174 Proceedings of the Royal Irish Academy. 



Now 



GB = GM+ MLe 3 = _ _ + ^^ — -_ = oj^ — - — = Jl. i, 



o o o o 



and 



777,^ TT^r ^4-r — ^ I -m Z + w'ot 2y- a- B 



GC= GM+MZe^ =- - + (d =w = -^ -. 



3 3 3 3 



The position of the vertex A is of course obtained by joining X to 

 G, and producing the joining line so that GA = 2GX. 



We see, therefore, that the solution of the cubic involves the tri- 

 section of the angle Z^GZr,, and the extraction of the cube root of 

 either of the quantities GZ-c . GZ^^ or GZ^ . GZ^, where GZ^ and GZ^ 

 are lengths only. 



Let us finally apply these results to the cases where the coeffi- 

 cients of the cubic are all real ; and firstly let us suppose the roots all 

 real, and therefore the points which are denoted by these situated on 

 axis of X. In this case the circles of Apollonius have two common 

 points, which are reflexions of each other with respect to that axis ; 

 and therefore as in the general case an angle has to be trisected, and 

 the cube root of a number extracted. 



Secondly, when two roots are imaginaries of the form^ ± >y - 1 q^. 

 it is easy to see that the points common to the circles of Apollonius 

 lie on the axis of x, and therefore the angle to be trisected is either 

 or TT. In other words, the solution of a cubic of this class differs from 

 a quadratic only in the extraction of a cube root. 



