the angles of crystals. 
125 
And hence we may arrange the faces in the order of their 
longitude and latitude. 
We might in the same manner find the position of the 
planes for other primitive forms, but what has been done will 
generally be sufficient. 
§ 9. On the angles made by edges. 
45. If we have two lines referred to any co-ordinates, of 
which the equations are y — Ax, z=zBx; y =. A!x, z = VI x ; 
and if the plane angles of the faces be known ; viz. the angle 
which X makes with y = <p, the angle which x makes with 
z=z-^ and the angle which y makes with z = u ; we shall 
find 6, the angle which the two lines make with one another, 
by the formula, 
1 4 AA'-f BB'-f- (A A') cos. -f (B -|- B') cos. 4 + (A' B 4 AB') cos. u 
d (i +A*4B^4 2 A cos. <p 42 B cos.4 4 2 AB cos. u) (i 4A'*4B'*'42A' cos. <p-}-2 B' cos.442jA' B'cos. u) 
(See Trans, of Camb. Phil. Soc. vol. ii ; P. I ; p. 202.) 
When we know the symbols of the planes, the co-efficients 
A, B will be found by eliminating y and 2; in the equations of 
the planes where intersection is considered. 
Ex. In a rhomboid it is required to find the angles made • 
by the opposite edges of a pyramid formed of planes (p, q, r). 
By referring to fig. 22. it will be seen that opposite edges 
are those which are produced by intersections of planes 
{p;q^r){q;p;r)^x\ 6 . (q\ r,p){r;q; p). 
To find the equation to the first line we have 
p^ + qy + ^'^ = 0 
qx+py -\-rz=:o 
whence y = z = — x. 
In the same manner we should find for the second line 
y = zz=-^l^x. 
