166 Proceedings of the Koyal Society of Edinburgh. [Sess. 
or, with the help of the fresh property, in the still more condensed form 
v=6 . 
2 ,2K- K'ww + • • • ) > 
;u. = 0 v = l ' ' ' 
where every suffix greater than 6 is to be made less than 7 by subtracting 
a multiple of 7 from it and duplicates are to be neglected. 
An entirely different mode of saving labour in the computation rests 
on the fact that all the terms which contain an element in the first 
power can be got by evaluating a zero-axial determinant of the next 
lower order. For example, since every term of C(a 0 , a x , a 2 , <x 3 , a 4 ) is 
o 
of this kind except we evaluate 
a i 
a 2 
a 3 
a 4 
a i 
a 2 
a 3 
a 4 
a 2 
CO 
8 
a 4 
and thus finding the cofactor of a 0 to be 
- 5 (a\a 2 + a\a± 4 a\a x + a\a 2 ) 4 5(a\a \ + a\afj - 
we obtain the full expansion 
~ 5 2 «o(^i a 4 + a 2 a 3^ + 5 2 a o(al a 3 + - 5 a 0 a 1 a 2 a 3 a 4 . 
In the case of the seven-line continuant the saving is almost quite as 
great, since in addition to the term 14&a\a \a\ is the only one that 
does not involve a first power of an element. 
Mum, T. (1885). 
[Detached theorems on circulants. Trans. Roy. Soc. Edin., xxxii, 
pp. 639-643.] 
The first of the theorems is that if in a circulant the element in the 
place (p, q) be by the nature of the circulant the same as the element in 
the place (r, s), the complementary minor of the former element is to that 
of the latter as ( — l) r+s : ( — l) p+q . But more than this is intended to be 
brought out, namely, that by cyclical transposition of rows and columns 
the element in the place (r, s) may be made to take the place (p, q) 
and the determinant be in outward form exactly the same as before. 
The second theorem is Stern’s of 1871, although not so credited. The 
mode of establishing it is to alter the first column of C(a, b, c, d, e) so 
as to make it possible to remove a + boo + Cco 2 + dw 3 + e^ as a factor, and 
leave 
1 , CO , <D 2 , CO 3 , O) 4 
