p. 40: Math, from Educ. Times, 
lxix, p. 85.] 
The result here recorded is that the skew circulant of the fourth order, 
C'(x, y,z, w), whose final expansion is 
x 4 + y 4 +. z 4 -f- w 4 + 2 (x 2 z 2 + y 2 w 2 ) 4 4 (x 2 yw - y 2 zx — z 2 wy 4- w 2 xz ) , 
is expressible in each of the forms 
a 2 + b 2 , c 2 - 4- 2d 2 , e 2 -2/ 2 
where a, b, e, d, e,f are definite quadratic functions of x, y , z, w, namely, 
x 2 + 2 yw — z 2 , y 2 - 2 zx - w 2 , 
x 2 — y 2 + z 2 - tv 2 , xy -yz + zw + wx , 
x 2 + y 2 4- z 2 + w 2 , xy +.yz + zw -wx . 
This is reached by taking the four linear factors 
x + £y + % + t?y - tfz + tw , x - ty 4- i^z - £% , x - tfy - £ 2 z - £iv 
of the circulant and multiplying them in the three different ways as a pair 
of pairs. For example, the product of the first and second factors being 
x 2 - y 2 + z 2 - w 2 + (£ + £ 3 )(xy - yz -\-zw + vjx) , 
and the product of the third and fourth being 
x 2 — y 2 4- z 2 - w 2 - (£ 4- C 3 )(xy -yz + zw 4- wx ) , 
the product of the four is got in the form 
( x 2 - y 2 + z 2 - w 2 ) 2 4- 2 {xy - yz + zw + wx) 2 . 
