26 SEQ. 2. GEOMETRY. 



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

 curious ruled surfaces, but these are not in 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 a 

 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 

 when the rings are horizontal, and centre to centre, that we get surfaces of 

 revolution. 



89. Hyperboloid of one Sheet, with its asymptotic cone. 



90. Hyperboloid of one Sheet, with its asymptotic cone. 



The tangent plane to the cone is also drawn. It meets the hyperboloid in 

 two parallel right lines. 



One of these right lines is the line of contact of a hyperbolic paraboloid 

 with the hyperboloid, and the tangent plane is one of the director planes of 

 the paraboloid, both systems of generating lines of which are exhibited. 



91. Hyperboloid of one Sheet. 



A slight variation from No. 90. The paraboloid only shows one system of 

 right line generators, and the tangent plane is made by parallel instead of 

 radiating lines. 



92. Hyperboloid of one Sheet, and its tangent para- 

 boloid. 



This shows the transformation of a cylinder and its tangent plane into a 

 hyperboloid and its tangent paraboloid. 



93. Conoid, with its director plane. The director curve is a 

 plane curve. 



By shifting the position of the brasses the conoids deform into different 

 conoids or other allied surfaces. 



94. Conoid, with a director cone. The director curve is of 

 double curvature. 



By shifting the position of the brasses the conoids deform into different 

 conoids or other allied surfaces. 

 i 



95. Conoid, showing both sheets of the surface. 



By shifting the position of the brasses the conoids deform into different 

 conoids or other allied surfaces. 



96. Conoids. Model showing the transformation of a cylinder 

 into a conoid and back again. Also model showing the trans- 

 formation of a cone into a conoid and back again. It is to 



