ON THE ANTIQUE HOUR-LINES. 71 



path it had described in the first ; this is the ease with the 

 three undulated cones which contain the third, fourth and fifth 

 hectemorial lines, figures 6th and 12th, 7th and 13th, 8th and 

 14th. 



The number of degrees between the meridians of two adja- 

 cent upper apices, is equal to twice the distance between the 

 two adjacent >ides of each undulation measured on the equa- 

 tor, or to four times the distance of the equatorial point of 

 the hectemorial line from the meridian measured on the equa- 

 tor. 



Where this number of degrees between the meridians of the 

 upper apices is not a divisor of 360, but of a multiple of 360, 

 the revolution is completed in as many circumferences as there 

 are units in that multiple ; because the generating diameter 

 does not enter into its former path till it has described that 

 number of circumferences. The undulated cone which con- 

 tains the second and tenth hectemorial lines, having an inter- 

 val of 16 equinoxial hours, or 240° between the adjacent up- 

 per apices j the describing diameter completes its revolution, 

 and comes to the point from which it set out at the end of two 

 circumferences or 720° ; the number of the upper apices being 

 3, which, multiplied by 210, is equal to 360 X 2 ; this is seen 

 in figures 5th and 11th. 



The undulated cone which contains the first and eleventh 

 hectemorial lines, figures 4th and 10th, having an interval of 

 20 equinoxial hours, or 300° between the upper apices of the 

 undulations, the describing diameter must go over five circum- 

 ferences before it comes to the point from which it set out, be- 

 cause 300 multiplied by 6, which is the number of upper api- 

 ces, is equal to 360 X 5.. 



The- 



