1896.] certain Properties of the Tetrahedron. 103 



We will suppose (x , y , z ) to be the coordinates of some point 

 lying on the line of intersection of the two planes defined by our 

 two triads of points. Then the equation of any plane passing 

 through the said line of intersection may be written in the form 



( x -x )(X + kX') + (y-y )(Y+kY') 



+ (z-z )(Z+kZ') = (17). 



Now consider the cone 

 A (x - x o y + B {y - y o y + G (z - z f + 2F(y - y ) (z - z ) 



+ 2G(z-z )(x-x ) + 2H(x-x t) )(y-y ) = (18), 



where 



A = bc -f 2 , B = ca—g 2 , G =ab — h", 

 F = gh-af, = hf-bg, H=fg-ch. 



To find the values of k corresponding to the two tangent 

 planes to (18) passing through the line of intersection of the 

 planes defined by our two triads, we have to express the con- 

 dition that the normal through (x , y , z ) to (17) lies on the 

 normal cone of (18). This gives us 



a(X + kXJ + b (Y+kYJ + c(Z + kZy 



+ 2/( Y + k Y') (Z + kZ') + 2g(Z + kZ') (X + kX') 



+ 2h (X + kX ') ( F+ kY') = 0. 



If this be expanded, it will be seen that it can be written in 

 the form 



( Wmf + 2kLmm' + ¥ ( WmJ = (19). 



If ki and k 2 be the roots of (19), we have 



2Lmm' 





(WmJ' 



(Wm)* . 

 (Tfm') 2 ' 



and therefore 



1 (7&i\ 2 fk 2 \ *) Lmm' 



2 \\kj ' \kj Wm. Wm 



Now kjko is one of the anharmonic ratios of the set of four 

 planes constituted by the two planes defined by our two triads 

 and the two tangent planes to (18) through the edge in which 

 they meet. If we replace this ratio by the symbol e 2ie , we have 



Lmm' — — Wm Wm' cos 6. 



