135. In making one of our measurements with care, it is unusual to have
an error of more than a few tenths of 1 lb. and it is hardly possible
that any of the _mean frictions_ we have found should be in error to
so great an extent as 0·5 lb. But with the value of the coefficient
of friction which is used in Table III., the discrepancies amount
sometimes to 0·9 lbs. With any other numerical coefficient than 0·27,
the discrepancies would have been even still more serious. As these are
too great to be attributed to errors of experiment, we have to infer
that the law of friction which has been assumed cannot be strictly
true. The signs of the discrepancies indicate that the law gives
frictions which for small loads are too small, and for large loads are
too great.
136. We are therefore led to inquire whether some other relation
between _F_ and _R_ may not represent the experiments with greater
fidelity than the common law of friction. If we diminished the
coefficient by a small amount, and then added a constant quantity to
the product of the coefficient and the load, the effect of this change
would be that for small loads the calculated values would be increased,
while for large loads they would be diminished. This is the kind of
change which we have indicated to be necessary for reconciliation
between the observed and calculated values.
137. We therefore infer that a relation of the form _F_ = _x_ + _y R_
will probably express a more correct law, provided we can find _x_ and
_y_. One equation between _x_ and _y_ is obtained by introducing any
value of _R_ with the corresponding value of _F_, and a second equation
can be found by taking any other similar pair. From these two equations
the values of _x_ and of _y_ may be deduced by elementary algebra, but
the best formula will be obtained by combining together all the pairs
of corresponding values. For this reason the method described in the
Appendix must be used, which, as it is founded on all the experiments,
must give a thoroughly representative result. The formula thus
determined, is
_F_ = 1·44 + 0·252 _R_.
This formula is compared with the experiments in Table IV.