20 SEC. 2. GEOMETRY. 



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. 14. 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 n 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. 



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 

 be noticed that the head lines of the two conoids, that is to say, 

 the right line in which the two sheets of each conoid meet, are 

 perpendicular to one another. 



The transformation is effected by making the upper semicircle turn through 

 two right angles. 



97. Conoids. 



Intersection of two equal conoids having a common director plane. The 

 horizontal intersection is a plane ellipse. 



98. Conoid, in contact with a hyperbolic paraboloid. 



99. Conoids. Two equal circles in parallel planes, divided 

 equidistantly, are connected by threads, so as to form four surfaces. 



A cylinder, A conoid. 



A cone. A second conoid. 



