APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 95 



and 



v,—v^ = v,'—v,'. (29) 



Hence v/ and v^' form a couple of the same class, and we have the 

 theorem : — 



The straight lines joining the points of a Steinerian couple 

 with any point of the curve cut the curve in a new couple of the 

 same class,' all couples of the same class are obtained hy letting v 

 describe the whole curve. 



The case where 7^=2, i. e., of quadrilaterals, is of particular 

 interest. Formula (27) now becomes 



'o^=v^—\ (2mw?-|-2m,Wj), 



in which m and m, may assume the values and 1. The original 



point together with the possibilities (0, 1); (1, 0); (1, 1) gives the 



four points 



v^^v^—w^^Vi—w,Vy—{w-\-Wi). (30) 



Substituting any of these arguments in (27) it is seen that any 

 two of these points form a Steinerian couple. 



Four points with this property may be called a Steinerian 

 quadruple. 



If at a point •w of a cubic a tangent is drawn, 2'W+aj=0, hence 



x=—'lu. (31) 



Substituting the values of (30) in (31), it is found that the 

 tangents at the points of a quadruple are concurrent. 



The rays joining a point v' of the curve with the quadruple (30), 

 or any other quadruple as it may easily be verified, intersect the 

 curve in another quadruple whose arguments in this particular case 

 are 



v,'= -v^—v\ 



-y,' = — v,— -y'+i^J, (32) 



v^ = -v^—v' -^w^, 



vj = —v^—v'-\-w-\-Wy. 



