8G LECTURES TO SCIENCE TEACHERS. 



giving an illustration of the two methods of generation 

 of the surface. 



The next figure will at first appear strange ; but, 

 if you examine it, you will see it is exactly similar to 

 the first one I showed you. It is a hyperbolic paraboloid 

 tangent to a twisted surface. At the same time you will 

 see here a plane tangent to the same surface, showing the 

 relation between a plane tangent and a hyperbolic para- 

 boloid tangent. I should remind you that plane tangents 

 are not necessarily tangents to the whole surface. You 

 will see that the plane tangent cuts right through the 

 surface ; but the hyperbolic paraboloid is a complete 

 tangent to the surface considered. 



These figures are called of one sheet or of two sheets, 

 according as they are of one or two parts. This is called 

 a hyperboloid of one sheet. Supposing you make the 

 hyperbola which I here describe on the board rotate round 

 its vertical axis thus, you will have the hyperboloid of one 

 sheet ; if, on the contrary, you make it rotate round the 

 horizontal axis, you will have two complete surfaces of 

 revolution, or a hyperboloid of two sheets. 



Besides these figures, there is another very pretty 

 model of intersection I have to draw your attention to. 

 Beginners are very apt to confuse the various intersections 

 of cylinders and cones ; and here are two very beautiful 

 examples of the intersection of two cylinders ; one where 

 there is but one intersection, and one where one cylinder 

 passes right through another. There are then two intersec- 

 tions. Again these other models are very interesting, showing 

 the change from the plane to the hyperbolic paraboloid. 

 A brief way of describing the paraboloid of one sheet is, 

 that it is a figure described by the sliding contact of a line 

 upon three others, no two of which are in the same plane, 

 and with which it remains constantly in contact. 



These next figures are cylinders with cones inside them, 

 and by a simple rotation you change the cone into a 

 cylinder, and the cylinder into the hyperboloid of one 

 sheet. 



In conclusion, Prof. Osborne Reynolds exhibits some excel- 

 lent folding models, showing the traces and the intersections 

 of planes by different coloured lines. Without some such 

 models as these before you, it would be almost impossible 



