Fig. 607 - Plan and Elevation of Transition Piece - First Case.

Fig. 608. - Diagrams of Triangles Based upon the Solid Lines of the Plan, Fig. 607.

611 have not been given. While in reality they would differ somewhat from those shown in Figs. 608 to 610, they would have the same general appearance, and in method of construction would be exactly the same, and therefore have not been considered necessary to the study of the problem.)

N P O Q represents a plan of the shape described, above which A B D C shows an elevation of the front of the same, or as Been when looking toward Q, while to the left is shown a view obtained by looking toward the side N. in which E G corresponds to P X of the plan and F H to Q1 Q.

Divide one-half of the plan of the top or round portion of the article into any convenient number of equal spaces, in this case 13. Since by the conditions of the problem one-half of the round end corresponds to the semicircular end of the oblong part, divide the semicircle J N L into the same number of equal parts.

Fig. 609. - Diargrams of Triangles Based upon the Dotted Lines of the Plan, Fig. 607.

Then connect points in the two lines of the plan of the same numerals. For example, L with 1, 2 with 2, 3 with .3, etc. In like manner connect the points in the end of the oblong portion with points of the next higher number in the round end, as shown, as, for example, 1 with 2, 2 with 3, 3 with 4, etc. Upon all of these lines drawn in the plan it will be necessary to construct sections or triangles in which these lines form the bases and in which the vertical hight of the article W V is the altitude. The various hypothenuses thus obtained will then represent the true distances across the finished article upon the lines indicated in the plan. The triangles corresponding to the solid lines of the plan are shown in Fig. 608, while those corresponding to the dotted lines of the plan are shown in Fig. 609. En order to avoid confusion each of these sets has been divided into two groups, as shown, and are constructed as follows: Lay off at any convenient place the line A B (Fig. 808), equal to V W of Fig. 607. From the point B, and at right angles to A B, draw the line B C and upon it set off the lengths of the several solid lines connecting the two outlines in the plan. Thus make B 1 equal to the distance 1 1 or J P of the phut. B 2 is equal to the distance 2 2 of the plan, and B 3 is equal to the distance 3 3 of the plan. etc. As already explained, D E is a duplicate of A B, and E F is drawn at right angles. On E F the spaces B 10, E 11, E 12 and E 13 are set off, being equal respectively to 10 10, 11 11, etc., of the plan. From the points thus established in the base lines B C and E F draw lines to the apices A and D, thus completing the triangles. Then the hypothenuses A 1, A 2, A 3, etc., D 13, D 12, etc.,.correspond to the width of the pattern measured between points indicated by like figures in the plan.

In the same general manner construct the triangles shown in Fig. 609, which correspond to sections on the dotted lines across the plan. G H and L M of Fig. 609 correspond to the hight V W of the elevation. H K and M N are drawn at right angles to the perpendiculars, and on these base lines spaces are set off, measuring from H and M respectively, corresponding to the length of the dotted lines across the plan. Thus H 1 corresponds to 1 2 of the plan, and M. 12 corresponds to 12 13 of the plan. From the points thus established in the base line lines are drawn to the apices G and L, thus completing the triangles. These hypothenuses are equal to the width of the pattern measured between points connected by the dotted lines in the plan. By the conditions of the problem, inasmuch as there are straight portions in the oblong end, there will be portions of the pattern that will correspond to triangles the liases of which are equivalent to the length of the straight portion in the plan and the hights of which are equal respectively to the distances E G and F H of the side view.

Fig. 610. - Pattern of Transition Piece Shown in Fig. 607.

Therefore, to describe the pattern proceed as follows: At any convenient place, as shown by A B in Fig. 610, draw a line equal to the width of the pattern at a point corresponding to Q1 Q in the plan. This would be the same as F H of Fig. 607 or 611. To complete the triangular portion referred to set off from B the distance B C equivalent to Q L of Fig. 607, thus obtaining the point C. The dotted line A C in the pattern is drawn to show the portion obtained by this means. From C as center, with the space 13 12 of the plan of the oblong end as radius, describe a small arc. as shown to the left. Then from A as center, with radius L 12 of Fig. 609, corresponding to the width of the pattern measured on the dotted line 13 12 of the plan, de-rscribe another arc intersecting the one just drawn, thus establishing the point 12 in the lower edge of the pattern. From 12 as center, with D 12 of Fig. 608 as radius, being the width of the pattern on the line 12 12 of the plan, describe a short arc, as shown at 12 in the upper line in the pattern. Intersect this with another arc drawn from A as center, with 13 12 of the plan of the round end as radius, thus locating the point 12 in the upper line of the plan. Proceed in this manner, using in the order described the stretchout of the semicircular end of the oblong section, the hypothenuses of the triangles corresponding to the dotted lines in the plan, the hypothenuses of the triangles corresponding to the solid lines in the plan and the stretchout of the circular end, reaching finally the points D and E of the pattern, representing one side of the remaining triangular section to be added. From E as center, with K J of the plan as radius, describe an arc. From D as center, with radius equal to D E of the pattern, strike a second arc intersecting the one just drawn, thus locating the point F. Connect F and E and also F and D, thus completing the pattern of the part corresponding to K J.P of the plan. The dotted line DG drawn across the pattern corresponds to the line X P of the plan, and D A B G will be one half of the finished pattern.

Fig. 611. - Plan and Elevation of Transition Piece - Second Case.