points instead of at two, while the arm of the moment of resist- ance is doubled, so that the resistance of each single key is only one-fourth as much as in the former case; the placing of the keys near the corners of each side of the shaft is much better also than placing them in the centre.
A comparison of Figures 53 and 62 shows at once how it is in
Fig. 85.
Fig. 86.
general so much more difficult to restrain turning than sliding, namely, that while one point of restraint can prevent sliding throughout a field of 180, it can do nothing more to prevent turning than to divide the whole field into two parts, from points in one of which right-handed and in the other left-handed turning can take place.
§20.
Simultaneous Restraint of Sliding and Turning.
In proceeding now to apply the results found above to cases where sliding and turning take place simultaneously, we may first state the following propositions relating to plane figures:
(1) Neither the sliding nor the turning of a plane figure in a plane can be prevented by two points of restraint.
(2) By three suitably placed points of restraint (a) sliding can be prevented, but not turning at the same time; and (5) turning can be prevented, but not sliding at the same time.
