Caption: Mathematical Models Catalog of a Collection of Models of Ruled Surfaces
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18 2„ The inclination of the tangent plane i constant at any s giyen distance from the axis of z » 3. A n y cylindric annulus of the surface, with the axis of z passing through i s centre, may be measured by a t parabolic arc, 4. If the projection of a portion of surface be any figure symmetrical to the projections of any two right line generators, the corresponding cylindrical volume shall be the product of the projected area into the ordinate at i s middle point. t 5. Gauss's measure of curvature is constant at a constant distance from the axis of z. The Hyperboloid of One Sheet. This in its most general form may be defined as the surface traced out by a line which meets any three fixed lines in space. It will be simplest to take the three lines as three edges (which do not meet) of a parallelepiped, to take axes parallel to them, and to take the centre of the parallelepiped as origin. Calling the lengths of these edges 2a, 2b, 2c, their equations may be written as y — hz x = a,y = - b In order that an arbitrary line x — a _^y — P ^ z ~~ y I m n may meet these three lines, we must have b — p ___ c_+_y m c — 7 __ a -\~ a n a — a __ b + P I n multiplying these together we get ( a - a) (b - P) (c - y) + (a + a) (b + P) ( + y) = 0, c or, a p + bya + cap 4- abc ^. 0. ,y ~~ I n
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