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made to lie when bent, without leaving any hollow spaces. Should this not be possible, then the surface is non-developable. From this definition, it will at once be seen, that all ju>?awe-surfaced solids are developable, while of those having curved surfaces, only the cylinder, and the cone, with their frustums, fall within the same category ; the sphere, with the spheroids, ellipsoids, and many other solids of revolution, having surfaces which will not coincide with a plane when laid out flat, but would tear or crease, being non- developable. The figure of the developed surface of every solid, which when bent will cover it at any and every point, is called the "envelope " of that surface. In the cylinder and cone, as well as in every other solid of revolu- tion, any line drawn on their surfaces, in the same plane as their axes, is called a "meridian." If such a line is straight, the surface is de- velopable, but if curved it is non-developable. A surface which is generated by the motion of a straight line is called a " ruled " surface, and may be either developable or non- developable ; if the latter, it is a " twisted " surface, in that it cannot be laid out on a plane without being torn. A ruled surface may, how- ever, be curved, and developable, and yet form no part of a cylinder or a cone, as will be shown later on. The finding of the developments of plane-surfaced solids involving no difficulty, few problems are necessary in connection with them, as it will be seldom that any will occur in the practice of the student, with which he will not be able successfully to grapple. The first is the simplest possible, and hardly requires demonstration, but as it shows