152 Mr Haidinger on the Series of Crystallisation of Apatite. 
The transverse sections of the two prisms are therefore equal 
to each other as to the measure of their angles, but they differ 
n the relative situation of their more acute and more obtuse 
edges. In the combination, Fig. 16., which, besides It + x (e), 
and P -b X contains the faces of (P -f- x )| (c) situated to 
the left, and those of (P + x y (f) situated to the right of the 
faces of P + X , the transverse section is a figure of twenty-four 
sides, the angles of which are alternately 169° 6' 24'^', and 160° 
53' 36". 
If in Fig. 18. the more acute terminal edge of the scalene 
six-sided pyramid is called x, the more obtuse one y, the lateral 
edge z ; and we suppose the axis of the rhombohedron to be- 
come infinite, the pyramid is transformed into a twelve-sided 
prism, of which the alternating angles are equal. The edge y 
becomes vertical, and appears in the combinations contiguous to 
the faces of R + x (^), the edge z also becomes vertical, and 
appears contiguous to the faces of P -f- x (M), exactly as is 
mentioned in the preceding Table. The reversed equality of 
the angles in two different prisms, as given in this Table, de- 
pends, therefore, upon that of the geometrical expressions for 
these two edges. 
The general expression for the edge y is 
/(3w«4-6m--l) ^ 24.18 
cos y = — I ^ ' 
that for the edge z, 
cos z = _ — l)a^— 9 - 
V(3to* + 1) + 9, 
a being the axis of the rhombohedron, to which the pyramid be- 
longs, and m the number of derivation, or that which expresses 
the ratio between the axes of the two forms. 
If now, a being infinite, cos y of (P -f- x is supposed 
= cos z' of (P 4- X for a certain m in the first, and another 
m in the second expression, we obtain 
Sm^ 4 - 6m ■ — 1 Sm'^ — 1 
2[(37722 + 1 ) a^^9)] 
)• 
2 ( 3^2 4 - 1 ) "" 
and 
m 
1 4- 6m 4- 9 
'^^2— ,2m4- Ij’ 
1 4- Sm' 
or m' 
1 4- 3m 
3 (m — 1) 
, and 
3(m'—l) 
from the last, m = 
