A configuration is a finite collection of points, lines and planes with a number of each on each; any one of the three kinds may be empty.

An ordered pair whose second member (or value) is a number, while its first member (or object) may be anything, will be referred to as a quantity. By a consistent class of quantities—briefly, c.c.q.— we mean a class of quantities that does not contain two quantities with equal objects and unequal values. Reviving Newton’s term, we will refer to c.c.q.’s also as fluents. The class of all objects (of all values) of the quantities belonging to a fluent is called the domain (the range) of the fluent.

Since a link in a plane has three degrees of freedom the simplest closed plane mechanism with constrained motion must have four lower pairs, i.e. one more than the number of degrees of freedom. If the lower pairs are turning pairs, the mechanism is the well-known four-bar linkage. One or two turning pairs may be replaced by sliding pairs without affecting constrained motion, but not less than two turning pairs must remain in a closed link group.

The problem of the “tippe-top” has been discussed in several papers, one by Synge makes the assumption of rolling and discovers instability when the axis is vertical with the peg up, but requires as a necessary condition that the top be not a solid of revolution. Since it appears that as far as is constructionally possible the top is axially symmetric, in which case the rolling motion is always stable, this solution would seem to be unrealistic. In a second group of papers Fokker suggests from observations of peg traces that normal tops roll with no sliding, Braams shows that for the tippe-top sliding will probably take place in the “ rapid precession “, and Hugenholtz deduces general conditions in which such sliding will cause the peg of the top to fall. For the final rise on the peg these authors introduce a “ rolling friction “. Here results substantially in agreement with those of Hugenholtz are found from a simpler result by vector methods. It is further shown that the motion is such that during the final rise sliding must take place until the top is very nearly erect, and that the simple assumption of sliding friction is sufficient to explain the entire motion.