516 Proceedings of the Royal Irish Academy. 



Since in Hamilton's treatment of these coordinates, the origin is 

 taken in the plane and coinciding with the point U, instead of being 

 arbitrary, it may not be superfluous to remark that, if the lines 

 AU Q.m\ AP meet ^C in U, and P^, 



7? jj ftp 

 {A.BUCP) = {BU.CP,) = ~r^^ 



h/3 + cy hPy + cyz 



b + c b>/ + cz '' 



_l^J_cy_ b/3 i/ + c yz' 



b + c by ^ cz 



y 



and this reduces at once to -. 



z 



2. The two planes which are swept out by the extremities of the 

 variable vectors 



OP = ^ = ^^^^^fy + ^7! ^^^ op, ^ ^, ^ a'a'x^b'[i'y ^_^ 

 ax + by + cz a'x + b'y + c'z 



may be said to be anharmonically partitioned or divided.^ The point 

 P in one has the same anharmonic relations to its unit-point U and to 

 its unit-triangle ABC, as the corrpsponding point P' in the other has 

 to its unit-point U' and to its unit-triangle A'B'C ; or, in general, 

 corresponding points are projective. 



3. The system of lines joining corresponding points on the two 

 homographically divided lines 



0P=. = ^^^^ ^^^ Qp, ^ ^ ^ a'^^ ^ v^'y 

 ax + by a X + by 



generate a ruled hyperboloid, the vector expression for which is 



wc7 + rzD' s {aax + b/3y) + t {a' a'x + b'fi'y) 



P = 



u + V s {ax + by) + t {a'x + b'y) 



Analogy suggests the consideration of the system of lines PP' joining 

 corresponding points not on homographically divided lines, but on 

 homographically divided planes. 



' See ClifPoid's paper on "The General Theory of Anharmonics." Collected 

 "vvorks, p. 110. 



I 



