In Fig. 433, let G H K I be the elevation of the article, C A E D the plan of the larger end and L M

Fig. 433 - Plan and Elevation

Fig. 434. - Pattern.

The Envelope of the Frustum of a Square Pyramid.

0 N the plan of the smaller end. Produce the hip lines C L, A M, etc., in the plan to the center P.

Construct a diagonal section on the line A P as follows: Erect the perpendicular P F, making it equal to the straight bight of the article, as shown by R K of the elevation. Likewise erect the perpendicular

M B of the same length. Draw F B and A B. Then P A B F is the diagonal section of the article upon the line P A. Produce A B indefinitely in the direction of X, and also produce P F until it meets A B extended in the point X. Then X is the apex of a right cone and X A the side of the same, the base of which, if drawn, would circumscribe the plan CAED. Therefore, from any convenient center, as X1 of Fig. 434, with X A as radius, describe the arc C1 D1 E1 A1 C2, and from the same center, with radius X B, draw the arc L1 N1 O1 M1 L2, both indefinitely. Draw C X', cutting the smaller arc in the point L1. Make the chord C1 D1 equal in length to one side, C D, of the plan, and D1 E1 to another side, D E, of the plan, and so on, until the four sides of the base have been set off. Draw D1 X1, E1 X1, etc., cutting the arc L1 L2 in the points N1, O1, etc. Then D1 N1, E1 O1 and A' M' will represent the lines of the bends in forming up the pattern. Draw the chords L1 N1, N1 O1, etc., thus completing the pattern.

PROBLEM n6.

The Envelope of the Frustum of an Octagonal Pyramid.

Fig. 435. - Elevation.

Fig. 436.-Plan.

Fig. 437.-Pattern.

The Envelope of the Frustum of an Octagonal Pyramid.

Fig. 435 shows the elevation and Fig. 436 the plan of the frustum of an octagonal pyramid. The first step in developing the pattern is to construct a diagonal section, the base of which shall correspond to one of the lines drawn from the center of the plan through one of the angles of the figure, as shown by G. B. Erect the perpendicular G C equal to the straight hight of the frustum, as shown by N M of the elevation, and at b erect a perpendicular, b A, of like length. Draw B A and A C. Then G B A C is a section of the article as it would appear if cut on the line G B. Produce B A indefinitely in the direction of X, and likewise prolong G C until it intersects B A produced in X. Then X is the apex and X B the side of a right cone, the plan of which, if drawn, would circumscribe the base of the frustum. From any convenient center, as X1, Fig. 437, with radius X B, describe an arc indefinitely, as shown by the dotted line E1 E2 of the pattern, and from the same center, with X A for radius, describe the arc e1 e2 of the pattern. Through one extremity of the arc E1 E2 to the center draw a straight line, as shown by E1 X1 cutting the smaller arc in the point e1. Set off on the arc E1 E2 spaces equal to the sides of the plan of the base of the article and connect the points by chords. Thus make E1 P1 of the pattern equal to E P of the plan, and so on. Also from these points in the arc draw lines to the center, cutting the arc e1 e2 as shown. Connect the points thus obtained in this arc by chords, as shown by e1 p1 p1 d1 d1 o1, etc. Then e1 E1 E2 e2 will be the pattern sought.

The Envelope of the Frustum of an Octagonal Pyramid Having Alternate Long and Short Sides.

In Fig. 438, let I M B N O P K L, be the plan of the article of which G H F E is the elevation. The first thing to do in describing the pattern is to construct a section corresponding to a line drawn from the center to one of the angles in the plan, as S B. At S erect the perpendicular S R, in length equal to the straight bight of the article, as shown by C D of the elevation. Upon the point b erect a corresponding perpendicular, as shown by b A. Draw R A and A B. Then B A R S is a section of the article taken upon the line S B.

Fig. 438.-Plan and Elevation.

Fig. 439.-Pattern.

The Envelope of the Frustum of an Octagonal Pyramid Having Alternate Long and Short Sides.

Produce S R and B A until they meet in the point X. Then X is the apex and X B is the side of a cone, the base of which, if drawn, would circumscribe the plan of the article. From any convenient center, as X', Fig. 439, with radius equal to X B, describe an are. as shown by M1 M1. Draw X1 M1 as one side of the pattern. Then, starting from M', set off chords to the arc, as shown by M1 B1, B1 N1, etc., equal to and corresponding with the several sides of the article, as shown by M B, B N, etc., in the plan. From these points. B1, N1, etc., in the arc, draw lines to the center X1.

From X1, with X A as radius, describe an arc cutting these lines, as shown by m1 m2. Connect the points of intersection by straight lines, as shown by m1 bl, b1nl, n1o1, etc. Then m1 m2 M2 M1 will be the pattern sought, and the lines B1 b1 N1 n1, etc., will represent the lines of bends to be made in forming up the article.