mechanically to touch the projection of the curve at the given point.

§ 1. SECTIONS OF CURVED SURFACES BY PLANES.

PROBLEM 1.

Intersection of a cylinder perpendicular to the horizontal plane by a plane perpendicular to the vertical plane. 47. The construction will consist of three distinct parts. First, To determine the projections of the intersection and those of its tangents.

Secondly, To construct this intersection of its true mag

nitude, and the tangents to it at the same points. Thirdly, To develop the cylinder and to describe the

curve which will represent the section developed, and to draw the tangents to this developed curve. 1o. The horizontal trace of the cylinder may be any curve at pleasure abc . . . and its generatrixes are perpendicular to the plane by the hypothesis. The vertical trace of the cutting plane may be any straight line qp', while its horizontal trace qp, must be at right angles to the ground line xy. Since every line on the cylindrical surface has abc... for its horizontal projection, and every line on the cutting plane has p'q for its vertical projection, the two projections of the intersection are determined.

Take any point m in abc... at pleasure, and through it draw mm"m' perpendicular to ay to cut p'q in m'; it is obvious that m"m' is the vertical projection of the generatrix which passes through m, and that the point in which this generatrix cuts the plane pqp' has m, m' for its projections. m,m' are, therefore, the projections of a point in the curve of intersection. By repeating this construction, the projections of as many points may be obtained as are requisite.

The tangent at any one of these points, as for example m,m', is easily obtained. The tangent plane to the cylinder at this point is vertical, and has for its horizontal trace the tangent mn, to the base abc.... Now the tangent required is the intersection of this tangent-plane with the given one; the projections therefore of this tangent are mn and qp'.

2°. To determine the true magnitude and form of the section let us suppose the plane pqp turned down on one of the planes of projection by rotating on the trace qp'. The point m, m' is situated in space on a perpendicular to qp' at a distance from m' equal to mm": hence by drawing mм at right to qp' and making it equal to mm", м will be one point of the curve sought. By the same construction as many points may be found as may be deemed neces

M

sary to allow the curve ABC . being drawn. To draw the tangent to this curve, at the point м, it is only necessary to observe, that in the rotation of the plane the line qn will fall in qn, perpendicularly to qp', so that by drawing MN, we obtain the tangent at the point м.

If the plane pqp' be turned round on its horizontal trace Pq, the curve of the section may also be drawn of its true magnitude in the horizontal plane of projection. Draw mr perpendicular to pq; if we then conceive a line in space between and the point m, m', this line will be perpendicular to pq and will be projected of its true magnitude in qm': if then m'r, in mr, produced, be made equal to qm', the point м' will be one in the curve of the section, when turned down on the plane of projection, and on the same principles any requisite number of other points may be obtained; nм will obviously be the tangent to the curve in mm' when this curve is turned down.

There is yet another construction for this purpose, which may be occasionally advantageously employed. Let the plane pqp be supposed to turn round on a line perpendicular to the vertical plane, as on dd", d' till it assume a position parallel to the horizontal plane; the curve, lying in this plane, will then be projected on the horizontal plane of its true form and magnitude: to obtain points in this projection, it is only necessary to make dм" equal to ď'm', as will be readily understood from the figure by what has been already explained. The point of contact of the tangent is now in м"; to draw the tangent at that point it is only necessary to deduce a second point in it, as N", on the same principle as that by which m" is obtained, the line nN" being drawn parallel to ay for the horizontal trace of the plane, in which the trace of the tangent lies during the rotation of the plane pqp' on dd", d'. N" will conse

quently lie in nn" thus drawn. It will also be observed that the point o in which the tangent nm, qm' cuts the axis of rotation dd", d', not being changed during the rotation of the plane, the tangent N"M" will pass through o.

3°. If a cylinder be considered as a prism with an infinite number of faces, it is clear that if this surface be developed, or unrolled, every section perpendicular to the generatrixes will become a right line on the developement at right angles to the edges, or arrises, of the prism when developed; and the length of these arrises, comprised between the oblique and perpendicular sections will not be

8

γ

B

a

μ'

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altered. If therefore the curve abc... be divided into any number of arcs ab, bc, ed, &c., and a line be drawn, and made equal to the curve abc by setting off along it the equi-distances aß, By, yd... respectively equal in number and magnitude to the arcs ab, bc, cd,... then by setting the true lengths of the respective arrises along perpendiculars, aa', BB', vy', 88'... to the line a... d, the curve a'B' drawn through these points will be the developement of the curve of the obliqué section. The true lengths of the arrises are furnished by their projections a'a", b'b”.......... on the vertical plane.

Let the point m, m' be represented by u' on the developed curve; then, since the elements of the original curve

do not change their angle of inclination to the generatrixes, or arrises, by being developed, the tangents, which are these elements produced, preserve their inclination to the arrises when developed. Now the tangent at the point mm' passes through n, and the angle made by this tangent with the arris of the cylinder is an angle of the right-angled triangle, of which mn is the base, and m'm' the altitude: if, therefore, μv be made equal to m'n, and μv be joined, this line will be the tangent to the curve.

Scholium. In the figure the curve abc... is a circle, but the construction above described is not the less general: whatever may be the curve abc..., it will always be found that the tangents in m and m', to the curves abc ... ABC..., cut the trace pq in the same point n, or, which is the same, the tangents in m and м' to the curves abc... ABC........., cut the right line dd" in the same point o; hence this remarkable

Theorem. If any, whatever, cylinder be cut by any number of planes, passing through a right line perpendicular to its generatrixes, and all the sections are brought into one plane by being rotated on this common line, the tangents to the different curves of the sections, at points in them, situated in one line perpendicular to the common one above-mentioned, will all meet this common line in the same point.

This property is analogous to the well-known one of ellipses described on a common axis; and is true in this case only inasmuch as these ellipses are the intersections of the same cylinder by planes fulfilling the above-stated conditions.