1896.] 



certain Properties of the Tetrahedron. 



105 



From this we immediately deduce the equation 



Km x + Km t + Km 3 + Km x = (21). 



From (20) we have 



m x Km x + mJZm^ + m^Kt^ + w^TSf?/^ = 0, 

 and from (21) we deduce 



m l Km 1 + m^Km^ + mJZ.m-s + m^Knii = 0. 



Adding these two equations, and making use of equations (3) 

 and (10), we obtain 



( Wniif + Lniyiiu + Lm{tn z + Lm 1 m i = (22). 



For simplicity we will denote the edges of the tetrahedron by 

 the symbols (12), (13), (14), (34), (42), (23), the numbers denoting 

 the subscripts of the coordinates of the points joined by these 

 edges. We shall also denote the angles of the anharmonic ratios 

 corresponding to the edges (12), (13), (14), (34), (42), (23) by the 

 respective symbols a, /3, 7, 6, </>, yfr. Further, we shall write 



aX 1 s + b I? + cZ* + 2fY 1 Z 1 + 2gZ 1 X l + 2hX, Y, = P\ 



and shall denote by the symbols Q, R, S the quantities corre- 

 sponding to P when the subscripts 2, 3, 4 are substituted for the 

 subscript 1. 



This being premised, we immediately obtain for the interpre- 

 tation of equation (22), the relation 



P = Q cos 6 + R COS (f) + S COS -v/r. 

 Similarly we should obtain 



Q = R cos 7 + S cos /3 + P cos 0, 

 R= S cos a + P cos <f> + Q cos 7, 

 S =P cosyjr+ Q cos/3 +R cos a. 



Eliminating P, Q, R, S from these four equations, we obtain 



— 1, cos 6, cos cf), cos ty =0. 



cos 6, — 1, cos 7, cos /3 



cos<£, cos 7, —1, cos a 



cosyjr, cos/3, cos a, —1 



Thus the six angles a, /3, 7, 6, (f>, ty are connected by a relation 

 exactly similar to Cayley's relation connecting the dihedral angles 

 of a tetrahedron. 



