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Envelope

draw, lines, plane, parallel, axis, surface, curve and describe

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ENVELOPE, (French,) the covering of a portion of the surface of a solid, by means of a thin pliable substance, which comes in contact in all points or parts with such surfiice.

To develope the surface of a solid, is to find the envelopes that will cover its different parts.

A few examples of the developement of surfaces will be here given, and for further information we refer the reader to the article SOFFIT.

Problem 1.—To derelope that portion of the curved surface of a cylindroid, which is contained between two parallel planes, and another plane passing through the axis at right angles with the parallel planes.

Plate I. Figure 1.—Let MNLC be the plane passing through the axis, terminated at at c and x L ray the parallel planes, and by the surface to be developed, at At N and c L; the four lines at c, c L, L N, N M, forming a parallelogram, C, L N. Draw c A c, at a right angle with c L, and pro 'Inca N M to A ; then A c is one of the axes of the elliptic section, at right angles to the axis of the cylindroid. On A C describe the semi-ellipsis A a c, having its other semi-axis equal to that of the cylindroid ; divide the curve A a c into any number of equal parts. say eight, and extend them from A to which coincides with the termination of the eighth part, marking the respective points as 1, 2, &c., at e, y. &c. Through the points of division, 1, 2, 3, &c., in the arc, draw the straight lines 1 E F 2 a n K, parallel to A at : also, from the points e, g, &c., draw lines e f g h &c., parallel to A. Ni. Transfer the distances. E F. n n, &c., to g h, &c.; through the points al, f, h., &c., to draw a curve ; and c, at N 1 will be the envelope required.

Problem 11.—To derelope the surface of a cylinder con tained between two other concentric cylindric surfaces and a plane, in such a manner that the axes of the two cylindric surfaces may rut the a.ris of the first cylinder at right angles, and that the plane may pass along the axis of the first cylinder, and rut the axes of the two cylinders at right angles.

Figure 2.—Let A C L N be the plane terminated by the arcs A c and N L, which are the intersections of the concentric cylindric surfaces, and by the parallel straight lines A N and c L, which are formed by the curved surface of the cylinder intersecting the plane.

Proceed in every respect as in Figure 1, and the envelope will be obtained ; the referring letters being alike in both Figures.

Problem 11I.—To derelope that portion of the surface of a cone contained between two parallel planes and a third plane, so that the axes of the rode may be cut at right angles be the parallel planes, and that the third plane may pass along the axis of the roue.

Figure 3.—Let AKKr be the plane passing along the axis. terminated at A c and E F by the parallel planes, and at A E and c F by the curved surface of the cone. A B C is a section of the cone, perpendicular to the axis. Produce A E and c F to meet in ; and with the radii n E and D A describe the arcs E 0 and A c; extend the semi-circumference of A a the arc A c, from A to C, and draw C c D; and A c a will be the envelope required.

Problem IV.-7o derelope that portion of the suViice of a cone contained between two concentric cylindric suiftces and a plane passing along the axis, so that the plane may cut the common axis of the cylindric surfaces at right angles.

Figure <1.—Let Aran be the portion of the plane passing along the axis, and the arcs A c and r K the intersections of the cylindric surfaces ; the straight lines A I and c a, the intersections of the conic surface. A a c is a section upon the chord A c. From D, with the radius D A, describe the arc A M ; divide the semi-circumference A B c into any number of equal parts, and extend them upon the arc A at, from A to NI, at the points 1, 2, 3, &c. to at : draw 1 D. 2 n, 3 D, &c. to m n included; also, through the points I. 2. 3, in the are A a c, draw lines perpendicular to A c, cutting it in as many points : from these points, draw lines to n, cut ting both the concave and convex curves ; from the points so cut, draw lines parallel to A c, cutting A u ; then from the points of intersection, made by the parallels drawn from one of the curves, describe the several arcs drawn from the point D, to cut the respective straight lines. Proceed in the same manner with the other curve, and through the points so obtained, draw the two curves ; amid AN1Lr will form the envelope required. But to show more particularly how the successive points in the curve of the envelope are found, we shall only describe a single point, and the remain ing points will he obtained in the same manner ; thus, to find the point rt in the envelope, draw '2 a perpendicular to A c, cutting it at a ; draw a a, cutting, the curve A C at draw F a parallel to A c, cutting A D at a ; from the centre D, describe the are o n, cutting 2 D at n, which is a point in the curve, as before stated ; but by drawing the several systems of perpendicular lines, of lines going to a centre, and parallel lines, at once, much time will be saved in the operation.

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