In Fig. 400, let A B C D represent the frustum of a cone, the base of which is to be so out as to make it lit against a roof of two inclinations, as indicated by P R D. Continue the lines of the sides of the cone A B and D C upward until they meet in the point X, which is the apex of the complete cone. Through the apex of the cone draw the line X R, representing the axis of the cone, meeting the ridge of the roof in the point R, and continuing downward in the direction of Y, as shown. At any convenient distance below A D draw a horizontal line, G H, as a base, and immediately below it draw a plan of the same, as shown by E SFY.

Subdivide this plan into any convenient number of spaces, as indicated by the small figures 0, 1, 2, 3, etc. From the points thus established carry lines vertically until they cut the base of the cone G II, and from this line carry them in the direction of the apex X until they cut the line of the given roof. From the points established in the roof line A R draw lines at right angles to the axis of the cone X Y, continuing them until they strike the side of the cone A B. From X as center, with X G as radius, describe the arc GK, upon which lay off a stretchout of the plan.

As the pattern really consists of four equal parts or quarters, the divisions of the plan have been numbered from 0 to 4 and from 4 to 0 alternating, the points 0 representing the lowest and the points 4 the highest points of each quarter. Therefore in numbering the points of the stretchout G K, any point can be assumed as a beginning which is deemed the best place for the joint (in this case 4), numbering from 4 to 0 and reversing each time, all as shown. From these points established in the arc G K draw lines to the apex X. Then, with X as center, and with radii corresponding to the points already established in the side B G of the cone, strike arcs as shown by the dotted lines, cutting measuring lines of corresponding number. Then a line traced through the points of intersection, as shown by O L, will be the shape of the pattern at the bottom and O N M L will constitute the entire pattern of the frustum of a cone adapted to set over the ridge of a roof, as indicated in the elevation.

Fig. 490. - Pattern for the Frustum of a Cone Fitting Against a Surface of Two Inclinations.