Russell — The Geometry of the Cubic. 171 



other words, z - a defines the length and direction of AZ. It is 

 equally plain that 



= - e-'^ 



% - a r 



defines a length the reciprocal of AZ, and a direction -which is its 

 reflexion with respect to axis of x — in fact it defines Aq'Zo. Similarly 



2 -a' s-^' s- 



7 



define the distances and directions of three points A^', Bq Cq from Zgy 

 and these four points Aq\ £o', Co', Zq from a figure the exact reflexion 

 of Z, and the inverses of A£ C with respect to a circle of unit radius 

 round Z as centre. Any geometrical relations, therefore, which hold 

 between the points Aq, Bq, CV, Zq in virtue of some algebraic relation 

 between the quantities 



% — a 2 — yS z — y 



will also hold between the points A\ B', C, Z. If the cubic equation 

 be a%^ + 3^2^ + 3cs + <? = 0, where a, I, c, d are quantities of the form 

 p Jr ^/-l q, it is well known that it can be reduced to the form 



l{%- Si)' + m{% - Sa)^ = 0, 



where Si and z^ are roots of the Hessian 



{ac - ¥) z- + {ad - he) z + {hd - C') = 0. (1) 



This quadratic breaks into two factors which, expressed in terms 

 of the roots, give for %i and %2 the equations 



1 (0 (1)^ 



+ ^ + = 0, 



Zi - a 2i-/? Zi-y 



; (2) 



1 w- 0) „ 



+ ^ + = 0, 



Z2 - a Z2 - p Zo - 7 



