. 57. OF SIMPLE FORMS IX PARTICULAR. 49 



And 



cos. y = 4 (1 + cos. z 



+ 2. J [_ (cos. x + cos. z) (1 + cos. x)}) ; 

 cos. x = ^ 4- cos. z 



-f 2. ^ [ (cos. y + cos. z) (1 + cos. y)]) ; 

 cos. z as (3 + 2. cos. y + 2. cos. x 



-f 2. V [2 (1 + cos. y) (1 + cos. x)]). 



. 57. THE TETRAHEDRON. 



The Tetrahedron, Figs. 13. 14-., is contained un- 

 der four equilateral triangles. 



1. The plane angles a,a,a of the tetrahedron are = 60 ; 

 the angles of incidence, at the edges A, A, &c. (their magni- 

 tude) = 70 31' 44". 



2. The sections of the tetrahedron are rhombohedral and 

 prismatic; one of the latter, through the centre, is a 

 square. 



3. The principal axes are rhombohedral ; they join the 

 solid angles with the centres of the opposite faces ; their 

 number is four, and they intersect each other at angles of 

 109 28' 1C", and 70 31' 44". These angles of intersection 

 are general for the rhombohedral axes^ whenever more than one 

 occur in the same form. The subordinate axes are prisma- 

 tic ; they join the centres of opposite edges ; their number 

 is three, and they are perpendicular to each other. 



4. The tetrahedron is a regular solid of geometry. * 



5. This form occurs, either by itself or in combina- 

 tions, in tetrahedral Copper-glance, in dodecahedral Gar- 

 net-blende, &c. 



* The principal sections and horizontal projections of 

 the forms of several axes being of comparatively little use, 

 and besides very easily ascertained, I have thought it 

 superfluous to enter here into a greater detail. 



VOL. I. D 



