Repository: UIHistories Project: Mathematical Models Catalog of a Collection of Models of Ruled Surfaces [PAGE 11]

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any selected obliquity. It will, of course, give the radii of the outside circles, and the line joining the points at which it cuts the inside semicircles will be a generator of the surface. This line will evidently pass through the director line, because it is in the same plane with it. In stone or brickwork ^ the sides of the voussoirs will be given by the auxiliary plane in question. W h e n the openings are parallel the voussoir joints are therefore plane, and the simplicity thus gained is the chief reason for adopting this form of skew arch. It is usual to take the right line director perpendicular to the openings, and symmetrical to them; that is to say, passing through the middle point of the parallelogram of the springing plane. W h e n the openings are not parallel, the voussoir joints shown by the model are deformed into hyperbolic paraboloids. This deformation, is, however, very slight, and in practical work would be avoided altogether by adhering to the principle of drawing a plane through the director line. T h e spacing of the voussoirs is usually determined by dividing the outer semicircle into equal parts. This form of arch is inconvenient w h e n the obliquity, and the length of the barrel are excessive, for the generators are not generating lines of the cylinder containing the opening semicircles, but chords of it, and, therefore, at the middle, falling considerably inside it. T h e arch therefore droops in the middle, and this would be ugly and inconvenient if the proportions were excessive. It is interesting to compare this surface with the skew vault of Marseille {arriere voussure de Marseille)^ an example of which is shown in the set of plaster models contributed by the 6 Brothers of the Christian Schools," and another, in imi€ tation brickwork, a m o n g M . Schroder's models of furnaces. In this case the curvilinear directors do not tally with one another, although they remain parallel, and the right line director is a vertical line behind the smaller arch. T h e construction for the right line generators is the same for both, namely, to consider an auxiliary plane pivoting about the right line director. 28. Staircase Vault for a square well (vis St Gilles carree). 29. Staircase Vault. Model for exhibiting some properties of this ruled surface, by showing h o w it is obtained from the deformation of a cylinder (douelle de la vis St* Gilles carree)*