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Caption: Mathematical Models Catalog of a Collection of Models of Ruled Surfaces
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35 while that of developable surfaces is rt = s2. It will be observed that this, is the condition that the first equation should give two equal values for c, Cones and Cylinders, T h e differential equations previously obtained are not those which distinguish these surfaces from other ruled or developing surfaces. O n the contrary, they are restricted by conditions which settle the position of the vertex of the cone, or the direction of the generators of the cylinder. T h e y are consequently expressed by differential equations of the first degree. If w e desire to eliminate the constants of the vertex, or of direction, w e must proceed to third differentials. Writing the equation of the cone as *-C=(,-«)F(f-^) and regarding x and y as independent variables, it is easy to verify that ay — /32 _ a$ — /3y __ ffi — y2 # r ~" "~2s . "" i ' but these equations are also derivable from the general equation of developable surfaces, rt = s2, and do not distinguish conical surfaces from all others. If again w e write the equation of the cylinder as ny — mz = F (mx — ly) w e find that the numerators of the above functions vanish, or, as the result m a y be more simply stated a j 3 y ~P == 7~= I and this appears to distinguish the cylindrical from other developable surfaces. T h e theory of this part of the subject is recent, and far from complete. Meanwhile it is certain that what goes before is strictly true as stated; but the student must be cautious of drawing inferences which go beyond the text. For instance, M r , Oayley has shown that the equations a [3 y do not represent cylinders only, or even ruled surfaces only, It is w h e n taken in connexion with the equation of developable surfaces that they represent cylinders. 29992. D
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