396 



Mr. M. Gardiner on Undevelopable [May 14, 



r'r' pierce the plane K ; draw any plane P parallel to K, and through the 

 points fx' in which it cuts r^r^ and/r' draw lines parallel to the diameter 

 rr to pierce the plane K in points 0' ; through the centre o draw (see 

 * Section of Ratio' of ApoUonius) the two lines ox^x', oi^i' so that 



p^x^ : p'x' : : p^i^ : pi' : : p'</)' ; 

 through and draw lines parallel to rr to cut r.^i\ and r'r in points x 

 and ^ ; through and ^" draw lines parallel to rr to cut rj'^ and rV in i 

 and 2. Then will the lines xx and n be reciprocals piercing the quadric in 

 the four points which are first extremities of the inscribable closed w'gons. 



Second method. — In the closing chord c^c^ of any inscribed open w'gon 

 assume any point j9, and for the moment regard it as n-\-\t\\ point of a 

 series having the n fixed points as first n points ; choose four points 5^, c?^ 

 e^, in the surface so that no two of the five c^, b^, d^, e^,/^ lie on one ge- 

 nerator; find the final points b^,d^, ^2'/2 ^"^^^^ibed (?i+l)'gons whose 

 sides pass in order through the n+l points, and which have 5^ d^, e^,f^ as 

 first extremities. Then (representing tangent-planes at points by capital 

 letters of like names, and subscript numbers as the small ones representing 

 the points of contact) in the chord d-^d^ find the point d^ such that 

 d^ ^ . . 



in the chord e^e^ find the point e^ such that 

 in the chord f^f^ find the point such that 



(the points d^, e^,f^ must be so determined as that a re«^ tangent-plane can 

 pass through either of them) ; draw the plane d^e^f.^ and it will touch the 

 quadric in a point ; draw the line c^a^y and find the point q in it which is 

 conjugate to p. Now if we regard the n given fixed points and the point q 

 as the n-\-l points of a series, every point in the surface will be the first 

 extremity of a closed 2{n+ l)'gon. Find (by last case) the two reciprocal 

 lines yy, sz which pierce the quadric in first extremities of closed 

 (w+ l)'gons whose sides pass in order through these n-\-\ points ; find the 

 line^'j' reciprocal to pq ; draw the lines xx, ii, each of which cuts the four 

 non-planar lines yy zz, pq, p'q. Then will xx and ii be reciprocal lines 

 piercing in four points which are the first extremities of the inscribable 

 closed 7^'gons. 



Third method. — On the closing chord c^c^ of any inscribed open w'gon 

 assume any point jo, and regard it for the moment as the n-\- 1th point of a 

 series to which the 7i fixed points belong. Assume three points b^, d^, e^ 

 in the surface so that no two of the points b^, d^, e^ are on one generator, 

 and find the final extremities 6.^, d^, e^ of inscribed ^z'gons having 5^, d^, e^ 

 as first extremities ; draw the tangent-plane at the point ; put D^, Dj, 



