Space Curves.—There remain, however, still further types of example wherein space curves of the 3rd order are represented by the developable surfaces of their tangents.
Such a series would comprise models showing:— I) the curves with their asymptotes, 2) the developable tangent surface, 3) the curves as partial sec tions of cones and cylinders, 4) the two dual generations of the curves, their developable sur face, etc.
An example shown in Pl. I. fig.
13, illustrates the involute of the planes which touch two conic sections possessing a common tangent.
These models would demon strate the cases of singularity which can arise in a position of a space curve according as the point or flexion plane is a progressive or regressive element and also the relation of the position to infinite distance.
Examples of the problems connected with the theory of cubic space curves—cubic ellipses, hyperbolas, parabolas, etc.,—are shown in fig. 3, viz., the tangent surface of the cubic ellipse, the surface which separates the points of the first case from those of the third, and in fig. 4 the horopter,—a symmetrical cubic ellipse lying on a circular cylinder both of them of special application in physiological optics.
A physical-science application is given in fig. 5 which illus trates the form of equipotential lines and lines of force cor responding to two electric conductors charged to the same sign.
Helical Surfaces.—Helical surfaces may best be demonstrated by either shaped wires or small surfaces of tinplate hinged to gether, the former providing the cheaper but a less flexible medium. Typical examples are the helical surfaces of P1. I. fig.
14, where generators and prin cipal tangent curves are picked out in different colours to ren der them distinguishable, and in Pl. I. fig. 15, that of a model composed of small hinged sections, we have an illustration of the same problem solved by the application of the idea of polyhedra to the theory of the bending of surfaces.
The same model also exempli fies the Voss surface demon strated by finite plane elements of surface hinged together to en able them to be bent in two conjugate systems of geodetic lines.
Cardboard Models.—In Pl. II. fig. 1 is illustrated an example of model made up of thin sheets, e.g., cardboard circles of reg ularly varying diameters set equally apart in parallel vertical planes, whereby it is possible to evolve the whole series of sur faces of the second order (ellipsoid, hyperboloid, paraboloid, etc.).
A further advantage of this type of model is that the sections may be interlocked across an axis and thereby deformed at will, a feature which may be reached in another way as in the deform able circles of figs. 2 and 3 of P1. II. In this type a number of
different sized wire circles are loosely jointed together across a diameter by a special form of hinge—Wiener's limited joint— which allows at once an extraordinary freedom and restraint.
The figure shows the limiting positions of circle and sphere and the formation of prolate and oblate ellipsoids. Similar models may be readily made to illustrate the elliptic paraboloid, and paraboloids of one sheet or of two sheets, and of double cones, etc., the method of construction with its property of semi-trans parency enabling a clear idea to be obtained of the constant rela tionship of the asymptotic cone and that the lengths of all seg ments of generating lines remain unaltered.
Surface Models.—The method of representing the surfaces of the 2nd order by thin sheet cir cles arranged in parallel planes suggests the means of producing what is probably the most gen erally useful of all types, viz., surface models of wood, clay or plaster. A model of a cubic surface for example may be con sidered as built up of a number of parallel horizontal sections each of which is a plane cubic curve.
In order to produce such models it is in the first place essential to prepare templets, which embody the particular function to be illustrated. If then a series of such be erected in the appropriate coordinate planes, a surface will ultimately emerge which may be definitely outlined by narrow strips of thin muslin or fabric fixed into position by a plastic medium such as claywash, wax or plasticene, from which subsequently a plaster cast may be taken. On the permanent surface may be marked appropriate axial and geodetic lines and to it tangent planes of say transparent celluloid, etc., may readily be applied. Surface models may alternatively be made up of thin layers of wood suitably shaped, the smooth con tour being filled in by wax, or they may be evolved by applying templets after the manner employed in shaping a model ship's hull. Such a model may of course represent a function of pure mathematics, e.g., f (x, y, z) = o or some physical function say of the pressure volume and temperature of a gas, as in the case of Prof. James Thomson's model of 1871, made to illustrate the data obtained by Prof. Andrews in his classic experiments on the relation between temperature, pressure and volume of a constant mass of carbonic anhydride when the values were plotted, with temperature as the x, pressure as the y, and volume as the z coordinates respectively.