408 



Then, the sum of the ten products formed by multiplying the fourth 

 power of the line joining any two points by the continued product of 

 the squares of the sides of the triangle of which the three remaining 

 points are vertices is equal to the sum of the twelve products formed, 

 each, by multiplying together the squares of the sides of each of the 

 twelve pentagons of which the five points are the vertices. 



17. Supposing the points A, B, C, D, E, of Art. 14 to be on a 

 plane, and that spheres whose diameters are B, 8', B", B r " } B"'\ touch the 

 plane in those points ; then inverting the whole from any arbitrary 

 point in space, and denoting the common tangents to the inverse spheres 

 by the same notation as that of Art. 14, viz., the common to the 



inverse of the spheres at B t C, by 1*; 



and so on ; then we have from equation (21) the following theorem :— 

 If five spheres, A, B, C, D, E, touch a sixth sphere, 2, the relation 



0, 

 P> 



n, m, p, a, 



0, I, q, ft 



1, 0, r, 7, 

 q, r, 0, B, 



A 7, o, 



(27) 



holds between the common tangents of the five spheres, the common 

 tangent to any pair of spheres being the direct or transverse, according 

 as the pair of spheres to which it is drawn have contacts of the same or 

 of opposite kinds with the sixth sphere, 2. 



18. The theorem of Art. 17 is an extension of the theorem of 

 Art. 15, analogous to the extension which the theorem in Art. 1 is of 

 Ptolemy's theorem, and an analogous use can be made of it. 



Tor, supposing the sphere at the point E to reduce to a point, and 

 denoting the other four spheres by S, S\ S", S"\ then we get the equa- 

 tion of the pair of spheres touching S, S," S'", 



0, n, 



n, 0, 



m, I, 



P> 2i 



& S' 



m, p, S 



I, q, S' 



0, r, S" 



r, 0, S'" 



S" S"' 0 



(28) 



precisely in the same way as the equation of the pair of circles touching 

 three circles was derived in Art. 2.— q. e.d. 



