In Fig. 497, let H F G A be the shape of the article as seen in side elevation. The plan is shown by I L N O. In order to indicate the principle involved in the development of this shape, it will be necessary first to analyze the figure and .ascertain the shape of the solid of which this frustum is a part. Since by the conditions of the problem the base is drawn from centers and the sides have equal flare, it follows that each arc used in the plan of the base is a part of the base of a complete cone whose diameter can be found by completing the circle and whose altitude can be found by continuing the slant of its sides till they meet at the apex, all of which can be seen by an inspection of the engraving. Thus those parts of the figure shown in plan by K U T M and R U T P may be considered as segments cut from a right cone, the radius of whose base is either O K or L R, and whose apex E is to be ascertained by continuing the slant of the side L2 C1 till it meets a vertical line erected from O1 of the plan, which is the center of the arc of the base, all as shown in the end view. Also those parts of the plan shown by KUR and M T P are segments of a right cone whose radius is U I or T N and whose altitude is found, as in the previous case, by continuing the slant of its side G,A (which is parallel to C1 L2) till it meets a vertical line erected from its center T, as shown in the side view.

To complete the solid, then, of which FGA II is a frustum, it will only be necessary to take such pails of the complete cones just described as are included between the lines of the plan and place them together, each in its proper place upon the plan. The resulting figure would then have the appearance shown by H D CB A when seen from the side, and that of O2 C1 L2 when seen from the end. The lines of projection connecting the various views together with the similarity of letters used will show the correspondence of parts. This figure is made use of in the second part of Chapter V (Principles Of Pattern Cutting), Principles of Pattern Cutting, to which the reader is referred for a further explanation of principles.

Divide one-half of the plan into any convenient number of equal parts, as shown by the small figures, and from the points thus established carry lines vertically, cutting the base line H A, and thence carry them toward the apexes of the various cones from the bases of which they are derived. That is, from the points upon the base line H A derived from the arc K M draw lines toward the apex E, and from the points derived from the arc I K carry lines toward the apex D, and in like manner from the points derived from the arc M N carry lines in the direction of the apex B, all of which produce until they cut the top line F G of the article. From the points in F G thus established carry lines to the right, cutting the slant lines of the cones to which they correspond. Thus, from the points occurring between F and f draw lines cutting B A, being the slant of the small cone, as shown by the points immediately below W. In like manner, from the points between g and G carry lines cutting the same line, as shown at G. The slant line of the large cone is shown only in end elevation, and therefore the lines corresponding to the points between f and g must be carried across until they meet the line B1 L2

Commence the pattern by taking any convenient point, as E1, for center, and E1 L2 as radius, and strike the arc L2 S indefinitely. Upon this arc, commencing at any convenient point, as K4, sot off that part of the stretchout of the plan corresponding to the base of the larger cone, as shown by the points 5 to 13 in the plan, and as indicated by corresponding points from K4 to M2 in the are. From the points thus established draw lines indefinitely in the direction of the center E1 as shown. From E1 as center, with radii corresponding to the distance from E1 to points 5 to 13 inclusive, established in the line B1 L2 already described, cut corresponding radial lines just drawn, and through the points of intersection thus established draw a line, all as shown by f1 g1 Next take A B of the side elevation as radius, and, setting one foot of the compasses in the point K4 of the arc, establish the point D1 in the line K4 E1, and in like manner, from M2 with the same radius, establish the point B2 in the line M2 E1. which will be the centers from which to describe those parts of the patterns derived from the smaller cones, Prom D1 and B1 as centers, with radius B A, strike ares from K4 and M2, respectively, as shown by K4 F and M2 N1, upon which set off those parts of the stretchout corresponding to the smaller cones, as shown by the arcs K I and M X of the plan. From the points thus established, being 5 to 1 and 13 to 17, inclusive, draw radial lines to the centers D1 and B2, as shown.

Fig. 497. - Pattern of the Frustum of a Cone, the Base of which is an Approximate Ellipse Struck from Centers, the Upper Plane of the Frustum being Oblique to the Axis.

