24 SEC. -2. GEOMETRY. 



plane ; but Avhen the bars are not in the same plane, the surface is the hyper- 

 bolic paraboloid. 



The surface is sometimes called the twisted plane. But it must not be 

 supposed that it can be made by bending a plane. On the contrary, when 

 the surface is twisted, no two of the strings lie in the same plane, and, there- 

 fore, no part of the surface is plane. It can neither be flattened nor made 

 from a plane, without stretching or contraction. 



The hyperbolic paraboloid is the natural surface proper for a ploughshare. 



78. Hyperbolic Paraboloid. 



Two bars, pierced -with holes at equal distances, the holes being connected 

 by two different systems of strings. The surface, as well as the arrangement, 

 is very nearly the same as in No. 77, only that there are two paraboloids in- 

 stead of one. As the movable bar swings round, one paraboloid opens out 

 while the other closes up. If the -bars are swung so as to be in the same 

 plane, one system of strings describes a plane by parallel lines, and the other 

 by lines radiating from a point. If one bar is now turned so as to be end for 

 end, we still get a plane, the set of parallel lines now passing through a point, 

 while the set Avhich previously passed through a point has now become 

 parallel. 



The pair of paraboloids intersect in three right lines. There is also a 

 fourth intersection on the " line at infinity." 



79. Hyperbolic Paraboloid. 



Two bars equally spaced ; each turns on an arm perpendicular to itself, 

 and one arm swings on a pillar. These arms can be ranged in one plane, and 

 also turned end for end. 



80. Hyperbolic Paraboloid generated by two systems of 

 right lines. 



A skew quadrilateral with four equal sides, each pierced with the same 

 number of holes, equally spaced. The model exhibits the double generation 

 of the surface. The plane containing two of the sides turns about hinges 

 connecting it with the plane of the other two sides. By closing or opening 

 this hinge the paraboloid opens out or closes. When completely open, it 

 forms a plane divided into diamonds. When completely closed it again forms 

 a plane, but the division is no longer uniform. The strings then become 

 tangents to a plane parabola. 



81. Hyperbolic Paraboloid. 



A skew quadrilateral turning upon four hinges with parallel axes or pins. 



The difference between this and the last is not in the kind of surface or 

 mode of generation, but in the manner of deforming the surface. In No. 80 

 the lengths of the strings alter ; while in this model they remain unaltered. 

 Moreover, although the surface flattens in two ways, yet in both ways the 

 strings become tangents to a plane parabola instead of parallel. 



This model is well adapted for showing the leading sections of the solid 

 All sections parallel to the pins of the hinges are plane parabolas, which de- 

 generate into right lines when taken also parallel to the brass bars. Any 

 other sections, whether perpendicular to the hinges or inclined to them, give 

 hyperbolas, which degenerate into a pair of right lines when the plane of 

 section is a tangent to the surface. 



It may b? worth while to remark that there is nothing absurd in the 

 tangent plane to a surface cutting that surface, as a student unaccustomed to 

 those subjects might at first think. On the contrary, when a surface is bent 

 one way in one direction, and the other way in the opposite direction, the 



