9‘1 
the angles of crystals, 
{pi being understood, that when quantities are only 
separated by commas, they are to be taken in all the ways 
in which they can be permuted. In the same manner 
{p,q \ r) may represent the two planes {p,q^ r) {q,p; r), 
the permutations not extending to r, which is separated by a 
semicolon. In the case of the rhomboid, however, the per- 
mutations always include all the three quantities, in conse- 
/ 
quence of the similarity of its three edges. 
4. We have hitherto considered only the planes produced 
by cutting off the upper angle ; but we may represent in the 
same manner the plane produced by truncating any other 
angle. It maybe observed that the angles oc,y,%,fig. 3, 
which are separated from the superior angle A by an edge, 
.are called lateral angles. The angles od , f , which are 
separated from A by a diagonal, are called inferior angles. 
Let/) q r, fig. 3, be a plane produced by a truncation at the 
lateral angles : ccp, xq, xr being h, k, I respectively. Produce 
r A beyond A, and take AP = xp, AO = xq, AR = xr ; then 
the plane P Q R will be parallel to p q r, and may be taken 
instead of it. Now it is manifest that the equation to this 
plane is + A + 
and therefore its symbol is | — T’ T’ 
^ = y , r = -k, the equation is p r + qy + rz z= rn, and the 
symbol ( — p ; q; r). Hence a plane which cuts off the 
lateral solid angles is distinguished by having one negative 
index. 
In the same manner let p q r, fig. 4, cut off an inferior 
angle x', so that x' p zsz h, x' q=zk, x* r:=z I : and taking 
