1905-6.] Dr Muir on the Theory of Circulants. 
393 
The procedure, however, is rather perverse, the theorem of § 13 
being forced into service. This gives 
A =(-l)"- 1 l A', 
n 
where A ' is the determinant of the system 
s 
a x 
-a 2 
a 2 — 
a 3 . . . 
. (i n _ i 
- a, 
s 
a 2 
-a s 
i 
CO 
a 4 . . . 
. a n 
~ a i 
s 
a B 
-a. 
a 4“ 
a 5 . . . 
. Oj 
- a ( 
s a n — rtj a Y - a 2 a n _ 2 — a n _ x , 
after which A ' js is partitioned into determinants with monomial 
elements, and certain more or less evident reductions made. The 
result is “ Le determinant du systeme propose s’obtiendra en 
multipliant a l + a 2 + ... +a n (i.e. s) par le determinant du 
systeme 
a x - a 2 
a 2 
a 3 . . 
. . a„ 
_i - a, 
a 
to 
1 
a 3 
- « 4 • • 
. . a n 
, - a- 
e 
e 8 
a n 
. . a n 
3 _ 
a theorem which afterwards came to be written in the form 
C(flh , a 2 , . . . , a n ) = (a 1 + a 2 + ... + a n ) 
• V{a l -a 2 , . . . , a n _ x -a n , a n -a 4 , . . . , a n _ 3 -« n _ 2 ) 
the symbol P(cc, y , z,w, v) being used to stand for the “per- 
symmetric ” determinant 
x y z 
y z w 
z w v 
Spottiswoode (1853). 
[Elementary theorems relating to determinants. Re-written and 
much enlarged by the author. Crelle’s Joum ., li. pp. 209- 
271, 328-381.] 
In the section (§ xi.) which did appear in the first edition, and 
which bears the title “ Miscellaneous instances of determinants,” 