For that part of the pattern shown from F1 to f1, set the dividers to radii, measuring from B, corresponding to the several points immediately below W of the side elevation, and from D1 as center cut the corresponding radial lines drawn from the are. In like manner, for that part of the pattern shown from G1 to g1, set the dividers to radii measured from B, corresponding to the points in the line B A at G, with which, from B2 as center, strike arcs cutting the corresponding measuring lines, as shown. Then F1 G1 N1 I2 will be one-half of the pattern sought - in other words, the part corresponding to I K L M N of the plan. The whole pattern may be completed by adding to it a duplicate of itself.

PROBLEM 147. The Envelope of a Right Cone, Cut by a Plane Parallel to Its Axis.

Let B A F in Fig. 498 be a right cone, from which a section is to be cut, as shown by the line C D in the elevation. Let G L H K be the plan of the cone in which the line of the cut is shown by D1 D2. For the pattern proceed as follows: Divide that portion of the plan corresponding to the section to be cut off, as shown by D1 G D2, into as many spaces as are necessary to give accuracy to the pattern, and divide the remainder of the plan into spaces convenient for laying off the stretchout. From A as center, with radius A B, describe an arc, as M N, which make equal to the stretchout of the plan G L H K, dividing it into the same spaces as employed in the plan, taking care that its middle portion, D3 D4, is divided to correspond with D1 D2 of the plan. From the points in M N corresponding to that portion of the plan indicated by D1 G D2 - namely, 8 to 16 inclusive - draw lines to the center A, as shown.

From points of the same number in the plan carry lines vertically, cutting the base of the cone, as shown from B to D, and thence continue them toward the apex A, cutting C D, as shown. From the points in C D carry lines at right angles to the axis A E cutting the side of the cone, as shown by the points between C and B. From A as center, with radii corresponding to the distances from A to the several points between C and B, cut lines drawn from points of corresponding number in the stretchout, to A, and through the points of intersection thus obtained trace a line, as shown by D3 C2 D4 Then the space indicated by D3 C2 D4 is the shape to be cut from the envelope M A N of the cone to produce the shape to fit against the line C D in the elevation.

To obtain the pattern of a piece necessary to fill the opening D3 C2 D4 in the envelope, and represented by C D of the elevation, draw any vertical line, through which draw a number of horizontal lines corresponding in hight to the points in C D. The width of the piece upon each of these lines may be found by measuring across the plan upon lines of corresponding number, as 11 13, 10 14, etc. Such a section is properly called a hyperbola (see Def. 113, Chap. I).

Fig. 498. - The Envelope of a Right Cone Gut by a Plane Parallel to its Axis.

PROBLEM 148. The Pattern for a Scale Scoop, Having Both Ends Alike.

In Fig. 499, let A B C D represent the side elevation of a scale scoop,of a style in quite general use, and E F H G a section of the same as it would appear cut upon the line B D, or, what is the same, so far as concerns the development of the patterns, an end elevation of the scoop. The curved line A B C, representing the top of the article, may be drawn at will, being, in this case, a free-hand curve. For the patterns proceed as follows: From the center K, by which the profile of the section or end elevation is drawn, draw a horizontal line, which produce until it meets the center line of the scoop in the point 0. Produce the line of the side D C until it meets the line just drawn in the point X. Then X is the apex and X O the axis of a cone, a portion of the envelope of which each half of the scoop may be supposed to be.

Fig. 499. - The Pattern for a Scale Scoop.

Divide one-half of the profile, as shown in end elevation by E G, into any convenient number of spaces, and from the points thus obtained carry lines horizontally, cutting the line B D, as shown, and thence carry lines to the point X, cutting the top B C, as shown. With X D as radius, and from X as center, describe an arc, as shown by L N, upon which lay off the stretchout of the scoop, as shown in end elevation. From the points in L N thus obtained draw lines to the center X, as shown. From the points in B C drop lines at right angles to O X, cutting the side D C, as shown. With X as center, and radii corresponding to each of the several points between D and C, describe arcs, which produce until they cut radial lines of corresponding numbers drawn from points in the arc L N to the center X. Then a line traced through the points thus obtained, as shown by L M N, will be the profile of the pattern of one-half of the required article.