1915-16.] The Theory of Circulants from 1880 to 1900. 173 
We note for ourselves that by the multiplication-theorem we obtain 
also 
%.e. 
and 
%.e. 
X 
y 
z 
w 
2 
e 
f 
-f 
- w 
x 
y 
z 
f 
e 
f 
— z 
— w 
X 
y 
f 
e 
f 
-y 
- z 
— w 
X 
~f 
f 
e 
e 
f 
V 
e 
f 
e 
2/ 
-f 
f 
e 
= (e 2 — 2Z 2 ) 2 • 
X 
y 
z 
w 
X 
-y 
z 
- w 
c 
-d 
-d 
- w 
X 
y 
z 
- IV 
- X 
y 
- z 
-d 
- e 
d 
- z 
— w 
X 
y 
— z 
w 
X 
-y 
d 
c 
-d 
-y 
- z 
- w 
X 
-y 
z 
— 10 
- X 
-d 
-d 
- c 
C'(x, y,z,w).(- l) 2 C'(x, y, z, io) = (c 2 + 2d 2 ) 2 ; 
X 
y 
z 
w 
X 
10 
- z 
y 
a 
b 
— 10 
X 
y 
z 
- w 
z 
-y 
X 
-b 
a 
— z 
-10 
X 
y 
- z 
y 
— X 
- w 
b 
- a 
-y 
- z 
- w 
X 
-y 
X 
w 
— Z 
a 
b 
C’(x , y , z , w) . ( — 1 ) 2 C'(x , y , z , w) = ( a 2 + & 2 ) 2 
LIST OF AUTHORS 
whose writings are here dealt with. 
1880. Studnicka, F. J. 
1880. Scott, R. F. 
1880. Glaisher, J. W. L. 
1880. Weihrauch, K. 
1880. Weihrauch, K. 
1881. Muir, T. . 
1881. Muir, T. . 
1881. Muir, T. . 
1882. Torelli, G. 
1882. Muir, T. . 
1882. Bhut, A. B 163 
1883. 
Cesaro, E. 
PAGE 
163 
1883. 
Neuberg, J. . . 
. 163 
1884. Forsyth, A. R. 
. 164 
1884. 
Muir, T. ... 
. 164 
1885. 
Muir, T. ... 
166 
1889. 
Robinson, L. W. 
. 168 
1891. 
Rogers, L. J. 
. 168 
1894. 
Woodall, H. J. 
. 168 
1896. 
Muir, T. ... 
169 
1898. 
Bickmore, C. E. 
. 172 
1899. 
Thyagaragaiyar, Y. R. . 
. 168 
PAGE 
151 
151 
152 
153 
154 
154 
157 
158 
159 
163 
{Issued separately October 14, 1916.) 
