III. MODELS. 25 



tangent plane must cut it. In this case, the plane passing through any two 

 intersecting strings is a tangent plane, and evidently cuts the surface along 

 each string. 



If we imagine two planes parallel to the hinge pins, and each bisecting a 

 pair of opposite bars, we obtain the asymptotic planes of the paraboloid, each 

 of which is the assemblage of the asymptotic lines of the hyperbolas parallel 

 to the principal hyperbolic section. Their being asymptotic has reference to 

 these hyperbolas, and not to the parabolic character of the surface. 



82. Hyperbolic Paraboloid. 



A skew quadrilateral, with its opposite sides equal in length, and pierced 

 with holes at equal distances. 



Nearly similar to No. 81, but differently mounted, and with the sides of 

 different lengths, the alternate sides only being equal. It is virtually a slightly 

 different aspect of the same surface as No. 81. 



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. 81. 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. 83 represents one quarter of what is here shown. The upper corners 

 ofNos. 83 and 84 correspond; but the lower corner of the former corre- 

 sponds with the middle ring of the latter. 



85. Hyperbolic Paraboloid. 



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

 Nos. 83 and 84, 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 cy Under 

 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. 



