BussELL — The Geometry of the Cubic. 



173 



ties, and the point G which is the centre of gravity of equal weights 

 placed at the points A, B, C, then we easily see that 



—> a + jB + y Py-¥ya(ii + af3<j)^ _ (a + l3<j/ + yoi)' _m'^ 

 ^ ~ 3 a + ^0} + yor ~ 3 (a + ySw + yur) ~ 3l ^' 



> P 



and Z2G = -— ; 

 3m 



and if the angle Z^GZ-^, be trisected by the lines GM, GL, and mean 

 proportionals inserted, i.e. GZ^. GL = GM^, GM . GZ^= GL", then 

 the lengths and directions oi LG and MG will be defined by the 

 quantities 



a + /3ci) + yco^ _l J *"• + P^^ + yco _ w 



_ = — • anci — ~~ • 



3 3' 3 3 



"We shall see now that if two equilateral triangles be described on 



Fis 



ML, their vertices will be denoted by two roots of the cubic /?, y. To 

 prove this, it is only necessary to show that 



GB^ji 



a+yS+y 2/3 -a-y 



