JoLY — tomographic Bmsions of Planes, 82)heres, and Space. 525 



The homographic plane 



xaa + i/h/3 + %cy + wdh 

 p = 



X + y + z - 8w 



is divided with reference to the three points represented by aa, h/3, 

 and cy, and their mean point - ^dS. 



As regards the "equation of congruence" ("Elements," Art. 71), 

 it is sufficient to remark that (xt/ztv) and (tx + s, ty + s, t% + s, tw + s) 

 represent the same point on the surface of the sphere if t is positive, 

 but opposite points if ^ is negative. The symbol (1111) denotes no 

 definite point because the versor of a null vector is wholly arbitrary. 



16. As Hamilton does not seem to have given the types of points 

 of "Third construction" for a plane net, it may be useful to give 

 them here. The points of first construction derived from the four 

 points Pg = (1000) are Pi = (1100) and the lines constructing 

 them are Aj = [1, - 1, 0, 0]. The lines of second construction are 

 A2 = [1, - 1, 1, - 1] and the points are P2 = (1? - 1> 0; 0)- 



These points P2 lie by threes on A3, 1 = [111 - 3] and connect 

 with Pq by A3, 2 = [Oil -2]. The intersections of As,! aJid A, 

 determine the type P3, 1 = (3100) ; those of A3, 2 and Ai, the type 

 P3, 2 = (2100); A3, 2 and A2 give P3, 3 = (3210); As,! and A3, 2 

 give P3, 4 = (- 3, 2, 1, 0), and the lines A3, 2 mutually intersect in 

 P3,5 = (4210) in addition to the old points. 



