In Fig. 334, A B C D shows the plan of a square pyramid, whose apex, if completed, would be at E. F H I J shows the horizontal section of an elliptical pipe, against the sides of which the sides of the pyramid are required to be fitted. From the side A B (or D C)of the plan is projected a front elevation, in which KLMN shows the broad side of the pipe. To the right another or side elevation is projected, in which the narrow view of the pipe is shown by O P Q B. An inspection of the plan will show at once that two patterns will be necessary, one for A B G S to fit against the broad side of the pipe, and another for B G T C to fit against what might be termed the edge or narrow side of the pipe. To obtain the pattern of the side of the pyramid shown by B G T C (or B E C, if the pyramid were complete), first divide that portion of the profile of the pipe from G to H by any convenient number of points, as shown by the small figures, from which, together with B and E, project lines vertically to the elevation above, cutting that side in profile as shown from E1 to B1. At right angles to E1 B1 carry lines from each of the points indefinitely, as shown. At any convenient distance away, cut these lines by any line, as E2 V3, drawn parallel to E1 B1. Upon each of the lines drawn from the points in E1 B1, set off from E3 V3 the distances upon lines of corresponding number in the plan measured from E V. Thus upon line 5 of the pattern set off either way from its intersection with the line E3 V3 a length equal to the distance of point 5 of the plan from the line E V. Upon line 4 of the pattern set oil-distances equal to that of point 4 from E V of the plan, etc. Also make V3 C3 and V2 B3 equal to V C and V B of the plan. Lines drawn from C3 and B3 toward E3 will meet the points previously set off on line 5 of the pattern, indicated by T1 and G1, and will constitute the sides or hips of the pattern, and a line traced through the points set off on lines 1 and 5 inclusive will give the shape of that portion of the pattern to fit against the pipe.'

Fig. 334. - The Patterns for a Square Pyramid to Fit Against the Sides of an Elliptical Pipe.

An exactly similar course is to be pursued in obtaining the pattern of the side of the pyramid A S G B

(or A E B of the complete pyramid), whose profile is shown by B3 E2 of the side elevation, showing the narrow view of the pipe. The pattern is shown at A4 B4 G2 S2 and the operation is clearly indicated by the lines of projection.

If it is desired to complete the elevations by showing the lines of intersection of the sides of the pipe with the sides of the pyramid shown respectively in each elevation, as from c to b and e to f, it can be accomplished as follows: To obtain the line c b, erect, lines vertically from points (6, 7 and 8 (not shown), passing through the space between c and b in the front elevation. upon each of which set off the hight of each point as measured upon lines of corresponding number from B2 C2 to B2 E2 as shown from R toward e, in the side elevation; then a line traced through the points thus obtained will give the line cb. In the same manner lines from the points 1. 2, 3 and 4 can be carried at right angles to B2C2 into the side elevation, upon which to set off hights of corresponding points as measured from A1 B1 to B1 E1. as shown between a and b in the front elevation; then a line traced through the points will give the line e f.