A coal-vase sometimes follows the shape shown in Fig.

174, the top being elliptical and the bottom round. If the top were made oval instead of elliptical, the pattern might be set out by one of the methods shown in Chapter XIX (Irregular Tapering Articles. Oblong Bottom).

In this case, however, the method of triangulation will have to be used. It will at the same time further explain its application to articles of this description that are irregular in shape.

An elevation and a quarter plan are drawn as shown, the quarter ellipse and quarter circle each being divided into three equal parts. The points on the plan are connected up, the lines thus representing the plans of the six triangles that make up a quarter of the complete body surface. To set out the pattern we shall require to get the true lengths of all the lines shown in plan. The first line of the pattern to set down should be 3 D, this being made equal in length to 3' d in the elevation. To obtain the true length of the required second line (3 C), set 3 c along the base lines from 3, thus marking the point c'; then 3' c' will give the length of 3 C. The small arc through C will be drawn by using point 3 as centre and the length 3' c" as radius. The compasses are now set to the length of one of the arcs on the quarter circles, and with D as centre the point C is cut. The line 2 c is now set along the base line from 2', thus fixing c", this latter point being joined to 2". The line 2" c" gives the length of 2 C on the pattern. As before, an arc is now described shown passing through 2, using 2" c" as radius and C as centre, this being cut by using 3 as centre and the length of one of the parts on the quarter ellipse as radius. Thus point 2 on the pattern is determined. The point b' is fixed by making 2' b' equal to 2 b, then 2" V will give the length of 2 B for the pattern. The length 1' b" is made equal to 1 b, line 1" b" then giving the length of 1 B. The line V a' is marked along equal to 1 a, thus giving 1" a' the length required for 1 A. And then the finishing line 0 A is obtained by setting 0 a along the base line from the foot of the centre line to give a", and measuring off 0" a". In this way the twelve triangles that build up the half pattern are set out. The points are joined up, and allowances put on as before. Fig. 173. Fig. 174.

The foot being a frustum of a right circular cone, the pattern will be marked by making the radii t n and t p on the pattern the same length as the letters denote in the elevation.

The methods of jointing can be the same as before, or the bottom can be knocked up on to the body, and the foot slipped over and riveted.

Before passing from the above example in the use of the method of triangulation, it is perhaps as well to point out that the true lengths of lines can be obtained by drawing a pair of lines at right angles, and setting along these the respective distances from plan and elevation; those from the plan being measured along the horizontal, and those from the elevation up the vertical, the slant lines then giving the true lengths for the pattern.