APPLICATIONS OF ELLIPTIC FCTNCTIONS TO PROBLEMS OF CLOSURE 107 



lines to their traces; from these draw tangents to L, and L, respec- 

 tively; the lines joining the points of contact on each circle are the 

 traces of T' and F". It is evident that in the arrangement of Fig. 

 4 these traces are the tangents to L, and Lj at Mj and Mj. To find 

 the tangent to the curve C* at any of its points P, find the traces of 

 PMj and PMj and in these drav^ the tangents to L, and L,; where 

 these intersect is the trace of the required tangent. The tangent to 

 C, at the projection of P also passes through this trace. If we now 

 consider the tangent to the C^ at C, it is in the first place clear that 

 its trace is a point of C,; secondly the tangents at M, and M, to L, 

 and L, pass through this point S^". 



Suppose now that in a general case the polar-planes of C, F' and 

 F' ' intersect each other in the line d. Pass a plane through C and 

 df intersecting the C^ in four points Dj, Dj, D^, D„ and connect one 

 of these, say D,, with C and produce to the intersection D,' with d; 

 let R and Q be the remaining points of intersection of CDj with the 

 cones, then, since d is common to both polar-planes it follows 

 (CD,D/R)= —1, (CD,D/Q)= — 1, i. e., R m'ust coincide with Q, or 

 CD, produced cuts the curve C^ in another point, say D^. Similarly, 

 CD, produced cuts the C^ in D^. Hence the theorem: — 



The polar-planea of the centre of projection with respect to two 

 cones of the second order intersect each other in a straight line 

 whose projection is the line containing the double-points of the 

 projected curve. 



In our case^ Fig 4, this line is the tangent to the O^ at C and 

 its projection is the point iS°. 



The trace h through S,, of an auxiliary plane through M,M, 

 gives the four points ABCD (this C is, of course, different from 

 the above C) as points of intersection of the four generatrices, two 

 through each, M, and Mj. 



ABCD is therefore a quadrilateral inscribed to the curve of 

 the third order and with its sides alternately passing through JIT, 

 and J/,, two fixed points of the curve C^. The same is true of all 

 quadrilaterals arising from choosing successively all possible traces 

 through Si2' Thus, /Steiner^s theorem of closure on the cubic in the 



