Problem V.—To develope the surface of a cuneoid con tained between two parallel planes, and u plane passing along the axis of the cuneoid, so that the parallel planes may be perpendicular to the plane passing along the axis, the section of one of the parallel planes being given.
Figure 5.—Let A nen be the plane passing along the axis, terminated by the straight lines A n and n c, which are the intersections of the parallel planes, and by the straight lines A B and c la which are the intersections of the euncoidal surface. Let the section B E c, standing upon I3 c, be a semi circle, and, consequently, the section A F D limited by the other parallel plane, will be a semi-ellipsis of the same altitude.
Produce A n3 and n c to meet in G. divide the are n E, which is the half of the semi-circumference, into any number of equal parts, say four, 11'0111 the points of division, draw the perpendiculars 1 2 I, 3 15, E L, cutting 13 C at K, a : draw a r perpendicular to A G ; on a r Blake a, c n, G a, and a v respectively equal to n 1, 1'2, is 3, and a E ; from the points in, a, o, r, as centres, with the respective distances G 11, G G E, and a a, describe the arcs h. r, i t, k v, and /.r ; extend the are B I, or 1, 2, which is the eighth part of the semi-circumference, from B to h, from h to i, from i to k. and from k to /; then drawing the eurve B It ik 1, will give the half of the envelope, which will coincide with the are n E, when the semicircle rl is c is turned peg wudicular to the plane, A B C a, upon its base B c : and since / a is the middle line, the other half, or counter part, will easily be found. To develope the elliptic edge, join in h, n o k, and P and produce them to u, s, f: transfer nt e, I 5, s T, to h. t, 1 j: and through the points A, u, t, y; draw a curve A a s i if which will be the envelope of the are A F, the half of the semi-ellipsis A F n ; then the counter part being found, will complete the envelope Aped of the curved surface, standing over A B C d.
Problem, V1.—To develope that portion of the surface of a euneoid terminated on two sides by a plane passing through, the axis, and by two concentric cylindric surfaces whose axis is peqeudiculur to the plane given ; given that portion of the plane terminated by the curved suiface of the cuneoid, and by the intersections of the two cylindric surfaces ; also, the semicircular section of the euneoid.
Figure 6.—Let A E and n G be the intersections of the euncoidal surface, and the arcs A E n and E F G the intersec tions of the concentric cylindric surf:ices.
Let n c be the intersection of the section, which is a semi circle, then final the envelope for the semi-circumference, as in the last problem for parallel planes ; then the lengths of the intermediate lines contained between the base of the circular section, and the intersections of the cylindric surfaces, being transferred upon the corresponding lines from the semicircular envelope, will form the covering of the cuncliidal surthee, as defined.
The reader will observe, that the two last constructions in Problems V. and VI., are only approximations near to truth; it is, we believe, impossible to find the true envelope by means of straight lines, or perhaps even to extend the true aineoidal surfitee on a plane at any event, any more than that Jr a sphere, which can only be represented by means of pro vetions. The only surfaces which can be extended on a plane. arc those to which a straight edge will everywhere apply through a certain point, or in parallel directions to a given line ; of this description are the surfices of planes, ..:ones, and cylinders: a straight line sill apply to all parts of the surface of a cone through the vertex. and to all parts If a plane through any given point, and to all parts of the -urfacc of a cylinder parallel to the axis.
The envelopes of all solids, to which a tangent plane to its surface parallel to the given plane will apply, have the same curvature or straight line at the edge, where the plane be comes a tangent, as the corresponding part of the edge of the given plane.
In describing the envelopes of•solids, the whole or a por tion of the section passing through the axis, is always sup posed to be given, as also a section of the solid, making a given angle ith the said plane, and the intersection in a given position.