CLOSED AND HIGHER PAIRS. 593

Strictly speaking the word " lower " should be added in speaking of the first two pairs, but we have seen that in most cases it may be omitted without fear of misunderstanding. For (Jf?) we commonly write ((7), calling the pair often a cylinder-pair ; we can, however, always return to the more general symbol when necessary.

The two pairs (N) and (P) may, as we saw in 3, be treated as special cases of the form (S). If we place the tangent of the pitch angle as an exponent to the symbol S (as was done, for instance, in the case of the plane hyperboloid, p. 254), we have at once (E) = (S ) and (P) = (*). We may also, in places where we require only to distinguish between classes of elements, include all the lower pairs under the symbol ($). We require, in other words, in such a general classification as is required for the synthesis, to consider only the one lower pair

�143.
The Simpler Higher Pairs.

The element C not only forms the closed pair ((7), but is used also in higher pairs, such, for example, as the cylindric friction- wheels, Fig. 363, which would be written (7, C or more generally

���FIG. 363, FIG. 364.

C. C. The class of pairs of which this one forms a special case is the pair of general hyperboloids, H, H. 5 * JSTon- circular cones, K,K, and non-circular cylinders, &,0 t are also special cases of the same class. We had numerous examples of the last-named elements in the higher pairs of 21 et seq., of which Fig. 364 represents a general case.

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