It is probable that of all the articles that are manufactured out of sheet or plate metals, the larger proportion are conical or circular equal-tapering in shape. It is, therefore, essential that a careful study should be made of the various methods that can be used to obtain the patterns for this class of article.

The simplest form of a conical-shaped object is that of a cap, as shown in Fig. 80. This is, of course, a complete cone, and in this shape is applied in the formation of ventilator and stove-pipe caps, pan-lids, pointers, strainers, candle-extinguishers, etc. The pattern for a conical surface is perhaps one of the easiest patterns that can be developed. Imagine a cone to roll on a flat surface as in Fig. 81, and that as it rolls along, the base of cone marks the outline as shown. Now if the joint line of the cone first of all lies on the line A C, and the cone is then rolled around until the joint comes on the flat surface again, say on the line B C. then it is evident that the whole of the curved surface of the "cone will have been in contact with the flat surface, and the sector of circle so marked out will be the development of the cone surface. The radius of this sector of a circle will, of course, be equal to the slant height of the cone, and the length of the arc A D B will be of the same length as the circumference of the base of cone. Thus, in Fig. 82, suppose the diameter of base of cap is 3 in., and slant height 2 in., then it is evident that the radius for describing the pattern will be 2 in. The length of the arc can be marked off in two or three different ways, as will now be shown. On the base line of half-elevation of cone construct a quarter-circle, and divide it into three equal parts, and carefully measure the lengths of one of these parts, and set it along the pattern curve twelve times. Join the points so found to the centre, and allow laps for grooving, riveting, or soldering as shown. Care must be taken that the lap lines are parallel to the end lines of net pattern. Fig. 80. Fig. 81.

The length of curve on pattern can be quite easily calculated from the following rule: - "Multiply diameter of cone base by 31/7" Thus, in the above example, the length of curve will be: Fig. 82.

3 x 3 1/7 = 9 3/7 in. This length is best set along the curve by a steel tape measure, piece of thin wire, or strip of sheet metal.

Sometimes it is convenient to calculate the length of arc on the piece cut out, and this can be done by either of the following rules: - Fig. 83.

(1) Deduct the diameter of base of cone from twice the slant height, and multiply the remainder by 3 1/7

2) "Multiply the difference of the diameters of pattern and base circles by 3 1/7."

By the use of either of the above rules it will be seen that the length of arc of the sector to be cut out in Fig. 82 will be -

(4 - 3) x 3 1/7 = 1 x 3 1/7 = 3 1/7 in.

The cut circle (Fig. 83) shows the pattern for a conical cap the vertical height of which is 9 in. and diameter 3 ft. The slant height of cone or the radius pf pattern can be calculated by bringing in the property of the right-angle triangle, previously mentioned, thus-

Slant height = = 20 1/8 in.

Having marked out the pattern circle to this radius, the length of arc to set along circumference to cut piece out can be calculated thus-

(40¼ - 36) X 3 1/7 = 4¼ x 3 1/7 = 13 5/14 in.

The end lines on the pattern for a conical cap may also be set out by the use of degrees. The following rule will give the angle that the end lines make with each other:-"Multiply 360 by the radius of the base, and divide by the slant height." Thus in Fig. 82 it will be seen that the angle -

= 360 x radius of base / slant height

= 360 X 1½ / 2 = 270°.

Sometimes it is more convenient to mark the angle on the piece that is to be cut out, and the degrees for this can be calculated by aid of the following rule:- "Deduct the radius of base from slant height, and multiply the remainder by 360 and divide by the slant height." Thus in Fig. 82 this angle will be-

360 X slant height - radius of base / slant height

= 360 x (2 - 1½) / 2 = 360 / 4 = 90o

In Fig. 83 the number of degrees in the piece to be cut out -

= 360 * 20 1/8 - 18 / 20 1/8 = 38o

The above examples are given to show the application of this particular method. The rules can be applied in all cases, and will usually give more accurate results than by measuring along the circumference of pattern circle.