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case we should only have to draw a series of planes perpendicular to these axes, each cutting the surfaces therefore in circles, the points of intersection of which must belong to the curve sought. But this method of proceeding is not applicable when the axes are not parallel ; for then the planes perpendicular to one will not be so to the other. Nevertheless, if the two axes meet, as by the hypothesis they are supposed to do, the points of the curve may be determined by the intersections of circles; the surfaces being cut, not by planes but by concentric spheres, having the point of intersection of the axes for their common centre. For each of these spheres will intersect the two surfaces in the circumferences of circles, the intersections of which will be points in the common section of those surfaces.
To render the constructions as simple as possible, let the vertical plane be assumed parallel to the two axes, and the horizontal plane perpendicular to one of them. On this supposition, one of the axes will have a point a, for one of its projections, and the other a line ab, parallel to xy. The vertical projection of the first axis will be a'a", perpendicular to xy, and the other may have any direction, as a'b', according to the data. The two surfaces are given by the vertical projections of two meridians, h'c'i'c" and h'd'i'd”, in the plane of the axes.
From a describe any circle at pleasure, c'd'c" " cutting the projections of the two meridians in c', c", d' and d". This circle will obviously be the projection of the section of a sphere of the same radius, having a, a' for its centre, made by the plane of the two axes. The points d'o”, d'd”, are therefore the vertical projections of points common to these meridians and to the sphere. If the meridian of the first surface be supposed to rotate on its axis, the points d'o" would describe a horizontal circle, the vertical projection of
which would be c'é"; this circle would be common to the surface and to the sphere.
For the same reasons, d'd" is the projection of a circle common to the sphere and to the second surface; the point n' in which the lines d'c”, dd" meet, being in the interior of the circle c'd'o"d", it follows that the perpendicular to the vertical plane through this point, cuts the spherical surface in two points, belonging to the two circles in question, and which therefore lie in the common intersection of the
given surfaces; these points obviously have only one common vertical projection. To determine their horizontal ones, N, N, the horizontal projection of the circle, corresponding to c'c”, must be drawn, and n'n drawn perpendicular to xy.
By repeating this construction with other circles described from a', as many points in the projections of the intersection may be determined as may be deemed requisite. Since the meridians, of which h'é'i' and h' d'i' are the projections, lie in the same plane, the points h' and i' must belong to the vertical projection of the intersection of the two surfaces; the corresponding horizontal projections are situated on the line ab.
The tangent at any point of the intersection of two surfaces has hitherto been determined by means of two tangent planes ; but it has been explained (p. 64), that such a tangent is also determined by the condition of its being perpendicular to the planes passing through the proposed point, and the normals to the two given surfaces ; and it was stated, that this conditiou is principally employed when those surfaces are surfaces of revolution, owing to the facility with which the normals to this class of surfaces are determined. The tangent plane being perpendicular to that of the meridian, through the point of contact, the normal itself must lie in this meridional plane, and is consequently easily determined.
Suppose, for example, that it was the tangent at the point n', n", which was to be drawn, this point lying on the two circles projected in c'c“", d'd"; the two normals at that point must first be obtained. To effect this it must be observed, that the normals drawn to any surface of revolution through the different points of a parallel, must all meet in one point of the axis. If, therefore, the normals c”k' and d'"'ľ, to the vertical projections of the two meridians are
drawn, cutting the axis in k' and l', and n'k', n'l be joined, these lines will be the vertical projections of the two normals sought; a and I are the two horizontal projections corresponding to k' and l', na, nl, are consequently the projections of the normals on that plane. It would next be necessary to construct the traces of the plane passing through these normals, and lastly to draw the perpendicular to this plane from the point nn'.
But it is unnecessary to construct the vertical trace of this plane, since the plane of the axes being parallel to the vertical plané, this trace will be parallel to k'l'. Instead of the horizontal trace, we may employ the line joining the points in which the normals meet any horizontal plane ; that, for example, of which e'é" is the vertical trace. Consequently, by prolonging n" k' and n" l' to their points of intersection r', s', with e'e”, the corresponding points r, s, on the horizontal plane can be deduced on the horizontal projections Na, nl, of the two normals, and rs must be drawn through these points. After having found the lines rs and k'l', to which the traces of the plane of the normals must be parallel, nt, n't' being drawn perpendicular to them from the point n, will be the projections of the tangent at n'n".
In general the projection of the tangent to a curve is a tangent to the projection of that curve; nevertheless there is one exception to this rule, in the case when the tangent is perpendicular to the plane of projection : for then the projection of the tangent is a point. This would happen with the tangents at the points projected in h' and i', for the tangent planes to the two surfaces at these points would evidently be perpendicular to the vertical plane. Hence if the tangent were to be determined by the intersection of the tangent planes, the construction would fail in this case as far as the vertical projection of that tangent is concerned,
and recourse must then be had to the normals as above described.
To determine the intersection of two surfaces of revolution,
the axes of which lie in different planes.
54. The horizontal plane must be again assumed perpendicular to the axis of the first surface, and the vertical plane parallel to the axes of both; the data will present the same