r 170 ] 



IX. 



THE GEOMETRY OE THE CUBIC. Bt E. EUSSELL, 

 II.A,, E.T.C.D. 



[Eead April 24, 1893.] 



If X and y be tlie rectangular co-ordinates of a point Z), it is custo- 

 mary to represent that point by the single symbol x + y^- \y = z. 

 We thus see that any relations between a number of such alge- 

 braic quantities may be interpretable into geometrical relations 

 between a corresponding number of points. 



As an example, let us] consider a cubic equation which may have 

 all its coefficients imaginary, and which therefore, in general, will have 

 three such quantities as its roots. The problem of solving the cubic 

 is the same as the geometrical problem of determining the vertices of 

 a triangle. 



Fig. 1. 



Now let a, /8, 7 be the roots of this cubic, and A, B, C the points 

 which they define. It is obvious that % - a = re^^, where r is the 

 length of AZ, and the angle which it makes with the axis of x. In 



