The class of subjects treated in this section will include all irregular forms which can be constructed from sheet metal by simple bending or forming, but whose patterns cannot be developed by the regular methods employed in the two previous sections of this chapter. These problems divide themselves naturally, in regard to the arrangement of the triangles used in the development of the patterns, into two classes, viz.: First, those in which the vertices of the triangles used in constructing the envelopes all terminate at a common point or apex, and, second, those in which the relative position of the base and the vertex is reversed in each succeeding triangle, or, in other words, those in which the vertices of alternate triangles point in opposite directions.

In the introduction to Section 2 (page 240), attention is called to the difference between a scalene or oblique cone and a right cone with an oblique base. The scalene cone may be called the type or representative of a large number of forms belonging to the first class above mentioned, since many rounded surfaces entering into the construction of various irregular flaring articles arc portions of the envelope of a scalene cone. The principles involved in this particular class of forms are explained in that part of Chapter V (Principles Of Pattern Cutting) referring to Fig. 271, page 94.

Inasmuch as triangulation is resorted to in all cases where regular methods are not applicable, it is not sur-prising that the forms here treated, especially those included in the second class above referred to, are more varied in character than those of any other class to be met with in pattern cutting. An explanation of methods and principles governing these will be found in the third subdivision of Chapter V (Principles Of Pattern Cutting). beginning on page 86. The last few problems of this class are devoted to the development of the horizontal surfaces of arches in circular walls.

The arrangement of problems in this section will be in accordance with the above classification although no headings will be introduced to distinguish the classes.