Theorems 
of Crystal- 
lometry. 
394 Mr. WuHeEwELL on the Classification 
It is easily seen that pR is a triangular pyramid, and, when 
its faces are repeated, a rhombohedron; that pRm is a six-sided 
pyramid, the hexagonal base of which has alternately equal angles 
(V'AUBSCT, Fig. 1.). Also pP and pQ are four-sided pyramids: 
pPmisa four-sided pyramid (V'4'BC’, Fig. 2.), and pQm an eight- 
sided one, with the angles of its base alternately equal (/’’BSC, 
&ce. Fig. 2.). The derivations of O are less obvious, but need not 
here be explained. 
In the combinations of different forms, the intersections of the 
planes bounding the figure are called the Edges of Combination. 
These lines have various parallelisms and relations, which, with 
other properties of the forms, are enunciated ‘in the following 
theorems : 
A. Iftwo forms are derived by the first law according to different 
indices, their edges of combination are horizontal (the axis 
being vertical). 
B. If two forms are derived from the same form by the second 
law, their edges of combination are parallel to the slant edge 
of that form. 
C. In the following series, each form truncates the edges of the 
following one, making parallel edges of combination ; 
ane Wa ee OR Ry eRe, 
Jock Roo Ry SR, PRG ER aR, eR ey, 
1Qr,3Q, Qr, Q, 2Qr, 2Q, 4Qr, 
Px) sPorieP’rs Wee 
D. If a be the axis of O, P, Q, R, the axis of pO, pP, pQ, pR 
will be pa. 
E. If 6 be the principal semi-diagonal of the base of O, P, Q, 
pOm, pPm, pQm will have for their axes pma, and mb for the 
intercepted part of the principal semi-diagonal. 
F. The bases of pOm, pQm, are octagons with alternately equal 
