In Fig. 574, F G H J of the plan represents the bottom of the article and A B D E the top. Below the plan is projected a front elevation and at the right a side elevation, like points in all the views being lettered the same. An inspection of the drawing will show that each side of the article consists of a triangular piece whose base is a side of the rectangle and whose vertex lies at a point in the circle of the top, the four vertices marking the division of the circle into quarters, and four quarters of inverted oblique cones whose bases are the quarter circles of the top and whose apices lie at the corners of the rectangle. A comparison between this figure and the one shown in Problem 177 will show that the conditions existing in either one of the corner pieces in this case are exactly the same as in the former problem, but that while in Problem 177 the four corners are alike, in the present instance the four corners are all different, and that therefore the pattern for each corner piece, as well as that for each of the flat sides, must be obtained at a separate operation, all being finally united into one pattern.

Fig. 574. - Elevations and Plan of an Irregular Transition Piece.

Divide the plan of the top A B D E into any convenient number of equal spaces in such a manner that each quarter of the circle shall contain the same number of spaces and from the points of division in each quarter draw lines to the adjacent corner of the rectangle of the base, all as shown in Fig. 575. Thus lines from the points in E D are drawn to H and lines from points in D B are drawn to G, etc.

The next operation will be to construct the four diagrams of triangles (one for each corner piece) shown in Fig. 576, of which these lines are the bases. Accordingly lay off at any convenient place the line H L, Fig. 576, equal to the straight hight of article, as shown by J' X in Fig. 574. From the point H, and at right angles to L H, draw the line H M, and. measuring from H, set off the length of lines in E D

Fig. 575. - Plan of Irregular Transition Piece with Surface Divided into Triangles.

H of the plan, Fig. 575. Thus H 1 is made equal to H E of the plan, H 2 is made equal to the distance H 2 of the plan, and H 3 is equal to distance H 3 of the plan, etc. From the points thus established in M H draw lines to L, as shown. Then the hypoth-enuses L 1, L 2, L 3, etc., will correspond to the width of the pattern, measured between points in E D of top and H in the base.

The triangles for the corner piece DG B are constructed in the same general manner. N G corresponds to the hight J' X of the elevation. O G is drawn at right angles to N G, on O G are set off the lengths of lines in DGB of the plan, and from the points thus obtained lines are drawn to N. Thus G 5 of the diagram is equal to G 5 of the plan, G 6 of the diagram is equal to G 6 of the plan, etc. The triangles in P Q F correspond with the lines in A F B of the plan, as do those in S R J with the lines A J E in the plan. Before commencing to describe the pattern the seam or joint may, for convenience, be located at K E of the plan. The real length of the line K E of the plan is given by H" E" in the side elevation. Fig. 574. or the distance E K can be set off as shown by H K in Fig. 576. The dotted line K L will then be the distance from K in the base to E in the top.

As it is necessary in obtaining the pattern for the entire envelope that the patterns of the parts shall succeed one another in the order in which they occur in the plan, the method of development here adopted is that of constructing separately each small triangle, as in the preceding problem, instead of by means of a number of arcs, as in Problem 177 and others preceding it. To begin, then, with the pattern of the part corresponding to F B G of the plan. The length F G of the pattern, in Fig. 577, is established by the length F G of the plan in Fig. 575. With F of pattern as center, and P 9 of Fig. 576 as radius, describe an are, B, which intersect with one struck from G of the pattern as center, and N 9 of Fig. 576 as radius, thus establishing the point B of the pattern. Then F B G is the pattern for that part of the article shown by F B G of the plan. From G of pattern as center, with radii corresponding to the hypothenuses of the triangles shown in O N G of Fig. .576. strike, the arcs shown. Thus G 8 of the pattern is equal to.N 8. 6 7 of the pattern is equal to N 7, etc. With the dividers set to the same space used in stepping off the plan, with B or 9 of the pattern as center, strike a small arc intersecting arc 8 previously drawn, thus locating the point 8. From 8 as center intersect are 7, and so continue, locating the points 6 and 5. Through the points thus obtained can be traced the line B D. Then G B D is the pattern for that part of the article shown on the plan by G B D.

Fig. 576. - Diagrams of Triangles Obtained from Fig. 575.

Fig. 577. - Pattern of Transition Piece Shown in Fig. 574.

With G of pattern as center, and G H of plan as radius, strike a small arc, H, which intersect with one struck from D of pattern as center and L M of Fig. 576 as radius, thus establishing the point H of pattern. Connect G H and H D, as shown. Then G H D is the pattern for that part of the article shown in plan by G H D. With H of pattern as center, and the hypoth-enuses of triangles in M L H of Fig. 576 as radii, strike arcs, as shown, making H 4, H 3, H 2, H 1 of pattern equal to L 4, L 3. L 2, L 1 of the diagram of triangles,

With the dividers set to the same space as was used in stepping off the plan, and commencing at 5, intersect each succeeding arc from the point obtained in the one before it, as shown by the figures 4, 3, 2, 1. Trace a line through the points thus obtained, and connect E' H. as shown. Then E'D H is the pattern for that part of the article shown on plan by E D H. With H of pattern as center, and H K of plan as radius, describe a small arc, which intersect with one struck from E' of pattern as center, and L K of Fig. 576, or what is the same, E" H" of Fig. 574, as radius, thus establishing the point K' of the pattern. Connect H K' and

K' E', as shown, which gives the pattern for that part of the article shown on plan by H K E.

The radii for striking the arcs in A F B of the pattern are found in O P F of Fig. 576. The length F J of pattern is established by the length F J of the plan. The radii for striking the arcs in A J E of pattern are found in S R J of Fig. 576. J K of the pattern corresponds with J K of the plan, and E K of the pattern corresponds with L K of Fig. 576. Thus E A B D E' of the pattern is the stretchout of E A B D of the plan of the top, as K J F G H K' of the pattern is the stretchout of K J F G H of the plan of the base.