In Fig. 636, let K L represent the pitch of the roof, A B C D the elevation of the flaring flange, A J D the half plan of the base, and 13 E C the half plan of round top through which the pipe passes.

Fig. 636. - Elevation of a Flaring Flange to Fit Against an Inclined Roof.

It will be seen by comparison that this problem embodies exactly the same principles as do the two immediately preceding, with the slight difference in detail that its short side is not at right angles to either upper or lower base. Also, in this case the bottom of the article appears inclined instead of the top. It will be seen at a glance that if the shape be considered as anything else than a flange against an inclined roof the drawing might be so turned upon the paper as to bring the line K L into a horizontal position, when it would present the same conditions as those of Problems 193 and 194 with the slight difference in detail above alluded to.

The method of triangulation employed in this case is exactly the same as in the problem immediately preceding, and the operation is so clearly indicated by the lines and figures upon the four drawings here given as scarcely to need explanation, if the previous problem has been read. The plans of both top and bottom are divided into the same number of equal parts, and a view of the top as it would appear when viewed at right angles to the base line K L, and as shown by F G H, is projected into the plan of base, as indicated by the lines drawn from B C at right angles to A D.

Fig. 637. - Diagram of Triangles Based upon the Solid Lines of the Plan in Fig. 636.

Fig. 638. - Diagram of Triangles Based upon the Dotted Lines of the Plan in Fig. 636.

Points of like number in the two curves FHG and A J D are joined by solid lines, and the four-sided figures thus obtained are redivided diagonally by dotted lines. These solid and dotted lines become the bases of the several right angled triangles shown in Figs. 637 and 638, whose altitudes are equal to the hights given between the lines B C and F G, and whose hypothenuses give correct distances across the pattern between points indicated by their numbers. The pattern is developed in the usual manner by assuming any straight line, as C D in Fig. 639, equal to C D of Fig. 636, as one end of the pattern, and then adding one triangle after another in their numerical order; using the stretchout of BEC, Fig. 636, to form the upper line of the pattern, the stretchout of A J L to form the lower side of the pattern and the various dotted and solid hypothenuses in Figs. 637 and 638 alternately to measure the distances across the pattern.

Fig. 639. - Pattern of Flaring Flange Shown in Fig. 636.