407 



circle or sphere whose radius is iT and centre at E, and denoting the 

 connectors of the inverse points A', B', C, D', by the same notation as 

 those of the points A, B, C, D, only with accents, also denoting 

 EA', EB f , EC, ED', by a*, 7*, 

 Now it is evident that 





K'm' 

 m— , 



7 a 







ap 



? ad' 



jrv 



E*r' 



Hence, making these substitutions in equation (24), clearing of frac- 

 tions, and omitting the accents as being no longer necessary, we have 

 the following theorem : — 



If A, B, C, B, E, be any five points in a plane, or on the surface 

 of a sphere, and if the connectors in pairs of the points A, B, C, D, be 

 denoted by I*, p* ; mh, q$ ; r$, ; and the connectors EA, EB, EC, ED, 

 by ah, ft, 7*, then the relation 



Irqa? + mprfi 2 + npqr 2 + Imnh 2 + Qp -mq- nr) (lah +pj3«j) 

 + (mq -nr-lp) (mfib + qay) + (nr -Ip - mq ) (n^B + raft) - 0, (26) 



or its equivalent, the determinant, 



0, 



n, 



m, 





a 



n, 



0, 



I, 



2, 



P 



m, 



l, 



0, 



r, 



7 



P> 





r, 



o, 



B 



a, 





7> 





0 



holds between these connectors. 



16. The equation (26) between the connectors of five points on 

 the surface of a sphere is the analogue of Ptolemy's theorem for four 

 points on a circle, and can be enunciated in a very concise manner by 

 the help of the following considerations : — 



1°. The entire number of lines of connexion of five points 



■&-». 



2°. The entire number of triangles which can be formed by combin- 



5x4x3 



ing the five points, three by three, is — — — - = 10. 



1 *2'3 



3°. The entire number of pentagons which can be formed having 

 the five points for vertices = — - — ^ — — = 12. 



