newson: collineations in the plane. 



17 



(ILPR)= — I. If, therefore, we set up a perspective transforma- 

 tion of order 2 having I for its vertex and 1 for its axis, it will 

 transform C into itself. 



Take the point of inflection (o, i, — i) and its harmonic polar 

 y — z=o; choose any two points on this line as (o, i, i) and (i, i, 1). 

 The single cross-ratio of this transformation is k=^ — i. The equa- 

 tions of the transformation may be written down by means of the 

 following formulas:* 



p\, 



y z o 

 B C A 

 B, C, kA, 

 B3 Co k'A, 



X 



py-. 



y 



B 

 B 

 B. 



o 

 B 



kBj 

 k'B, 



pz. 



y z 

 ABC 



A, B, Cj kC, 

 A„ B„ C, k'C, 



(10) 



Substituting in these formulas the above values of A. B, etc., and 

 making both k and k' equal to — i, these reduce to 



Xj=X, 



yi=z. 



Zi==y- 



(lO 



There are nine transformations, one for each point of inflection; 

 they may all be written down by means of the same general for- 

 mula. Three of them will be real and six imaginary. If we make 

 any one of these nine substitutions in the equation of the pencil of 

 cubics, we find that every cubic of the pencil is transformed into 

 itself. 



Again, let us take a transformation whose invariant triangle is 

 the triangle of reference and whose cross-ratios are k and k'. 

 Writing down the equations of this transformation by means of 



formulas (10) we find -I yj=r:ky Making this substitution in the 



( Zi=k'z. 

 equation of the pencil of cubics we get 



x3-j-k3y3^k'Sz3+6mkk'xyz:=ro. 



*K. U. Quarterly, vol. viii, pp. 45-66. 1 have recently found that tliese formulas were 

 previously given in nearly tlie same form by Prof. Gabriele Torrelli in the Rciuiiconti 

 di Circolo Matematico di Palermo, Tome viii, pp. 41-54. 



