JoLY — Homogyaphic Divisions of Planes^ Spheres, and Space. 517 



The vector to a point on one of these lines may be written in 

 the forms 



P = 



U + V 



s{aax + IfBy + cy%) + t {a'a'x + h'(i'y + c'y'z) 

 s {ax + hj -\- c%) + t {a'x + Vy + c'z) 



x{saa + ta'a') + y{sh/3 + th'(3') + %{scy + tc'y') 

 x{sa + ta') + y {sh + th') + %{sc + tc') 



It is evident that these lines do not generate a surface ; they belong 

 to a congruency whose properties will be investigated later on. 



4. Some preliminary calculations may be completed by considering 

 the regulus of lines joining corresponding points on homographically 

 divided lines before dealing with the congruency of lines lately re- 

 ferred to ; and the results will be useful in pointing out analogies in 

 the more general discussion. 



If CT and ot' are the vectors to any two points on a line, the line is 

 determined by the auxiliary vectors^ 



o- = TO - ot' and T = Fotot'. 



In fact o-"^T = -o-~^Ftoo- = to -o-^'toct"^ is the vector perpendicular 

 from the origin on the line, and p = o-~^t + to- is an expression for 

 the vector to any point on the line. 



Quoting from Art. 3, the vectors to two corresponding points P 

 and I" on homographically divided lines, are given by 



xa (to - a) + yb (^ - ^) = 0, xa' (to' - a') + yb' (w' -ft') = 0. 



Operating on both, of these by F(to - to'), the equivalent forms in 

 o- and T are found to be 



xa (t + Fao-) + ijb{T+ V/3(t) = 0, xa'^r + Va'cr) + yh' (j + TyS'cr) = 0. 



aa + a'a' „ T ^^ + Vft' 



If — = a and — r -— = B", it is easy to show that 



a + a' b + b' ^ ' -^ 



aS (a - /3) T + Safta- = 0, S (a' - ft') t + Sa' ft'cr = and 



.S{a"-ft")r+ Sa"ft"a- = 0. 



1 The coiistitutents of these vectors are what Plucker has called the " six 

 coordinates" of the line. 



2 2 



