396 Proceedings of Royal Society of Edinburgh. [sess. 
V — nD 
i = »D a = 
The third 
Lemma 
is to : 
the effect 
that the quotient of the 
determinant 
m o 
do 
q x d . 
• • • q n - *d n 
—2 
m 1 
d 
di d 
q 2 d 2 . 
. . . g n -\d n 
-1 
m n 
_f n ~ l 
dn-ld n ~ 
1 do 
• . • q n -sd n 
-3 
by d is a 
rational function 
of d n . By multiplying the 2 nd , 3 rd , 
4 fch , . , . . 
columns by d n , d r> 
i—l Jn— 2 
. . . respectively, and then 
dividing the corresponding rows by d , 
d 2 , d*, ... , 
. respectively, 
there is obtained 
m 0 
q 0 d n 
<h dn • ■ 
, . . q n _f n 
m 1 
<di d " 
q 2 d« . . 
. . q n -\d n 
rn 2 
q 2 d n 
<hd n . . 
• • do 
m»-i 
Qn-id n 
do 
• • q n - s 
where no power of d occurs except the w th . But, if the original 
determinant be H, the latter is 
H • d n d n ~ x d n ~' 1 . ... d 2 . H Jn 
d<p* . . . ^ 1 - e - i' d ’ 
consequently H/<i is of the form asserted. 
In connection with this last lemma it is curious to find no note 
taken of the closely related and more attractive fact that 
C (cq , a 2 d , aft 2 , ... , a n d n ~ x ) 
is a rational function of d n . 
Bellavttis (1857). 
[Sposizione elementare della teoria dei determinant. Venezia, 
Mem. 1st. Veneto, viii. pp. 67-143.] 
Circulants are practically unconsidered by Bellavitis in his 
exposition, all that appears (§ 85) being two of Laplace’s expansions 
for C (a,b,c,d) obtained by means of Cauchy’s “ chiavi alge- 
briche,” namely, 
( a 2 - bd) 2 - (b 2 - ac) 2 + (c 2 - bd) 2 - (d 2 - ac ) 2 - 2 (ab - cd)(ad - be) 
and ( a 2 - c 2 ) 2 - (b 2 - d 2 ) 2 - 4 (ab - cd)(ad - be) . 
