PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 301 



(O'FQ') of the reciprocals the (00', OP', OQ') of the coinitial 

 vectors (OZ, OP, OQ), whose extremities ZPQ lie in a spherical 

 surface, are coplanar. That is to say, if the term P of a vector 

 OP move over the entire surface 1 of the sphere OQZ, the term P' 

 of its reciprocal will trace out an infinite circular plane. 2 Or again, 

 if a point P starting at the pole Z, move on the sphere so that its 

 polar distance (ZP = <£) bears a constant ratio to its longitude 

 (w say, e.g. 4> = 2mo>) the reciprocal of its distance to the opposite 

 pole PO, will trace out the curve r = b tan mw lying wholly in a 

 plane. 3 It is obvious that there is no limit to the complexity of 

 possible forms capable of geometrical development, and on the other 

 hand simple figures may be generated from the most complex. 

 Forms of generation in which a line is developed from a pure point 

 in any other way than by motion along it, or a surface from a line 



1 This is really an impossible supposition, since the term is a pure 

 point, and no scheme of motion can make it cover a surface. 



2 Obvious, since OQ . OQ'= pp' = a cos 6 . b sec 6 = ab. 



3 r becomes infinite only when P reaches O, i.e. when moo = \tt. It may 

 be noted that the infinity depends on the order of the infinitesimal 

 approach to O by the point P. 



