APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 103 



P+V,+^i + ^2 = 0. 



If P and V, remain fixed then P-}-y, = c'j and also — [u^-^Vj) = 

 c^ (constants). If the plane turns about P-Wj then the ray joining u^ 

 and -Wj describes a hyperboloid of one sheet passing through F. Keep- 

 ing P fixed and assuming another point v^ on T so that P+v^^Cj 

 and passing a plane through P, v.^ and u^^ this plane will cut F in a 

 fourth point u^ so that — ('Wj+^a) —^2- The ray u^u^ is on a second 

 hyperboloid through F. Assume a third point v^ on F, so that P+v, 

 = ^3; pass a plane through P, v^ and -j/,, cutting F in a fourth 

 point u^,, so that — {u^+u^ =<?3. The ray u.^u^ lies on a third hyper- 

 boloid. Continuing this construction up to a point v^^ on F, giving 

 finally a point -Wa^+i on F, the table results: — 



P+^'i + ^i + 'l/2 = 0, 



1. — (Wi+Wj)=C„ 



2. — (^/,2+?^3)=Cj, 



3. — (1/3+^^4) = ^3, 



(16) 



n—1. — (^<2n-lH-^'2n) = ^2n-l) 

 ^- — (W2n+^2n+l)=^2n• 

 The number 2n has been chosen for exactly the same reason as 

 in §3, 2. Subtracting the sum of the congruences of odd order from 

 the sum of those of even order, the congruence results 



n-l 



= 2c2k— 2^2^+1. (17) 



1 



The last point v^^ or any other fixed point may always be 

 uniquely chosen in such a inanner that the right-hand member of 

 (17) vanishes. Indeed, 



P+^i-fwi+Wj^O, 



V+v,+u,+u, = Q, (18) 



i*4-^'2n+^2„+^2n-fl = 0, 



