148 Mr Haidinger on the Series of Crystallisation Apatite. 
between a and' r, are parallel to the terminal edges of r, hence 
the latter is the scalene sid-sided pyramid belonging to R — 1 , 
that is to say P-— 1. The axis of this pyramid is equal to 
one-half of the axis of P, and the angles, as given above for 
the variety Fig. 12. In the same way in which r and a, and 
X and s are co-ordinate members of the two series of six- 
sided pyramids, and of the rhombohedrons to which they be- 
long, so likewise z and d belong together, being P + 1, and 
2 (R -f 1), as the consideration of the figure will sufficiently 
prove. The angles of P -f- 1 (s) are r=:129° P; US'* 48', 
those of 2 (R + 1) (d) 123° 31' ; 142° 18'. The faces per- 
pendicular or inclined to the axis are very smooth and splen- 
dent ; those which are parallel to it, commonly bear more or less 
deep striae. The intersections of the different forms with each 
other cannot, in every instance, be observed at one and the same 
angle, and it is here as in so many other cases, that the prac- 
tised eye of the crystallographer must supply the accidental de- 
ficiencies, owing to the irregular formation of crystals. 
Very interesting varieties of forms are met with among the 
white transparent crystals from St Gothard, as, for instance, 
those in Figs. 16. and 17. They exhibit, most distinctly pronoun- 
ced, the faces of the scalene six-sided pyramids belonging to R, 
marked in the figures with the letters u and h. They moreover 
contain faces of the limits of the series of these forms, unequi- 
angular twelve-sided prisms, which likewise partake in the re- 
markable property that they enter into the combinations with 
only half the number of their laces. A very beautiful speci- 
men, containing both the pyramids, u and 6, is in the excellent 
collection of Mr Allan. 
The crystallographic problem to be resolved in respect to the 
varieties represented, is, from the observed parallelism of the 
edges of combination, to find the geometrical relations of the 
simple forms towards each other, and to express these forms 
themselves by their crystallographic signs. 
From the preceding combinations, we know P to be — R — oo , 
r=P — 1, a=:2 (R— 1), sz=^{n), ^ = P-|-1, M = 
P -j- 00 , and ^ = R -f- go ; the forms to be determined are there- 
fore only those marked u, and b, whose general expression is 
(P-|-n)»^; and those marked c and expressed in a general 
