206 Mr Turnhull, Some Geometrical Interpretations, etc. 



the polars of {x) in f and /' cut the plane {u) satisfy the har- 

 monic relation for the complex C=(AB)(Ap)(Bp). 



Finally the last form [(ABYY, which is equivalent, except for 

 reducible terms, to (Abu)bx {Ab'u)hx (§ 2). is involved in the con- 

 dition that the line common to (a) and the polar of (x) inf shuuld 

 touch /. 



§ 23. Next there are four forms of order (0, 2, 2), such as 

 (Ap){Abu) (abp)apUp, (Ap)(AB){Bau){al3p)apUa, and four cor- 

 relatives of order (2, 2, 0). All of these have obscure geometrical 

 properties, though they present no difficulty to identify. 



After this there are twenty-four forms of order (1, 2, 1). The 

 simplest of these is (Abu) b^ (Ap) (Bp), which vanishes Avhen 

 u, X, p satisfy the following conditions : if the polar of (x) in /' 

 meets (p) at a point {y), and if the polar o{(y) in/' cuts the plane (u) 

 in a line (q), then p, q satisfy the harmonic relation {Ap) {Aq) = 0. 

 The remainder of these (1, 2, 1) forms are of like nature. 



Beyond this there are four forms (2, 1, 2), and two forms (3, 0, 3), 

 none of which present concise geometrical interpretations. 



i 



