A horizontal line APS, shown on a diminished scale in Fig. 103, is
to be neatly drawn upon a piece of cardboard about 14" × 6". A
scale which reads to the hundredth of an inch is to be used in the
construction of the figure. A pocket lens will be found convenient
in reading the small divisions. By means of a pair of compasses and
the scale, points are to be marked upon the line APS, at distances
1"·4, 2"·8, 4"·2, 5"·6, 7"·0, 8"·4, 9"·8, 11"·2 from the origin A.
These distances correspond to the magnitudes of the loads placed upon
the slide on the scale of 0"·1 to 1 lb. Perpendiculars to APS are to
be erected at the points marked, and distances F₁, F₂, F₃, &c. set
off upon these perpendiculars. These distances are to be equal, on
the adopted scale, to the frictions for the corresponding loads. For
example, we see from Table III., Experiment 3, that when the load upon
the slide is 42 lbs., the friction is 12·2 lbs.; hence the point F₃ is
found by measuring a distance 4"·2 from A, and erecting a perpendicular
1"22. Thus, for each of the loads a point is determined. The positions
of these points should be indicated by making each of them the centre
of a small circle 0"·1 diameter. These circles, besides neatly defining
the points, will be useful in a subsequent part of the process.
[Illustration: FIG. 103.]
It will be found that the points F₁, F₂, &c. are very nearly in a
straight line. We assume that, if the apparatus and observations were
perfect, the points would lie exactly in a straight line. The object
of the construction is to determine the straight line, which on the
whole is most close to all the points. If it be true that the friction
is proportional to the pressure, this line should pass through the
origin A, for then the perpendicular which represents the friction is
proportional to the line cut off from A, which represents the load.
It will be found that a line AT can be drawn through the origin A,
so that all the points are in the immediate vicinity of this line,
if not actually upon it. A string of fine black silk about 15" long,
stretched by a bow of wire or whalebone, is a convenient straight-edge
for finding the required line. The circles described about the points
F₁, F₂, &c. will facilitate the placing of the silk line as nearly as
possible through all the points. It will not be found possible to draw
a line through A, which shall intersect all the circles; the best line
passes below but very near to the circles round F₁, F₂, F₃, F₄, touches
the circle about F₅, intersects the circles about F₆ and F₇, and passes
above the circle round F₈. The line should be so placed that its depth
below the point which is most above it, is equal to the height at which
it passes above the point which is most below it.
From A measure AS, a length of 10", and erect the perpendicular ST.
We find by measurement that ST is 2"·7. If, then, we suppose that the
friction for any load is really represented by the distance cut off by
the line AT upon the perpendicular, it follows that
_F_ : _R_ :: 2"·7 : 10".
or _F_ = 0·27 _R_.
This is the formula from which Table III. has been constructed.