There is, however, still another difficulty. Although the lower end of the pipe, _o o_, fig. 40, can, with a ball-joint above, be moved in any direction horizontally, yet if the pipe is too long or too short it is obvious such a joint will not permit it to be moved up or down. A joint with a flat surface, like that shown in fig. 87, would, however, permit such motion in the pipe without leaking. If, for example, the steam-pipe were ¹⁄₈ of an inch too short, it might be drawn down that distance, and if the upper joint were then screwed up it would still be steam-tight. In order, then, to get both vertical and lateral flexibility in the joints of the steam-pipes, a ring, _a b_, fig. 89, is interposed between the pipes. One side of this ring is spherical and the other flat, so that the pipes can move either around the spherical part or slip up or down or sideways on the flat surface of the ring. In this way the pipes are flexible and adjustable in every direction, and for all kinds of motion caused by expansion, or which may be needed when the parts are put together. Sometimes the joints at one end only of the steam-pipes are made in this way, and the other is connected with a simple ball-joint. [Illustration: Fig. 89. Scale 1¹⁄₂ in. = 1 foot.] In designing these joints their form should be drawn with a radius, _c d_, fig. 88, from one centre, _c_, so that the surface of the joint will form a part of a sphere. If they are drawn from two centres, as is sometimes done, it is obvious that the surface of the joint will not be a part of a sphere, and therefore will not have the requisite flexibility. The surfaces of the joints are carefully turned to the proper form, and then made steam-tight by scraping or grinding them with emery and oil, and the pipes are then fastened together with bolts, _g_, _g_, fig. 89, and flanges, _f_, _f_, cast on the pipes. QUESTION 158. _How are the exhaust pipes constructed?_ [Illustration: Fig. 90. Scale 1¹⁄₂ in. = 1 foot.]