384 



However, when (e) is a sphere, the developable surfaces cut 

 it in two series of curves, (f'), (f"), which everywhere cross 

 each other at right angles ; and generally at any point p on 

 (e), the tangents to the two curves (f') and (f") are parallel 

 to two conjugate semidiaraeters. 



The centres of the three surfaces of the second order are 

 placed on one straight line; and every closing chord v^m i" is 

 cut harmonically at the points where it touches the two sur- 

 faces (e), (e"), or the two curves (c'), (c"), which are the 

 aretes of the two developable surfaces (d'), (d"), passing 

 through that chord P2„j p. In a certain class of cases the three 

 surfaces (e), (e'), (e") are all oi revolution, round one common 

 axis ; and when this happens, the curves (c'), (c"), (f'), (f") 

 are certain spires upon these surfaces, having this common 

 character, that for any one such spire equal rotations round 

 the axis give equal anharmonic ratios ; or that, more fully, if 

 on a spire (c'), for example, there be taken two pairs of points 

 c'l, c'g and c'3, c'4, and if these be projected on the axis b b' 

 in points g'i, a'a and g's, G'4, then the rectangle bg', .g'2b' 

 will be to the rectangle bg's . g'i b', as bg's . G'4 b' to BG'4 . g'sb', 

 if the dihedral angle c'iBB'c'2 be equal to the dihedral angle 

 C3'bb'c4'. In another extensive class of cases the hyperbo- 

 loid of two sheets (e") reduces itself to a pair of planes, touch- 

 ing the given ellipsoid (e) in the points b and b'; and then 

 the prolongations of the closing chords, PamP, all meet the 

 right line of intersection of these two tangent planes: or the 

 inscribed ellipsoid (e') may reduce itself to the right line bb, 

 which is, in that case, crossed by all the closing chords. For 

 example, if the first four sides of an inscribed gauche penta- 

 gon pass respectively through four given points, which are 

 all in one common plane, then the fifth side of the pentagon 

 intersects a fixed right line in that plane. 



An example of imaginary envelopes is suggested by the 

 problem of inscribing a gauche quadrilateral, hexagon, or po- 

 lygon of 2m sides in an ellipsoid, all the sides but the last 



