and Mineralogy. 169 
vian. Tinstone, Yellow Lead, Uraii-Mica, Rutile, Anataae, and 
others. 
46. Accompanying Series . — Upon this series depends also an 
accompanying series of four-sided pyramids, with square bases : 
its co-efficient is Supposing the horizontal projections to 
be equal, the axis of a member in the principal series is to the 
axis of a member in the accompanying series as II' 
P (Fig. 14. PI. VIII.) is put then is u (Fig. 15. PI. VIII.) 
rr: -f 3 : SO that the principal series is denoted by jo + ; the 
accompanying series by -f n. If c (PL xlvii. Fig. 70.) is put 
= /?, then is n (Fig. 73.) r= — 3. 
47. Scalene eight-sided Pyramid. — If in Fig. 183. — 185, 
(PL Ixxx.) the laces z., are enlarged till they alone compre- 
hend the whole solid, a scalene eight- sided pyramid results. Three 
pyramids of this kind depend upon each four-sided one with 
a square base, in the principal series ; and the axes of the former 
are to those of the latter as 3 : 1, 4 : 1, 5 : 1. In the order here 
assigned to them, like the six-sided pyramids formerly mentioned 
(33.), they are named the primary^ secondary^ and ternary. The 
manner in which these eight-sided pyramids originate from the 
four-sided pyramids, cannot be here developed for want of 
room. 
48. Their Series . — Three series of pyramids, therefore, the 
series of primary, of secondary, and of tei’iiary pyramids, de- 
pend upon the principal series of four-sided pyramids with 
square bases. These three series present among themselves a 
general series, the expression for which will be {p -b 
n -j- 2)', the general expression of these 
pyramids (34.) being {p jF n)^. This general series possesses 
properties analogous to those of the series of scalene six-sided 
pyramids. 
49. Limits . — The limits which bound this single series of 
eight-sided pyramids (45.) are eight-sided prisms, having the 
same unequal angles as the transverse section of the pyramid 
to which they belong. Since the horizontal projections of two 
consecutive four-sided square-based pyramids occupy different 
places, these prisms must be considered as having two different 
positions (39). 
