A somewhat later application of the method was made by Max well who as the outcome of a suggestion by Prof. Willard Gibbs used the quantities volume, energy and entropy in making his famous thermodynamic surface model in which the properties of a substance in its solid, liquid and gaseous or any conditions in which these states co-existed are indicated by the geometrical properties of the surface.
Maxwell showed how isothermal and isopiestic lines could be drawn upon it and that there is one position of the tangent plane in which it touches the surface in three points which represent the solid, liquid and gaseous states of the substance when the tempera ture and pressure are such that the three states can exist together.
Plaster Models.—Plaster casts can obviously be produced at less cost than the original mould, so that wherever feasible the method affords a convenient means of reproducing surface models of either constantly varying functions as in surfaces of revolution or of irregular or non-continuous form.
The former case is typically represented by Pl. II. fig. 4, the sur face of rotation of the tractrix about its asymptote, upon which may easily be scribed after moulding the body, the geodetic lines and principal tangent curves, or such surfaces of constant mean curvature shown in Pl. II. fig. 5 which illustrates (left to right) (I) onduloid, (2) nodoid, (3) ring of the nodoid arising by rota tion of the loop, and (4) the catenoid, a minimum surface whose constant mean curvature is null.
Surfaces of rotation having a constant negative measure of curvature as in Pl. II. fig. 6, that on the Jell being of the cone type and bearing geodetic and asymptotic lines, that on the right illustrating the hyperboloid type and being marked with parallel geodetic lines and geodetic circles.
P1. II. fig. 7 illustrates a surface of the third order showing four real conical node points and the principal tangent curves.
A form of Kummer surface (singularity surface of a complex of the second degree) is shown in Pl. II. fig. 8. It is of the fourth order of the fourth class and has sixteen real node points and the same number of double tangential planes.
A further example of a surface of the fourth order, four planes making contact along circles, is the so-called Roman surface due to Steiner and shown in Pl. II. fig. 9. It has three intersecting double straight lines and is of the third class. The asymptote lines are indicated.
An interesting example of a model illustrating a minimum sur face is shown in Pl. II. fig. Io. It contains a system of real para bolas the planes of which make a constant angle with a fixed plane of the space.
Fundamental examples in connection with the function theory are shown in P1. II. fig. I 1, where is shown: (I) (at top) Simply connected Riemann surface (two leaf) which contains in its interior one point of double inflexion of the first order. (2) (bottom left) A simply connected Riemann surface (three leaf) with an interior point of double inflexion of the second order. (3) (bottom right) A triply connected Riemann surface with a boundary line turning back upon itself. Fig. 6 illustrates the func
tion and the course of the elliptic functions p(n) and p/ (n) in the Weierstrassian series is shown in fig. 7.
Linkages and Kinematical Models.—Linkages may be de fined mathematically as systems of bars connected by pin joints or hinges, to allow deformability without sliding motion. All algebraic curves may be generated by such articulated linkages, Kempe, Darboux, etc., having analysed the position very fully, and numerous attempts have been made to solve by linkage sys tems the mathematically indeterminate trisection of an angle. These devices fall, however, into the classification of instruments rather than models, and space shortage forbids their treatment here, a qualification which applies also to a treatment of kine matical models dealing with related motion.
Stereoscopic and Optical Methods.—Another series of mod els has been developed to a limited extent by producing a solid effect from plane figures by means either of viewing bicoloured diagrams through absorption screens or the more common method of displaced image.
Examples of the first method are the plastographs or anaglyps by Mr. H. Richard of Chartres, and in England by Mr. G. F. Smith, who have produced examples illustrating the interpenetra tion of prisms, sections of a helicoid, etc.
The second method is represented by the series designed by Sir George Greenhill to illustrate gyroscopic movements, e.g., the locus of the axis of a spinning top or Maxwell gyroscope.
Quite recently a series of lantern slides illustrating certain alge braic curves, viz., the dual singularities, etc., involved in the theory of cubits and the construction of hyperelliptic and quartic curves has been developed by Prof. Arnold Emch at the University of Illinois, who has also produced a cinematograph film of the Poncelet polygon, i.e., showing the succeeding positions or con tinuous movement of a triangle remaining inscribed and circum scribed to two fixed circles respectively.
A novel method of treatment is that devised by Prof. Papperitz of Freiberg, viz., kinodiaphragmatic projection. The device con sists essentially of a variable speed gear box capable of imparting rotatory motion to a transparent diaphragm placed parallel to and immediately in front of the condenser of an optical lantern. Upon the diaphragm may be fixed any combination of thin polished wires which will reflect narrow beams of light into the focussing lens. Beyond and in front of the lens is placed a sec ond rotating axis which may be vertical or inclined, and carry a surface model—say, a sphere, cylinder, etc.,—built up of wires, spaced apart.
Shadowgraphs are thus projected on to a distant screen, the forms being continuously changed or dissolved into one another according to their relative axial speeds, and by the rotation of stereomatical bodies and simultaneous projection it is possible to produce three-dimensional images in space. (G. W. Cu.)