Fig. 603. - Plan and Elevations of Flaring Article Round at the Top and Oblong at the Bottom.

In Fig. 603 are shown the side and end elevations and the plan of an article which might form a transition between an oblong pipe below and a round pipe above. According to the conditions, as given in the engraving, the problem is capable of two solutions. Since the upper and lower bases are composed, either wholly or in part, of semicircles lying in parallel planes, those portions of the pattern of the article lying between the semicircles, as P O N O P, must necessarily form parts of the envelope of a scalene cone. Those portions of the pattern may therefore be obtained, if desirable, by the method employed in Problems 168 and 169. The other solution, which is perhaps the more simple, is given in Figs. 604 to 606. An inspection of the plan will show that the article consists of four like quarters, therefore in Fig. 604 is shown an enlarged plan and elevation of one-quarter of the article. Divide P T and O N of the plan each into the same number of equal parts, and connect the points in P T with those in O N, as indicated by the solid lines. Also connect points in the top with those in the bottom, as shown by the dotted lines. These lines represent the bases of right angled triangles, the altitude of which will be equal to the straight hight of the article. For a diagram of the triangles representing the solid lines of the plan, draw any vertical line, as J K in Fig. 605, which make equal in hight to the hight of the article, as shown by the dotted line D F. From K, at right angles to J K, draw K L, upon which set off distances, measuring from K, equal to the lengths of the solid lines drawn across the plan. Thus make K 6 equal to T N and K 7 equal to 2 7 of plan, and so on. Also set off from K the distance H P of plan, as shown by K H. From the points thus established in K L draw lines to J. The hypothenuses thus obtained will give the distances across the finished article, as indicated by the solid lines of the plan.

Fig. 604. - Quarter Plan and Elevation Enlarged, Showing Method of Triangulation.

Fig. 605. - Diagram of Triangles.

The next step will be to construct a diagram of triangles that will give the distances between points in the base and top, as indicated by the dotted lines in plan. This diagram is constructed in a similar manner, as shown at the right in Fig. 605. Draw the right angle V W X, making V W equal to the straight hight of the article, and from W set off on W X the lengths of dotted lines in plan. Thus make W 7 equal to T 7 and W 8 equal to 2 8, and so on. From the points thus established in the base draw lines to V. The hypothenuses of the triangles thus obtained will give the distance from points in the base to points in the top, as indicated by the dotted lines in plan.

To lay out the pattern first draw any line, as T N of pattern, in length equal to J 6 of first diagram of triangles, or, which is the same thing, D E of elevation. From N of pattern strike a short arc with a radius equal to N 7 of the plan, as shown. From T of pattern as center, with radius equal to V 7 of the second set of triangles, intersect this arc, thus establishing the point 7 of pattern. From T, with radius equal to T 2 of the plan, strike a small arc, as shown, and intersect it with another from point 7 of pattern as center, with J 7 of the diagram of triangles as radius, thus establishing the point 2 in pattern.

Fig. 606. - Quarter Pattern of Article Shown in Fig. 603.

Proceed in this manner, using alternately the hypoth-enuses of the triangles in V W X of Fig. 605, the spaces in plan of base O N, the hypothenuses of the triangles in J K L, Fig. 605, and the spaces in the plan of top, P T, in the order named, and as above explained. The resulting points, as indicated by the small figures in the pattern, will be points through which the pattern line will pass. For the pattern of triangle P H O of pattern, with O of pattern as center, and O H of plan as radius, strike a small are in the direction of H. With P of pattern as center, and J H of the diagram of triangles as radius, describe another small arc intersecting the one just struck. Draw O H and H P, thus completing the quarter pattern.

PROBLEM 188. Pattern for a Transition Piece Round at the Top and Oblong at the Bottom. Two Cases.

In Fig. 607 is shown the plan and elevations of a transition piece, constituting the first case, such as is frequently required in furnace work when it is necessary to connect a round pipe with another pipe of equal area but flattened into an oblong shape.

In Fig. 611 are shown the plan and elevations of a transition piece of the second case, answering the same general description as that given above, but differing only in the fact that the circle representing the top in the plan view does not touch the side of the line representing the bottom of the article. In other words, the side F H is slanting instead of vertical, as in the first case, shown in Fig. 607. The principle involved in developing the patterns of the two shapes is exactly the same, consequently the following demonstration will apply equally well to either Pig. 607 or 611, in each of which corresponding points are lettered the same. (It will be noticed that separate diagrams of triangles and a separate pattern corresponding to Fig.