In addition to the example given in Chapter XIV. of a conical hopper fitting squarely on to a cylindrical pipe, we have yet to deal with the more difficult case of a conical pipe fitting obliquely on to a round pipe.

Two applications of this would be in the case of the hopper on the slanting pipe, as in Fig. 113, and also the foot of the coal-scoop as seen in Fig. 116.

The developing of the pattern and the setting out of the hole are illustrated by Fig. 114. Before the pattern can be marked out, a side elevation of the cone and pipe must first be drawn, and on this the elevation of the joint line, or some points on the joint line, shown. To do this, mark down the outline of the cone and pipe on the side elevation, construct a semicircle on the cone base and divide it into six equal parts, after which, run lines square to the base and on to the cone apex as shown. Now to determine points on these lines which shall be on the joint line, it will be necessary to draw the half end-elevation.

Fig. 113.

From each point on the cone base, and also from the apex, run dotted lines along parallel to the centre line of pipe. Produce the end line of pipe both up and down, and using this as a base line to measure from, cut off the dotted lines equal in length to the lines with the same number on the cone-base semicircle. Thus the dotted lines on the half end-elevation numbered 1 1, 2 2, 3 3, etc., will be the same length as the lines 1 1, 2 2, 3 3, etc., on the cone-base semicircle. If the new-found points on the end elevation be joined up, a half-ellipse will be formed as shown. (There is really no need to do this in practice, as the fixing of the points is all that is required.) Join the points to C' the apex of the cone, as seen in the figure, and from the points where the lines cross the semicircle on end of pipe draw lines along parallel to the centre line of pipe. Where these lines intersect the respective lines having the same number on the cone in side elevation will give points on the joint line. Thus, take point 4 on the half-ellipse, follow the line up towards the apex of cone, and we come to point 4' on the semicircle. Now go along the line drawn through this point parallel to the centre line of pipe, until it intersects the line drawn through point 4 on the cone-base; this will give one point on the joint line. In the same manner the position of every other point can be followed out. In Fig. 114 the points are joined with a free curve and an elevation of the joint line thus determined. There is, however, no need to draw in this curve, the fixing of a few points being quite sufficient to enable us to obtain the lengths of lines necessary for the striking out of the pattern. Through each point on the joint line draw lines square to the axis of cone, and thus project the true lengths of lines on to the outside line of cone, as previously explained.

Fig. 114.

The pattern is set out by first marking down the development of the complete cone, dividing up into twelve parts and setting the lengths along from the sides of the cone in the side elevation. Thus lines C 0, C 1, C 2, etc., on the pattern will be the same length as the lines C 0, C 1, C 2, etc., measured from the apex down the sides of the cone on the side elevation.

For a hopper it will be necessary to lay out the shape of the hole in the pipe. The width of the hole can be obtained from the semicircle on the end of pipe, which should be set down by drawing a straight line and marking along it the lengths of arcs 0' 1', 1' 2', etc., as seen on the

Articles By cones cut obliquely 141 hole in Fig. 114. Through each of these points lines square to 3' 3' should be drawn. To obtain the distances to set along these we must again refer to the side elevation. Draw the line b b perpendicular to the centre line of pipe, and use it as a base line from which to measure in obtaining the lengths for the different parts of the hole. The centre line 0 6 of the hole will be made up by marking 0' 0 equal to b 0 and 0' 6 equal b 6. The line 1 l' on the hole will be of the same length as the line measured from point 1 on the joint line up to b b, and so the lengths of all the other construction lines for the hole will be obtained by measuring to the right or left of b b up to the points on the joint line. The points thus determined are carefully connected with an even curve, and the shape of the hole thus obtained.

The reader with little knowledge of geometry will think the above a somewhat complicated case; but with care in following the correspondingly numbered lines, anyone who can use a rule and a pair of compasses ought to be able to set out pattern and hole from the description given. Anyhow, the problem is well worth studying, for in all work where circular and tapered pipes have to be joined together the same principle is involved.