70 ON THE ANTIQUE HOUR-LINES. 



the bicrural branches of one right section are alternately placed 



in respect to the branches of the other right section. 



The undulated cone belonging; to each of the five hectemo- 



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rial lines is different from that of the others. The five upper 

 bases are represented in figures 4th, 5th, 6th, 7th, and 8th, 

 drawn according to the rules of the central projection, on a 

 plane parallel to the equator, for the latitude 66° 30'. The 

 full lines are the upper base of one of the two opposite undu- 

 lated cones ; the dotted lines are the upper base of the other 

 opposite undulated cone. These bases or right sections are 

 each made up of a pair of hectemorial lines for a given lati- 

 tude, forming one equicrural branch, and of similar and equal 

 pairs for other points of the same parallel of latitude and of the 

 opposite parallel. 



In order to present to the eye the image of one of the un- 

 dulated cones, figure 9th is drawn. It is a shaded view of one 

 of the two opposite conical surfaces to which the 3d and 9th 

 hectemorial lines belong, for the latitude 66° 30'. The point 

 of sight is in the plane of the equator ; the distance is two dia- 

 meters from the centre of the sphere. 



The sections of the undulated cones, by a cylindrical surface 

 circumscribed round the equator, are made up of the same 

 branches ; they are complete and re-entering, whereas the sec- 

 tions by a plane are always incomplete. These cylindrical 

 sections are laid down in figures 10th, 11th, 12th, 13th, and 

 14th. The cylindrical surface is here unrolled ; when restored 

 to its cylindrical form, the inscribed curve is re-entering, and 

 without a break. 



Where the number of degrees between the upper apices of 

 each undulation is a divisor of 360, the undulated cone is com- 

 pleted in one circumference, because at the beginning of the 

 second circumference, the generating diameter enters into the 



path 



