149 
1911-12.] The Theory of Circulants from 1861 to 1880. 
To establish the second part we have to take along with the n condi- 
tioning equations the n identical equations 
(<*» . 
5 n ) • • 
. , ’ ^3 > ' * 
• t ^2n) 
= 0) 
(«4 . 
— a % , a<2 , . . 
• . - “5 $ 
) 
= 0 
(a 2 n , 
“ a 2n-l 5 a 2n-2 j • < 
■ • . - “l i 
) 
= 0 
/ 
thus obtaining a set of 2 n equations having a Y , a 2 , a 3 , . . . , <x 2n as quasi 
unknowns and ( — 1) W C as the determinant of their coefficients. The 
usual expression for an unknown as the quotient of two determinants 
leads at once to the desired result. Thus when n — 3 the set of 
equations 
is 
( a 
-/ 
e 
-d 
c 
-b $ a,b ,c,d,e,f) = 1,0, 0,0, 0,0. 
b 
- a 
f 
— e 
d 
— c 
c 
-b 
a 
e 
-d 
d 
- c 
b 
- a 
f 
- e 
e 
-d 
c 
-b 
a 
-f 
f 
- e 
d 
- c 
b 
- a 
LIST OF 
AUTHORS 
whose writings are 
herein dealt with. 
PAGE 
PAGE 
1861. Roberts, M. 
. 136 
1877. Nicodemi, R. . 
. 142 
1862. Zehfuss, G. 
. 137 
1878. Scott, R. F. 
. 142 
1864. Baltzer, R. 
. 137 
1877. Glaisher, J. W. L. . 
. 143 
1870. Baltzer, R. 
. 138 
1878. Glaisher, J. W. L. . 
. 145 
1871. Stern, M. A. . 
. 139 
1878. Menesson 
. 147 
1872. Glaisher, J. W. 
L. . . . 140 
1878. Minozzi, A. 
. 147 
1875. Gunther, S. 
. 141 
1879. Lemonnier, H. 
. 148 
1875. Baltzer, R. 
. 142 
1879. Glaisher, J. W. L. . 
. 148 
(Issued separately May 17, 1912.) 
