Sir W. Rowan Hamilton on Quaternions. 203 



parallel to either of their two common tangent planes, the 

 sections will be two similar and similarly situated ellipses. 



[For example, if the original ellipsoid reduce itself to a 

 sphere, then the two points o^ contact, b and b', become two of 

 the four urnbilics on the itiso'ibed ellipsoid.] 



VI. The closing chords pp^,,, are also tangents to a certain 

 series or system of curves (c'), not generally plane, on the sur- 

 face of the inscribed ellipsoid (e') ; and therefore may be 

 arranged into a system of developable surfaces, (d'), of which 

 these curves (c') are the aretes de rebroussement. 



VII. The same closing chords may also be arranged into 

 a second system of developable surfaces, (d"), which envelope the 

 inscribed ellipsoid (e') and have their aretes de rebroussement 

 (c") all situated on a certain other surface (e"), which is, in its 

 turn, enveloped by ihe first set of developable surfaces (d') ; 

 so that the closing chords vv^m a^e all tangents to a secojid 

 set of curves, (c"), and to a second surface, (e"). 



VIII. This second surface (e") is a hyperboloid of fwo sheets, 

 having double contact with the given ellipsoid (e), and also 

 with the inscribed ellipsoid (e'), at the points b and b'; one 

 sheet having external contact with each ellipsoid at one of 

 those two points, and the other at the other. 



IX. If either sheet of this hyperboloid (e") be cut by a 

 plane parallel to either of the two common tangent planes, the 

 elliptic section of the sheet is similar to a parallel section of 

 either ellipsoid, and is similarly situated there'voith. 



[For example, the points ot contact b and b' are two of the 

 umbilics of the hyperboloid (e"), when the given surface (e) is 

 a sphere.'] 



X. The centres of the three surfaces, (e) (e') (e"), are situ- 

 ated on one straight line. 



XI. The two systems of developable surfaces, cut the ori- 

 ginal ellipsoid, (e), in two new series of curves, (f'), (f"), not 

 generally plane, which everywhere so cross each other on (e), 

 that at any one such point of crossing, p, the tangents to the 

 two curves (f') (f'') a^-e parallel to two conjugate semidiamefers 

 of the surface (e) on which the curves are contained. 



[For example, if the original surface (e) be a sphere, then 

 these two sets of curves (f') (f") cross each other everywhere 

 at right angles, upon that spheric surface.] 



XII. Each closing chord pPa^ is cut harmonically, at the two 

 points, c^ c", where it touches the inscribed ellipsoid (e'), and 

 the exscribed hyperboloid (e") ; or xiohere it touches the curves 

 (c') and (c"). 



XIII. The closing chords, or the positions of the last side of 

 the variable polygon (p), are not, in general, all cut perpendicu- 



