JoLY — Homographic Divisions of Planes^ Spheres, and Space. 519 



These equations, thougli apparently different, will now be shown 

 to he equivalent, and either of them may be taken as the equation 

 of the congTuency of lines joining corresponding points of the homo- 

 graphically divided planes. 



6, If p and p' are any two points on a line of the congruency, 

 and if o- = p' - p, then will t = Vp'p = To- p. 



The first equation of the congruency now takes the form, 



^,V<r{p-a)S<T{p-l3'){p-y') + ^,Vcr{p-/3)Sa{p-y'){p-a') 



+ ^^r<T{p-y)Scr{p-a'){p-(3') = 0, 



as may be verified by elementary transformations. 

 If 



<l><TS{p-a'){p-P'){p-y') = %-a)S<T{p-l3'){p-y') 



+ '-{p-y)Sa{p-a'){p-(B') 



■defines a linear vector function ^, the first equation of the con- 

 gruency may be written in the simple form 



Va-cjjcr = 0. 



Nowc^(p-a') = ^(p-a), <A(p-/3') = ^(p-;8), 4.{p-y') = j,ip-y); 



"whence it is not difficult to verify that the inverse function <^~^ is 

 defined by 



<^-V>S(p-a)(p-/3)(p-y) = ^'(p-a')>SV(p-/:i)(p-y) 



CI 



+ ^'(p-y8').Scr(p-y)(p-a) 



+ ^'(p-y')>Sor(p-a)(p-^) 



and, accordingly, that the second equation of the congruency may be 

 reduced to the form 



Vo-cji-^a = 0. 



This new form is of course equivalent to V<j4>(t = 0, since either 

 expresses that o- is parallel to an axis of — a linear vector function 

 varying with p. 



