Ill 
the angles of crystals. 
31. But since the solid angles at A and at x are symmetri- 
cal, for every plane at A we shall have a co-existent plane at 
xf of which we shall find the equation. 
We may as before suppose A x^ Ay, A 2:, to be co-ordi- 
nates, and with any plane r at A we shall have a co-exist- 
ent plane PQR at x, such that xP, jtQ, xR are equal to 
Ap, Aq, A r respectively. ^ 
Prop. The symbol of pqr being (piqir) to find the 
symbol of PQR. 
Let the small component parallelepipeds have the edge in 
direction A a: = and the edges in dir.ections Ay, A z each 
= c (these being equal). Also, let Ax==na, Ay Az = nc.f 
And let the plane be obtained by taking away h molecules 
in the direction Ax, k in the direction Ay, and I in the direc- 
tion A z. Therefore Ap =zha, Aq = kc, A r = I c : and the 
equation to the plane p qris 
-L . X L = 1- 
h a k c • I c ’ 
• The parallelepipeds of which the solid is supposed to be made up at x, are not 
in the same position with those of which it is supposed to be made up at A. Those 
at X are bounded by planes parallel to Axy, Axz,yxz, as those at A are by the 
planes which meet at A. If the crystal be divisible according to all the planes of 
a tetrahedron or octahedron, there are four different kinds of pardlelepiped of 
which it may be conceived to be composed, corresponding to the four angles A, x, 
y, z. And we may take any one of these kinds with equal propriety. In fact, the 
mode of conceiving secondary planes to be formed by removing parallelepipeds, is 
an assumption to be considered right only so far as it exhibits the dependence of 
secondary planes upon the simplicity of the ratios p : y : r. 
f If we suppose Axy z\.o be made up of parallelepipeds. Ax, Ay, and A z having 
equal numbers oi them, planes parallel to xyz will pass through all their angles. 
And if instead of parallelepipeds, we suppose that we have only points in space 
where the angles of the parallelepipeds would be, the planes which are determined 
by any adjacent three points will be the four planes, Axy, Axz, Ayz,xyi. 
