This shape is very frequently used in pans and plates, and therefore in Fig. 473 is shown an elevation and plan of what is familiarly termed an oval flaring pan. Let that part of the plan lying between H and L be an arc whose center is at U. and lot those portions between V and H and L and W be arcs whose centers are, respectively, R and S. A C D B represents an elevation of the vessel, and is so connected with the plan as to show the relationship of corresponding points.

The first step is to construct a diagram, shown in Fig. 474, by means of which the lengths of the radii to be used in describing the pattern are to be obtained. Draw the horizontal line H U indefinitely, and at right angles to it draw H A, indefinitely also. Make H U equal to H U of the plan, Fig. 471. Make H C equal to the vertical hight of the vessel, as shown in the elevation by D X. Draw the line C G parallel to H U, making G G in length equal to U N of the plan. Through the points U and G thus established draw the line U G, which continue until it meets H A in the point A. Then A U will be the radius by which to describe that portion of the pattern which is included between the points H and L of the plan. With A U as radius, and from any convenient point as center - as A, Fig. 475 - draw the arc H L, which in length make equal to H L of the plan, Fig. 473, as shown by the points 1, 2, 3, etc. From the same center, and with the radius A G of Fig. 474, describe the parallel arc N O From the points H and L of the arc first drawn draw lines to A, thus intercepting the arc N O and determining its length.

Fig. 473. - Plan and Elevation.

Fig. 474. - Diagram of Radii.

Fig. 475. - Pattern of One Half.

The Envelope of the Frustum of a Cone, the Base of Which is an Elliptical Figure.

In the diagram, Fig. 474, set off from H, on the line H U, the distance R H, making it equal to R H of the plan, Fig. 473. Also, upon the line C G, from the point C, set off C I, equal to R N of the plan. Then, through the points R and I thus established, draw the line R B, which produce until it intersects A H. Then R B will be the radius for those portions of the pattern lying between V and H and L and W of the plan. From the point H, on the line H A, Fig. 475, set off the distance H B, equal to R B of Fig. 474. Then, with B as center, describe the arc E H, and from a corresponding center, C, at the opposite end on pattern, describe the arc L K. 'From the same centers, with B I as radius, describe the arcs N M and O P, all as shown. Make H E and L K in length equal to H E and L K of the plan. From E and K, respectively, draw lines to the centers B and C, intercepting the arcs N M and O P in the points M and P. Then E K P M will be one-half of the complete patterns of the vessel.