eh ON THE ANTIQUE HOUR-LINES. 
The circumstances of each of the undulated cones with re- 
spect to the number of undulations and of circumferences, are 
as follows : 
| The undulated cone containing 
hectemortal lines 
—$ ———S- ————- 
In degrees. 
First A, and eleventh IA,| 300 
Second B, and tenth 1,| 240 
Third T, and ninth @,| 180 
Fourth A, and eighth H,| 120 
Fifth E, and seventh = Z, 60 
o the | intercepts between two 
adjacent upper apices, 
a 
In equinoc- 
tial hours. 
20° 
16 
12 
8 
4 
of upper 
apices 1s 
Awnw a” 
—— —— | 
The number | The number of 
circumferences in 
which a revolu. 
| tion is completed, 
ts, 
Hee bo & 
Four of the five undulated cones have opposite and similar 
undulated cones, formed by the remote half of the generating 
diameter; but in that undulated cone which contains the 
fourth hectemorial line A, and eighth H, the two opposite un- 
dulated cones coincide in one, as is seen in figures 7th and. 13th. 
The algebraic formula expres- 
ses the different branches of the 
right section of the undulated 
cone, by the change of sign of 
tan. pol. dist. in the values of @ 
and y; 
n : 
TS cosgs—cos8)) tan. pol. dist, 
y = sin. stan. pol. dist. 
For example, in the right sec- 
tion of the undulated cone which 
contains the third and ninth hec- 
temorial 
—+- 
