Mr Haidinger on the Series of Crystallisation of Apatite. 149 
manner by inasmuch as the latter are limits of the 
series of the pyramids themselves. 
According to the derivation employed in the crystallographic 
method of Professor Mohs, a scalene six-sided pyramid is ob- 
tained from a rhombohedron, by lengthening the axis of the 
latter on both sides to an indefinite but equal length, and join- 
ing the terminal points of the axis thus determined, and the late- 
ral angles of the rhombohedron, by straight lines, as in Pig. 16. 
The number m expresses the ratio of the lengthened axis A! X', 
to the original one A X. Two forms, a rhombohedron, and a 
scalene six-sided pyramid thus connected with each other, are 
considered as co-ordinate members, or such as belong together in 
their respective series. 
The edges of combination between s and u are parallel to the 
lateral edges of R ; so are those between s and h. Both the py- 
ramids, therefore, belong to R, and their general sign, n being 
equal = 0, becomes (P)®*, where the exponent m is still to be 
determined. 
The situation of u is exactly determined by the parallelism of 
the edges of combination between x and w, and those between 
u and e. Suppose in Fig. 19., AEB, ABD, ADC to be three 
faces of the isosceles six-sided pyramid P, ABKC one of the 
faces of the rhombohedron R. If a face of the scalene pyramid 
in question passes through the point B, its intersection with 
ABKC will coincide with the edge BK, the latter being the 
edge of combination between the faces of R and P -f x ; but 
the line BN, its intersection with ABD, will bisect the terminal 
edge AD of the pyramid P in the point N, because B'BNCC' 
denotes the direction of one of the faces of R -|- x (^), passing 
through the point B. The line KI, which joins the lower 
angle K of the rhombohedron R, with the angle of combination 
N, determines the situation of I, the apex of the required pyra- 
mid, one of whose faces, therefore, is the triangle IBK. The 
axis of the derived pyramid is = 2 AI -f- 1 AG, AG being = | 
of the axis of R, and AI the prolongation of the axis on one 
side of the rhombohedron, analogous to A A' in Fig. 18. Now, 
lA -f AH : HN lA -f AG : GK, 
and a being the axis of R, 
lA -\-\a\ I = I A 4- 1 a : 1 . 
