395 
1905 - 6 .] Dr Muir on the Theory of Girculants. 
where a r stands for a r and a for a primitive root of the equation 
x n - 1 = 0, he multiplies the determinant 
«o 
«1 
a 2 . , 
• • «»- 1 
cq 
«2 
a 3 * 
. . a 0 
a 2 
a 3 
a 4 • 
. . a x 
or D say, 
«»-i 
a o 
a Y . 
• • Un-2 
by the determinant 
1 
1 
1 
.. 1 
1 
a i 
a 2 
• • a n- 1 
1 
«; 
• • -Li 
or A say, 
1 
a n f r 
ar 1 
• • c; 
5 
and obtains a; product-determinant from whose columns, he 
says, 
the factors 0 1 . 
, 0 2 , . 
• * 5 
0 n may be removed in order, so 
that 
their results 
1 
1 
1 
. . 1 
1 
2 
n — 1 
a n- 1 
<-l * • 
• • a n _! 
DA = Of 2 
. . . o n 
1 
a n- 2 
2 
a n-‘2 * • 
n— 1 
• • %-2 
1 
2 
71 — 1 
a L . • 
. . a, 
— Of) 2 . • • • 
0 n • 
(-1) ! 
n(n- 1)^ ^ 
and .*. 
D = ( 
-l) 1 
n(n— 1) 
Of) % • • • 0 n . 
The proof, which is said to be due to Brioschi, is not improved in 
neatness by introducing the conception of 
a primitive root, nor by 
writing the root 1 in a different form from the other roots. 
The second lemma concerns the differential-quotient of D with 
respect to any variable of which the a’s are functions. Denoting 
this differential -quotient by D', and by D r the determinant got 
from D by substituting for each element in the r th column the differ- 
ential-quotient of that element, Cremona of course has at once 
D = Dj + D 2 + . . . + D n . 
As, however, Dj here can be shown by translation of a number of 
rows and the same number of columns to be equal to any one of 
the D’s following it, there results 
