120 
Mr. Whewell on calculating 
+ — 7«=i or px {p —q)y^r»=p. 
its symbol, or that of PQR, is {p; p — q\ — r). 
Similarly, aty, we shall have a plane [q — p ; q; — r). 
Also, since the edges A a: and Ay are symmetrical, we 
have a plane {q; p; r). And hence the co-existent planes are 
iP;q;r){p;p-^q;^r){q—p;q;^r)(q;p;r)(q;q--p;--r) 
{p — qi pi — r). Which may be included in the symbol 
f iP,p — q-^ — r) iq,q—p-, — r)] 
§ 7. The rhombic Dodecahedron. 
40. If we take a regular tetrahedron w x y z, fig. 20, and 
from its centre of gravity A draw lines Aw, Ax, Ay, Az, 
the angles made by any two of these lines will be the same. 
And by taking planes passing through any two of these 
lines we shall have six planes symmetrically disposed, each 
of which will make an angle of 120® with four others. A 
figure bounded by planes parallel to these planes, each taken 
twice, and symmetrically disposed, will be the rhombic 
dodecahedron. 
We may consider the three lines A jc. Ay, A « as axes of 
co-ordinates ; and any plane p q r which cuts them must 
have co-existent planes cutting any two of them and A w. 
Also, as the lines Ax, Ay, Az are similar, in a plane {p^q^r) , 
we may present the indices in any manner. 
41. Prop. To find the symbols of co-existent planes in the 
rhombic dodecahedron. 
Let a plane p qr cut w A produced in O. Let x, y, z be 
the co-ordinates of the point O. The equations of the line 
Aw are y = X, = X. And if the equation to the plane 
