APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSUKE 87 



2. To make this result familiar we shall apply it to the deter- 

 mination of the points of inflexion of a cubic and show its importance 

 for the solution of problems of closure afterwards. 



The tangent at a point of inflexion intersects the curve in three 

 consecutive points. Hence, if u is the argument of such a point, 



from which 



u= g '• (19) 



In this expression I and l^ may be given all integral values; but 

 two values of u which differ only by multiples of ^Iw and 2«Jj do 

 not give different points of inflexion. It is therefore sufficient to 

 give I and l^ the values 0, 1, 2, associated in all possible ways. The 

 cubic has therefore 9 points of inflexion, and their arguments are, 

 indicating the values associated to I and Z, in every case: 



2w, 4^, 



^0 0=0 u^ i=-y' '''o 2=-^' 



These points are three by three in straight lines; the straight 

 line joining two always passes through a third. For example 



shows, according to (18), that the corresponding points of inflexion 

 are coUinear. As p'(^o)=co, it follows that the first point of inflexion 

 -Mq is infinitely distant in the direction of the y-axis. 



3. If from a point n, a tangent is drawn to the cubic, then we 

 have, since ^2=^3=3? (to be found), 



2x-\-^l^^=27nw-{-2mJW^, 

 and 



