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touched by the geodetics. The cone (b), therefore, which 

 envelopes (B), anil has the same vertex as (a), is confocal wilh 

 (a), and intersects it orthogonally. The normal planes to 

 (a) along L and L", heing thus tangent planes to (b), inter- 

 sect in a right line drawn from F to the pole of the plane 

 LL" with relation to (B). Moreover, this right line lies in 

 a plane perpendicular to the internal axis of the cone (a), and 

 therefore makes equal angles with L and L". 



" Suppose now that the straight lines L and L" be replaced 

 by a continuous flexible and inextensible cord, which is kept 

 stretched by a style at F, and prolonged in the direction of 

 geodetic lines to two fixed points />, p", at which it is attached 

 to the surface : it is easy to show that the style will trace a 

 curve on an ellipsoid {A') passing through V, and confocal 

 with {A) ; whilst it moves in such a manner as to allow the 

 cord to roll on one, and off the other, geodetic line. In fact 

 the path described by the style at the beginning of its motion, 

 if any motion be possible under the prescribed conditions, must 

 be in the intersection of the two planes through L and L", 

 which are normal to (a) : and we have already seen that this 

 intersection is in a plane perpendicular to the internal axis of 

 the cone (a), that is, in the tangent plane to a confocal ellip- 

 soid passing through F. But further, motion is possible, 

 though the length of the cord remains unaltered ; since the 

 two straight parts of it are equally inclined to the line of 

 intersection of the two normal planes. From what has been 

 said above we may derive a simple mode of determining the 

 direction of the tangent to the curve traced on (A') at the 

 point F. For this purpose we must draw a right line from F 

 through the pole of the plane of the two straight portions of 

 the cord, taken with relation to the hyperboloid (B). 



*' What has been already proved with respect to L and 

 L" holds good in like manner for L' and L'". And it is to be 

 observed that the paths described by the style on (A'), corres- 

 ponding respectively to these two pairs of opposite sides, 



z 2 



