the projections a and a have a connexion which must be carefully observed and remembered.

Let a and a' be the original projections of A, before the motion of the plane xyu: we have seen that the perpendiculars from a' and a, upon xy, meet in the same point p. Now, while ayu revolves round xy, a'p is still to xy; and when xyu coincides with xyu, a ̋p, which is the position of a'p, is in the same plane with ap, and to the same line xy, and at the same point p in xy: :.ap and a'p must be in one and the same straight line; whence we see that the straight line joining the horizontal and vertical projections of the same point a is perpendicular to the ground line xy; and that the part a ̋p gives the elevation of the point a above the horizontal plane, and that ap is the distance of it from the vertical plane.

12. We now proceed to give problems relative to the straight line and the plane. The drawing that contains the construction of the problem is called the Draught.

The given quantities and the results of the problems are generally expressed by continuous lines, and the lines of construction by dotted lines.

And that the explanations of the drawings may be readily understood, a uniformity in the notation is very necessary. We therefore indicate points in space by capital letters, A, B, C, &c., but these rarely appear in the drawings. The small letters abc, &c., belong to the projections in the horizontal plane; the accentuated letters a'b'c', &c., indicate projections in the vertical plane.

The ground line is always designed by xy, and the Greek letters a, ß, y, &c., generally indicate points situated in that line.

The following abbreviations are also used :-The point [a, a'], designates the original point a, which has a and a'

for its projections; the line [ab, a'b'], that line which has for its projections ab and a'b'; and the plane aaa', that of which the traces are a a and a a'.

And when a point, a line, or a plane, is said to be known, it is meant that the projections of these are known. And conversely, when a point, a line, or a plane, is to be determined, it is sufficient to find the projections of the point, those of the line, or the traces of the plane.

PROBLEM 1.

Given the projections of a line to find its traces, or the points where the line meets the planes of projection.

13. The straight line in space is the intersection of its projecting planes; and the point of intersection of the horizontal traces of these planes is evidently in both planes, and thus is a point of the line required: similarly, the intersection of the vertical traces of the two projecting planes is the vertical trace of the line.

Let ab and a'b' be the two given projections, meeting the ground line in a and b A vertical plane through ab will have ab for its horizontal, and vo', xy at , for its vertical, trace.

A plane through a'b' to the vertical plane will be the other projecting plane of the line, and a'b'

will be its vertical, and b′b + to xy in the horizontal plane its horizontal trace.

The point b, where the horizontal traces of the projecting planes intersect, and e', where the vertical traces intersect, are two points of the line in space, and, being also in the co-ordinate planes, are the traces required.

Hence, we obtain this practical rule to find the vertical trace:-Produce the horizontal projection of the line to meet the ground line; from v, the point of intersection, draw vvxy, meeting the vertical projection a'b' in v':

is the vertical trace; and, to find the horizontal trace, draw from b', where a'b' meets xy, b'bxy, to meet the horizontal projection ab in b: b is the horizontal trace.

14. Hence, conversely, given the traces of a line, its horizontal and vertical projections, may be found.

For, given and 6, we have merely to draw 'e and bb' to the ground line xy: then join vb and v'b': these lines produced will be the projections required.

15. If we change the projections ab and a'b', the traces v' and can take a variety of different positions.

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In fig. 1, the horizontal trace is in front of the ground line, and the vertical trace above it.

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In fig. 2, the horizontal trace is still before the ground line, but the vertical trace is in the lower part of the vertical plane.

In fig. 3, the horizontal trace is behind the ground line, and the vertical trace above it.

In fig. 4, the horizontal trace is behind the ground line, and the vertical trace is in the lower part of the vertical plane.

PROBLEM 2.

Given the traces of two planes, to find the projections of the common section of the planes.

16. Let m be the point of intersection of the two horizontal traces ap and bq of the planes, n' that of the vertical traces, a'p and b'q.

These points are common to the two planes, and the line which joins them in space is evidently the intersection

of the planes. The point m being in the horizontal plane is the horizontal trace of the line; and if mm' be drawn to xy, m' will be the vertical projection of m. In the same way, n' is the vertical trace of the line, and n the horizontal projection of n'. If, therefore, we draw the lines mn and m'n', these are the projections required.

The following cases require attention:

1o. Let one of the planes have one of its traces, the horizontal trace, for instance, perpendicular to the ground line; this plane is perpendicular to the vertical plane, and its vertical trace is the vertical projection of its intersection with the other given plane. The rest of the construction is the same as above.

2°. Suppose two of the traces, viz., the horizontal traces to be parallel; the intersection of the two planes will be

parallel to these traces, and the horizontal projection mn, of the intersection will be also parallel to them, and its vertical projection m'n' parallel to the ground line.

3o. Let the two planes meet the ground line at the same point; and let the two planes be apa' and bpb'. The general construction will not apply to this case. The planes must be cut by another plane, and the projections of the two lines of intersection found. The points where these projections meet will be the projections of a point common to the two planes; and as the point p, where these planes meet the ground line, is also common to them, it follows that the projections of their intersection are completely determined.

In general, the third plane of projection is taken perpendicular to the ground line, the traces of the given planes upon it are determined, and thus the construction is referred to the general case. This solution is that given in the figure.

The traces of the third plane are aa, aa', both to xy: they meet the traces of the plane ap'a in a and a′; and, if we conceive the vertical plane of projection to be

placed, for an instant, in its true position, the line joining a and a' will be the trace of the plane apa' upon the third