72 



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 the 

 hcctcmorial lines 



intercepts between two 

 adjacent upper apices, 



The number 

 of upper 

 apices is 



The number of 



circumferences in 



which a revolu 



lion is completed, 



is, 



First A, and eleventh I A, 

 Second B, and tenth I, 

 Third r, and ninth 0, 

 Fourth A, and eighth H, 

 Fifth E, and seventh Z, 



In degrees.\In equinoc- 

 tial hours. 



6 

 3 

 2 

 3 



6 



5 



2 

 1 

 1 

 1 



300 

 240 

 180 

 120 

 60 



20 



16 



12 



8 



4 



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 

 rio-ht section of the undulated 

 cone, by the change of sign of 

 tan. pol. dist. in the values of x 

 and y ; 



x 



— I cos-s — coss j tan, pol, dist. 



n 



U 



sin, - s tan. pol. dist. 

 6 



et 



For example, in the right sec- 

 tion of the undulated cone which 

 contains the third and ninth hec- 

 temorial 



