PAIR-CLOSURE AT CHANGE-POINTS. 193
in the change-positions by pairs of elements formed so as to correspond to the nature of the motion which is required.
In order to do this we must, as we know from Chapter III, deter- mine the axoids or centroids of the bodies to be paired. These bodies are here, for example, the two cranks a and c, or otherwise the two links b and d ; two opposite links, in other words, of the four-linked chain containing four parallel turning pairs shown in Fig. 153, the centroids of which we have already examined in 8
��FIG. 153.
and 9. In that case the centroids were very complicated figures, here they are made extremely simple by the equality of the opposite links. Eemembering always that the cranks are to revolve in opposite directions, the centroids will be found to have the forms shown in Fig. 154.
For the links a and c, the shorter pair, they are ellipses, ha.ving their foci at the ends 1,2 and 3,4 of the cranks, and their major axes AB and C D equal in length to the links b and d. The instantaneous centre moves backwards and forwards along these links (being always at their intersection). For b and d the cen- troids are hyperbolae, their transverse axes E F and ZTlying on the links themselves, and being equal to a (= c) ; their foci are the points 2,3 and 1,4. The instantaneous centre traverses each branch of the curve to infinity, turning from - co along the other branches.
If it be required to pair two opposite links at their points of change, a higher pair must be employed in each case ; such pairs need not, however, go further than corresponds to the elements of the rolling conics in contact at the change positions. If the links chosen be the two shorter ones, a and c, these are the elements of the ellipses at the extremities of their major axes, viz. A, B> C and D. By putting a pin and a gab * at these points, as shown in
- I use the common technical word for a fork or open eye of this kind.
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