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Caption: Mathematical Models Catalog of a Collection of Models of Ruled Surfaces
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27 Surfaces of which the Directors are given, either explicitly or implicitly. In these cases the arbitrary functions must be determined by the conditions in question. In the case of ruled surfaces with given director lines, through which the variable right line must pass, it will in general be sufficient, for the determination of the arbitrary function, to m a k e the equations of the director satisfy the functional equation identically, for then the director will be a line traced upon the surface. Differential Equations, of Families of Surfaces. In the restricted cases in which we have obtained functional equations, w e m a y obtain the differential equations by the direct processes of partial differentiation. But they m a y also be obtained from the equation to the generating line, expressed in terms of the coefficients which w e wish to retain ; for our object is simply to obtain the relation between certain partial differential coefficients, and it is immaterial whether w e obtain these by implicit differentiation from a line in which only one variable is independent, or by partial differentiation from a surface in which there are two independent variables. T h e character of the restriction imposed on the motion of the line will determine what are the constants to be eliminated by the differentiation. For the motion of a point fixed while the line shifts, and the independent motion of the point along the line, each infinitesimal in amount, determine the tangent plane of the surface, and this tangent plane is one of the complete solutions of the partial differential equation, of which the functional equation of the surface is the general solution. A n y family of surfaces of which the actual surface is the envelope will be a complete solution of the differential equation of the surface, but the tangent plane is the only one in which w e are certain that the right line generator must be contained. Cylindrical Surfaces. Starting from the right line x — a __ y — / 3 I m z— y n and observing that our object is to retain Z5 m9 n} w e obtain by implicit differentiation dz dz dy __ n dy _ m dx dy dx I dx I
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