258 
PROFESSOR H. J. S. SMITH ON THE ORDERS 
Table I. 
A. — ^and F properly primitive. 
0 = 1, mod 2. 
0=2, mod 4. 
0=4, mod 8. 
0=0, mod 8. 
A=l, 
mod 2. 
¥ 
p-:t 
(_1)— ^ 
(-l)^,*( — 1 )T 
f-l F-l 
(-1>“. *(-!)“ 
p- 1 
(-i)~ 
A— 2, 
mod 4. 
F 2 — 1 
(-1)~¥ 
: 
* 
l\f 
7 
(-l)^,f(-l) F ^' 
F-l , F 2 — 1 
*(-!)—+— 
f-l F— 1 - F 2 — 1 
(-1)-, 
(-l)^t(-l)^ 1 
A=4, 
mod 8. 
/- 1 F-l 
•(-1)“. ("I)” 
P ~ 1 F-l 
(-1)“ 
f-l p-l 
*(_ipr + — 
f-l F-l 
(-1)-, (-1)“ 
f-l F-l 
(-ipr (-1)- 
/ 2 -i 
(-1)— 
A=0, 
mod 8. 
f-l F-l 
*<-iys (-i)~ 
F 2 — 1 
(-1)— 
f-l.P-l F-l 
*(-l)- + — , (-1)- 
P- 1 F 2 — 1 
t(-l)— , (-1)— 
f-l F-l 
(-1) % (-1)“ 
F 2 — 1 
(-1)— 
f-l F-l 
(_1)-F, (-1)2 
/2_1 F 2 — 1 
(-1)—, (-1)— 
B.— ^ improperly, F properly primitive. 
12=1, mod2 ; (-Ipt-f-l/t 
A=2, mod 4. 
F-l 
(-1)- 
A=0, mod 4. 
F-l F 2 — 1 
(-1)-, (-1)— 
C . — -f properly, F improperly primitive. 
A=l, mod 2; (-l/^=-(-lp\ 
0=2, mod 4. 
(-1)“ 
0=0, mod 4. 
/-I /*-! 
( — 1)~, (“!)“ 
In this Table the asterisk, prefixed to a supplementary character off, indicates that 
F— l . n'-i 
that character is attributed to f only when (— 1) 2 =(— 1) 2 ; prefixed to a supple- 
mentary character of F, it indicates that that character is attributed to F only when 
/- 1 A'— 1 
( — 1) 2 =(— 1) 2 , 12' and A' denoting the greatest uneven divisors of Q and A, taken 
