If each link is assumed to form two pairs with two adjacent links, then the relation between the number of pairs ( p ) forming a kinematic chain and the number of links ( l ) may be expressed in the form of an equation :

When the two elements of a pair are connected in such a way that one can only turn or revolve about a fixed axis of another link, the pair is known as turning pair. A shaft with collars at both ends fitted into a circular hole, the crankshaft in a journal bearing in an engine, lathe spindle supported in head stock, cycle wheels turning over their axles etc. are the examples of a turning pair.

When the two elements of a pair are connected in such a way that one element can turn about the other by screw threads, the pair is known as screw pair. The lead screw of a lathe with nut, and bolt with a nut are examples of a screw pair.

A kinematic chain which consists of two turning pairs and two sliding pairs is known as double slider crank chain and elliptical trammels are such pairs.

When four links are joined at the same connection, the joint is called a quaternary joint. It is equivalent to three binary joints.

In general, when n number of links are joined at the same connection, the joint is equivalent to (n – 1) binary joints.

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When the kinematic pairs are coupled in such a way that the last link is joined to the first link to transmit definite motion (i.e. completely or successfully constrained motion), it is called a kinematic chain. In other words, a kinematic chain may be defined as a combination of kinematic pairs, joined in such a way that each link forms a part of two pairs and the relative motion between the links or elements is completely or successfully constrained.

The piston and cylinder in a steam engine form a pair and the motion of the piston is limited to a definite direction (i.e. it will only reciprocate) relative to the cylinder irrespective of the direction of motion of the crank.

When the two elements of a pair have a line or point contact when relative motion takes place and the motion between the two elements is partly turning and partly sliding, and in mesh the gears have a point contact.