In Fig. 486 is shown, by means of elevation and plan, the general requirements of the problem. A B represents the pitch of the roof, G H K I represents the pipe passing through it, and C D F E the required flange fitting around the pipe at the line C D and against the roof at the line E F. The flange, as thus drawn, becomes a portion of the envelope of a right cone.

At any convenient distance below the elevation assume a horizontal line as a base of the cone upon which to measure its diameter, and continue the sides downward till they intersect this base line, all as shown at L M. Also continue the sides upward till they intersect at W, the apex. Below the elevation is shown a plan, and similar points in both views are connected by the lines of projection. S T represents the pipe and N O the flange. While the pipe is made to pass through the center of the cone, as may be seen by examining the base line L M in the elevation, and also

P R of the plan, it does not pass through the center of the oblique cut E F in the elevation, or, what is the same, N O of the plan.

For the pattern of the flange proceed as shown in Fig. 487, which in the lettering of its parts is made to correspond with Fig. 486, just described. Divide the half plan P X R into any convenient number of parts - in this case twelve - and from each of the points thus established erect perpendiculars to the base of the cone, obtaining the points 11, 21. 31, etc. From these points draw lines to the apex of the cone W, cutting the oblique line E F and the top of the flange CD, as shown. Inasmuch as C D cuts the cone at right angles to its axis, the line in the pattern corresponding to it will be an arc of a circle; but with E F. which cuts the cone obliquely to its axis, the case is different, each point in it being at a different distance from the apex. Accordingly, the several points in E F, obtained by the linos from the plan drawn to the apex W. must be transferred to one of the Bides of the cone, where their distances from W can be accurately measured. Therefore from the points 03 13 23 33 in E F, draw lines at right angles to the axis of the cone W X, cutting the side W M. as shown. With W as center, and with W M as radius, strike the arc P1 R1 indefinitely, and, with the same center and with W D as radius, strike the are C1 D1 indefinitely, which will form the boundary of the pattern at the top. At any convenient distance from M draw W P1, a portion of the length of which will form the boundary of one end of the pattern. On P1 R1, commencing with P1, set off spaces equal in length and the same in number as the divisions in the plan P X R, all as shown by 01 12 22, 32, etc. From these points draw lines to the center W,as shown. With one point of the dividers set, at W and the other brought successively to the points obtained in W M by the horizontal lines drawn from E F, cut the corresponding lines in the stretchout of the pattern, as indicated by the curved dotted lines. A line traced through these points, as E1 F1, will represent the lower side of the pattern. As but one-half of the plan has been used in laying out the stretchout, the pattern C1 E1 F1 D' thus obtained is but one-half of the piece required It can be doubled so that the seam can be made to come through the short side at C E, or through the long side at D P, at pleasure.

Fig. 486.-pian and Elevation.

Fig. 487. - Pattern.

A Conical Flange to Fit Around a Pipe and Against a Roof of One Inclination.