130. Having ascertained the values of the force of friction for eight
different weights, we proceed to inquire into the laws which may be
founded on our results. It is evident that the friction increases with
the load, of which it is always greater than a fourth, and less than
a third. It is natural to surmise that the friction (_F_) is really a
constant fraction of the load (_R_)—in other words, that _F_ = _kR_,
where _k_ is a constant number.
131. To test this supposition we must try to determine _k_; it may be
ascertained by dividing any value of _F_ by the corresponding value of
_R_. If this be done, we shall find that each of the experiments yields
a different quotient; the first gives 0·336, and the last 0·262, while
the other experiments give results between these extreme values. These
numbers are tolerably close together, but there is still sufficient
discrepancy to show that it is not strictly true to assert that the
friction is proportional to the load.
132. But the law that the _friction varies proportionally to the
pressure_ is so approximately true as to be sufficient for most
practical purposes, and the question then arises, which of the
different values of _k_ shall we adopt? By a method which is described
in the Appendix we can determine a value for _k_ which, while it does
not represent any one of the experiments precisely, yet represents
them collectively better than it is possible for any other value to
do. The number thus found is 0·27. It is intermediate between the two
values already stated to be extreme. The character of this result is
determined by an inspection of Table III.
The fourth column of this table has been calculated from the formula
_F_ = 0·27 _R_. Thus, for example, in experiment 5, the friction of a
load of 70 lbs. is 19·4 lbs., and the product of 70 and 0·27 is 18·9,
which is 0·5 lb. less than the true amount. In the last column of this
table the discrepancies between the observed and the calculated values
are recorded, for facility of comparison. It will be observed that the
greatest difference is under 1 lb.