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though not the same, are equally inclined to the lines of cur- 

 vature on {A') passing through V. This suggests the theorem, 

 that if the plane of L and L" be a principal plane of the cone, 

 the path of the style will touch a line of curvature on (A') 

 atF. 



" Let us next consider a pair of adjacent sidesof the cone, 

 such as L and L'. Normal planes to the cone (a) along these 

 sides intersect in a right line, which lies in a plane perpendicular 

 to that external axis of the cone (a), through which the plane 

 of L and L' passes. Hence it follows, as before, that two inter- 

 secting cords, L and L', may be both rolled on, or both rolled 

 off the geodetic lines upon which we suppose them prolonged 

 to fixed points, p and/)', in such a manner that the shortest dis- 

 tances between their intersection and the fixed points j9, p, 

 shall have a constant difference. And their intersection at V 

 will lie upon a hyperboloid (-B), confocal with the ellipsoids 

 (A) and (A'). 



" Lastly, considering the pair of sides L and L'", and 

 cords produced along them to fixed points p, p"\ on the geo- 

 detic lines touched by them, we see that if the difference be- 

 tween the lengths Vp, Vp" remain constant, V will trace a 

 curve on a second hyperboloid (C), which passes through V, 

 and is confocal with {A') and {B'). 



" It is obvious that the curve described by the point F, 

 under the circumstances considered above, is not, in general, 

 a geodetic line on a surface confocal to {A). We may, how- 

 ever, regulate the motion of the cords so as to effect this. 



" For instance, let the four cords, L, LI, L", LI" , be pro- 

 longed in the direction of similar geodetics until they touch 

 two opposite lines of curvature, along which they are thence- 

 forth applied and carried on to fixed points, p, p, p", p". Then 

 a style at F, stretching a continuous cord p Vp", which coin- 

 cides with two opposite sides of the cone, L and L", will trace 

 a geodetic line upon the confocal ellipsoid {A)y provided it be 

 made to move always in the plane of the two straight portions 



