The I. C. of the relative motion of two pieces is that pair of coincident points (one on each body, extended if necessary) having the same velocity, for their relative velocity is then zero; each body thus turns about the I. C. relatively to the other. Let (12) and (13) be known in Fig. 12. If (23) is at some such point as C the velocity of C, considered as being part of 2, is normal to (12) C; when it belongs to 3 its velocity is normal to (13) C. These velocities have different directions and cannot be vectori ally equal unless they are collinear, whence (23), the correct position of C, must lie on (12) (13). The utility of the theorem of three centres is illustrated in Fig. 13; the two crosses indicate fixed centres. By the theorem, (13) is collinear with (12) and (23), and also with (15) and (53) ; therefore it lies at the intersec tion of two lines through these pairs. The most detailed discussion of the theorem is given by Klein, 'Kinematics of Machinery' (1917).
Kinematically Determinate Mechanisms.— Any system of interconnected bodies is kine matically determinate when the velocity of one point determines the relative velocity of every other point. When the mechanism consists of P pins interconnected by 1 links the criterion of determinateness is found as follows: It takes 3 links to connect 3 of the pins into a rigid triangle; each remaining pin requires 2 links to fasten it rigidly to the frame already formed,
.•. 3 + 2 (p — 3) =2p — 3 links will connect p pins into a rigid coplanar net work. Each link represents a constraint. If one link is removed one degree of freedom will be introduced and the frame will be uniquely deformable. Hence 1=2p-4 is the relation between the number of links and pins in a kinematically determinate mechanism.
The criterion requires interpretation in special cases.
(i). Fixed pins are those fastened to some frame of reference. Two such pins are equiva lent to one link; for f fixed pins there are virtually 2f links. If f of the p pins are fixed and there are 1 actual links 1+21-3=2p-4.
(ii). One link having 3 pins on it counts as 3 links since it may be regarded as a collapsed triangle; a link with m pins counts as 2m 3 links.
(iii). A crosshead or sleeve is equivalent to a link of zero length • it must theiefore be counted as 2 pins and 1 link.
(iv). A point at which there is line contact, as in cams and gears, counts as 2 pins and 1 link.
locus in space of the in stantaneous centre is called the space centrode; the locus in the body is the body centrode. The term centrode is due to Clifford, 'Elements of (1878) ; the idea, however, is found in Poinsot, nouvelle de la rotation' (1851) and Reuleaux. The latter used it in place of the foregoing methods. In Fig. 14