1896.] 



certain Properties of the Tetrahedron. 



107 



The other relations of this set are 



pa + & _ 2PR cos <f> = S 2 + Q 2 - 2SQ cos /3, 

 P 2 + £ 2 - 2PS cos yfr=Q 2 + R-- 2QR cos 7. 



7. If we write down the first three results of Art. 5 in their 

 symbolical form, we have 



( Wm^ 2 + Lmjiu + Linjiiz + Lm^i^ = 0, 



Lm x m 2 + ( Wm. 2 ) 2 + Lm. 2 m 3 + Zm,m 4 = 0, 



Lm l m 3 + Lm. 2 m 3 +( Wm 3 ) 2 + Lm^n^ — 0. 



From these three equations we readily deduce 



+ 



Lm^iii, Lin^i,, Lm^rn^ 

 Lm % m i} (Wm. 2 ) 2 , Lm. 2 m 3 



Zm 3 m 4 , Lm. 2 in 3 , ( Wm 3 ) 2 



0. 



(Willi) 2 , Lniiiiu, Lm 3 mi 

 Lniiiiu, (Win*) 2 , Lm.m,. 

 Lm 3 m,i, Lm. 2 m 3t (Wm 3 ) 2 



Thus, taking account of equation (20), we obtain 



(Wnii) 2 , Lmim. 2 , Lm 3 nii 



Lmiin,, (Wm. 2 ) 2 , Lm 2 m 3 



Lm s nii, Lm 2 m 3 , (W?n 3 ) 2 



L (m 2 + m 3 + m 4 ) m 4 , L (m. 2 + m 3 + m 4 ) m,, L (m, + m 3 + m 4 ) m 3 



LmtfUt, (Wm 2 y, Lm. 2 m 3 



Zm 3 m 4 , Lm. 2 m 3 , (Win 3 ) 2 



(Wm 4 ) 2 , Lmgiiz, Zm 3 m 4 

 Lmjiu, (Wm. 2 ) 2 , Lm. 2 m 3 

 Lin-nii, Lm. 2 m 3 , (Wm 3 ) 2 



(Wm 2 ) 2 , Lm 2 ?n 3 , Lniitiu 

 Lm. 2 m 3 , (Wm 3 ) 2 , Lm s m 4 

 Lm i m is Lmtftito (Wm^f 

 This result may be written in the form 



P 2 



— 1, cos 6, cos cf) 

 cos#, —1, cos 7 

 cos<£, cos 7, — 1 



= S 2 



— 1, cos 7, cos /3 

 cos 7, — 1, cos a 

 cos/3, cos a, — 1 



