in. MODELS. 19 



83. Hyperbolic Paraboloid. 



A skew quadrilateral, with all its sides equal, and pierced holes at equal 

 distances. 



As far as the curved surface is concerned, the same as No. 5. But the 

 hinges are altered in direction, and the model shows plans and elevations of 

 the right line generators of the surface. The rings also show parabolic sections 

 of the surface. 



In consequence of the alteration in the direction of the hinges, the spacing 

 of the inclined bars, although equidistant, is at a different pitch from that of 

 the horizontal bars. 



84. Hyperbolic Paraboloid. 



A skew quadrilateral, with all its sides equal, and pierced with holes at equal 

 distances. It shows the plans and elevations of the right line generators. 

 The rings show the parabolas of the principal sections. 



No. 7 represents one quarter of what is shown in No. 8. The upper corners 

 of Nos. 7 and 8 correspond ; but the lower corner of No. 7 corresponds with 

 the middle ring of No. 8. 



85. Hyperbolic Paraboloid. 



A skew quadrilateral, with all its sides unequal. The surface is the same as 

 Nos. 7 and 8, but the proportions and the portion of the surface chosen for 

 representation are different. The quadrilateral base being irregular, the 

 strings alter in length as the surface is deformed by closing the hinges. 



86. Hyperbolic Paraboloid. 



Skew quadrilateral, pivoting on a single hinge. Intended to show the con- 

 struction of the parabola connecting two roads which meet obliquely. This 

 construction is used by engineers in laying out roads. 



87. Hyperboloid of one Sheet. 



Two rings or circles, in parallel planes, are pierced with equally spaced 

 holes. In a certain position the threads give, 1st, a cylinder; and 2ndly, a 

 cone. 



The upper ring turns round a pin at its centre. In turning it, the cylinder 

 closes in and the cone opens out, each altering into a hyperboloid of one 

 sheet. We can go on turning the ring until these coincide in one hyperbo- 

 loid, of which we thus get both systems of generating lines. 



If the rings are set on a slope the hyperboloid is elliptic. If the rings are 

 horizontal the hyperboloid is one of revolution. 



Sloping one ring, so as not to be parallel with the other, gives rise to some 

 curious ruled surfaces, but these are not m general hyperboloids. 



88. Hyperboloid of one Sheet. 



Two rings of different radius, in parallel planes, are divided into the same 

 number of equal parts. The smaller and upper ring turns round a pin at its 

 centre. In a particular position of the rings, the threads give two cones. 

 Turning the ring transforms each of the cones into a hyperboloid, and 

 when the two hyperboloids coincide, we get the two systems of right line 

 generators. 



The same stand also has a model of a hyperboloid with only one set of 

 strings. By turning the upper ring either way it deforms into a cone ; in the 

 one case with its vertex between the rings, and in the other with its vertex at 

 a considerable height above the rings. 



Both these can have their upper rings moved along the top bar so as 

 to incline the surfaces. We still get cones and hyperboloids, but it is only 



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