Repository: UIHistories Project: Mathematical Models Catalog of a Collection of Models of Ruled Surfaces [PAGE 24]

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M o n g e writes it in the form

z = x F (ax + by + cz) + yf (ax + % -f c?)

which conies to the same thing, as F and f are arbitrary functions. Monge's form m a y be obtained directly by taking the equations of the line as ax -j~ by + cz = ^ /# -f ^ — 2. if the plane of x y be taken for the director plane, a and b vanish, and w e m a y write the equation as x F z + yfz — 1 or z. Ruled Surface with right line Director, This again requires two arbitrary functions. Taking the' line for the axis of z, the projection of the variable line on the plane of (x, y) will be of the form x — ky = 0, while its projection on the other plane, say of (x, z)> will be of the form

mx -- n ~ z. \

Now making m and n arbitrary functions of k or — wre -have y for the equation of the surface

--**(*) +/(!)•

To this family of surfaces belong the hyperboloid of one sheet, all conical surfaces, conoids, skew helixo'ids, the skew arch, k n o w n by the French n a m e of the " biais passe" (Model N o . 27 of this series), and the splay vaulting, k n o w n as the • Arriere voussure de Marseille" O f the last, there " are two specimens in the education m u s e u m at South K e n sington, although it is not represented in this series. T h e ruled surface with a director plane is the particular case of the ruled surface with a right line director, in which the right line has moved off to infinity. It follows that conoids are a particular case of surfaces with two right line directors. These directors must not meet in a point, or the surface will reduce to a plane. Ruled Surfaces with two right line Directors, The most symmetrical form in which the two directors Can be referred to rectangular axes is to take the shortest distance