522 Proceedings of the Royal Irish Academy. 



equation of the plane found in the last article to envelope the focal 

 surface. This corresponds to a property of the hyperboloid — viz., 

 its double generation. If in its equation, as given in Art. 3, s and t 

 are supposed to be constant, the equation is to be interpreted as that 

 of a generator of the second system, and the locus of these second 

 generators is of course the hyperboloid. 



Through a given point three of these planes pass, and it is easy 

 to see, as in Art. 8, tliat each of them contains an infinite number of 

 lines of the congruency.^ It would appear from this that their three 

 lines of intersection are the lines of the congruency which pass 

 through the given point. 



12. Just as a plane was divided anharmonically, space may be 

 divided anharmonically. Take a pyramid, A B C D, and let the 

 vectors to its vertices be a, /?, y, and S respectively. The vector 



OP = zs = ^-S- ', — 



a;» + yo + zc + wd 



is capable of representing any point in space. The plane through 



x(i- OL 4- yh S 



C, D. and P meets the edge AB in the point —^—. The plane 



xa + yo 



through the unit point U, which is represented by the vector 



_ aa + h^ + ey + dB 

 U U — V — - 1 , 



fl + 4- c + a 



and the edge CD meets AB in /-. The anharmonic of the 



a -{- b ^ 



pencil of planes joining AB to C, U, B, and P, is equal to - ; or 



s 

 lAB ' CUBP) = -. Similar results hold for planes through the 

 w 



remaining edges. 



i!^ow, if two flifFerent unit pyramids are taken and two arbitrary 

 unit points, the lines joining P and P', corresponding points in the 

 anharmonic divisions, belong to a complex. 



Suppose 



xa'a' + yb'ft' + zc'y' + wd'8' 



OP' = To' = 



xa' + yb' + zc' + wd' 



' These lines envelop a conic since, reciprocally, the chords of a twisted cubic 

 which pass through a fixed point on it lie on a quadric cone. 



