PHILLIPS AND MOORE.— LINEAR DISTANCE AND ANGLE. 71 



If we use A, B, C for the angles and a, b, c for the sides opposite 

 this becomes 



. a -\-h + c 



^ = ^^ • ■ ■ ■ (26) 



Similarly we have 



A+B-\-C 



j^ n ' ' ' ' (-' ) 



In the tetrahedron if a, /?, 7, b are the planes opposite the vertices 

 A, B, C, D we have for the angle 



-^^ (p-BCDCDA) ^ (pCD){BCDA) 

 (FBCD)(FCDA) {FB~CDy{F C D A) 



CD BCD A 



BCD CD A 



This gives for the volume 



jrwyn ^^^ CD A a/? 



BC D A = = L ^ _ (23) 



CD ^ ' 



That is the volume of a tetrahedron is equal to the product of the 

 areas of two faces and the dihedral angle between them divided by the 

 length of the common edge. 



The trihedral angle a /9 7 is given by 



aj5 7 = a;5 + /37 + 7a 



££j;££^ DACDAB DBADBC 



bcdc1)a'^cWa^dTb IWbcTb 



= W7^p1( ^^^BCD^CD-DA + CDAD B 



lfcl)ClDAD~AB 



(29) 



This formula solved for B C D A will also express the volume in terms 

 of the trihedral angle and the three face triangles and three edges 

 which meet at its vertex. 



The volume can also be expressed in many other forms. 



