522 Proceedings of the Royal Irish Academy. 



important part in the following investigation. It can readily be shown 

 that if a point 0' lie in the plane corresponding to a point 0, then 

 lies in the plane corresponding to 0', and the line joining and 0' 

 possesses a certain invariant relation to the curve. Also the locus of 

 the corresponding points of planes passing through a given line is a 

 right line, which may be called the corresponding line to the given 

 one, their relation being reciprocal. 



Let us now consider the binary cubic — 



f= aoX-^ + ZttiX-^x'- + da^XiX.? + a^ xi = a/ = b/ - c/, Sec, 



adopting Clebsch's notation. 



By our transformation the binary cubic is transformed into a plane 

 /, the equation of which is — 



«oX+ Sffi Y + 3a^Z+ a, W^ 0. 



Now it can be easily shown that the corresponding point of this 

 plane is given by the equations 



We shall call it the point 0. 



II. 



It is known that through any point in space can be drawn one 

 chord, meeting the curve in two points. Let us now determine these 

 points, being given the point 0. 



The co-ordinates of the line joining the points on the curve, the 

 parameters of which are Xi : X2, y\ : y-ii respectively, are easily found 

 to be — 



a = Ao Aa, /= 4 Ai- - Ao A2, 



^ = 2AiA2, ^--2AiAo, 

 c = Ao'^, h = Ao^, 



where 



Ao = xoy2, ^\ = ^iy2 + yi^2, ^2 = Xiyi. 



Now take two equations of the chord, viz. : — 



aX+bY+cZ =0, 

 hY-fZ+aJF=0, 



and these furnish us with — 



AoX+2Air+Ao^=0, 



Aor+2Ai^+A2 7r-o. 



