ON HEXAGONAL FACETS IN A CIRCLE. 85 
hexagonal facets in a given hexagonal arrangement of the 
same. 
This is a necessary step towards finding the number of 
such facets contained in a circle described about this hexago- 
nal group. 
VII. We know indeed that the number required must be 
(by § II) less than N? x ‘9 (=C) and not less than H, or H, 
(or H). 
The question therefore is,—Can we find a proportion of the 
difference C — H, which being subtracted from C, would give 
a remainder equal to the number required ? 
Now proportion, in the case of N = 164, is found, viz., by 
actually calculating all the ordimates in the segment of the 
circle beyond each side of the hexagonal figure, erected, on 
the versed sine, at intervals equal to the diameter of a facet, 
and then finding the number of facets that may be arranged 
between each pair of ordinates in succession to be so nearly 
L C — H) that we may well be contented with this approzi- 
4 y p 
mate rule for finding the number (8) of entire hexagonal facets 
contained in the given circle, viz., 
1 9 1y9 7 
=C — -(C—H) = — N?—-4 —N?— Her. 
ah (Cc H) rie era J 
