The solution of this question differs only from the pre ceding, in haying the generating right line, instead of being alwas parallel to itself, passing constantly from the apex whose two projections are given. We think it unnecessary to enlarge in this place, and leave the reader to examine for himself. with the assistance of Figure 13, should he stand in need of such aid.
'• Third Question.—From an imaginary point of a surface revolving upon a vertical axis. and given in the horizontal projection, to draw a plane tangent to such surface.
"Saiution.—Figure 14. Let A be the given horizontal projection of the axis, a a' its vertical projection, a c DE 1' the given generating curve, considered in a plane drawn from the axis. and a the given horizontal projection of tho point of contact.
It' from the point of contact, and from the axis, a vertical plane be conceived, whose projection would he the indefinite horizontal 1111C A G. such plane must cut the resolving in a eurve, which will become the generator, passing thr?mgh the point of contact : if' from the point G. a vertical line be conceived, it will meet the generating curve, and consequently the surface, in one or several points, which will become so many points of contact, of which o will he the common horizontal projection. All these imaginary points of contact will be found in the plane of the generator, by carrying A G upon L frotn a to e, and drawing through the point e a line parallel to a a' ; all the points, a, c, in which this line cuts the curve 11CDEF, will be the intersections of the curve with the vertical line drawn through the point a, and will indicate the altitudes of as many points of contact above the horizontal plane. To obtain the vertical projections of these points of contact, draw through all the points, E, c, indefinite horizontal lines, which will contain such projections ; and as they are also contained in the line perpendicular to L NI. drawn from the point e, the intersec tions, >7 y', of this line with the horizontal lines, will be the projections of the several points of contact.
if from each point of contact, a section be con ceived, made by a plane. such section, which may be considered as a second generator, will be the eireumfer (Nice of a circle, whose centre will be in the axis, and of which the tangent, which must he perpendicular to the extremity of the radius, will also be perpendicular to the aertical plane drawn through A a, in which the radius is finind theretbre the tangent plane which must pass through this tangent, will be also perpendicular to the same vertical plane, and trill have, upon the horizontal plane, its trace perpendicular to A G. We only want, therefore, the trace
of each of the tangent planes, to enable us to discover its di-tance flom the point A : now, if through the points E, c, we draw to the first generator the tangents E 1, c u, produced till they meet L at in the points T. II, the lines a 1,a u, will lie equal to those distances; therefore, if these lines be trans ferred from a to and from A to h, and if through the points i and h, the perpendiculars i o, h e, be drawn to A a, and produced till they meet the line L NI, we shall have, on the horizontal plane, the traces of all the tangent planes.
"To find on the vertical plane, the traces of the same planes, we must suppose for each point of contact, and in the correspondent tangent plane, a horizontal line produced to the vertical plane of projection ; this line, which is the tangent to the circle, will determine, on this plane, which belongs to the trace. Now, for all the points of contact, these lines have the same horizontal projection. viz. the line a ic drawn from the point G, perpendicularly to a o, and terminating in the right line L al. If, therefore, from the point K, an indefinite perpendicular, a k A.', be drawn upon L M, it will contain all the points of coincidence of the horizontal lines with the vertical plane of' projection. But as these points of coincidence will also be found on the respective horizontal lines drawn through the points E, e ; the intersections k, of such horizontal lines with the ver tical line a. k', will be each a point of the trace of one of' the tangent planes. Thus the line o k will be on the vertical plane, the trace of one of the tangent planes ; the line P will be that of another ; and so of the rest, were there a (Treater number.