117 
the angles of crystals. 
pyramids which compose the octahedron being right ones 
and equal, it is evident that the four lines DE, EG, GH, HD 
will be equal, and the four lines DF, FG, GK, KD. Now 
Ax is double of FH, and xy of HK. Hence Kx=y%. 
Similarly Ay = x%, and Ax = xy. Hence it appears that 
the four triangles which form the sides of the tetrahedron 
have their sides equal respectively, and are therefore equal 
and similar. Hence the four solid angles A, x,y, x, are con- 
tained by equal angles, and are symmetrical. Thus the 
angles x hy, y Kz, z Kx are equal to A x x, y x x, A xy. 
And this tetrahedron may be called an inverse symmetrical 
tetrahedron. 
From the law of symmetry, whatever plane is formed at 
the angle A, we must have a coexistent plane at each of the 
angles x, y, z, the equal and opposite edges being similarly 
affected. 
37. Prop. A plane {piq\r) being known, to find the co- 
existent planes. Fig. 17. 
Let A X, Ay, Azhe n a,n b^n c. 
Kp, Aq, Ar=ha, kb^lc; and 7 = -L, / = -i- 
X P, X Q, X R are ha, kb, I c. 
Draw y O, X M parallel to PR. 
AM = Ax.4^ = -^ = -^ 
AO h r 
‘-7 
Similarly if x N be parallel to PQ, AN = — . 
‘-T '-J 
Hence the equation of the plane x NM, which is parallel 
to PQR, is 
