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 unaltered; the changed arrangement of the ports presented some difficulties, the slide-valve also required to be somewhat dif- ferently arranged for convenience' sake, but in essence they have remained as before.

As another example I may mention the reversing link of Humphry and Tennant, Fig. 47 (or more rightly of Naismyth),* as compared with the older and more common one of Stephenson, Fig. 48. Here inversions of two pairs have taken place. First, Humphry's bar-link A B is an inversion of the slotted link A l B r used by Stephenson, and hence the element paired with the link in the latter case, the slide C^D^ becomes in Fig. 47 a hollow block C D, in which A B can slide. Naismyth has also changed the cross pin F l into a body F F having a cylindric hole, and the piece E v which in Stephenson's has such a hole, into a solid cylinder E E, of such a size as is necessary to allow the link to pass through it in the way shown. Kinematically the pieces C D E and C l D^ E are completely identical, both having for their element-forms a curved sector having a prismatic cross-section, and a cylinder normal to it.

These inversions frequently afford great advantages in construc- tion, and on this account they are matters of considerable import- ance in machine-design. In kinematic science they are examples of the application of a simple general law, which as we have seen affects the simplest element-pairs generally and a priori.

§17. The necessary and sufficient Restraint of Elements.

While in the course of our examination of closed pairs we considered the forms screw, revolute, and prism, and examined the relation between the corresponding solid and hollow pieces, we took no notice of the fact that the mutually enveloping geo- metrical forms were not always equally large or equally extended in the cases we used as illustrations. We found, and find almost always in practice, the nut to be much shorter than its screw,


 * Cf. Practical Mecli. Journ., 1862-3, p. 232.