APPENDIX I.
The formulæ in the tables on p. 73 and after can be deduced by two methods,—one that of graphical construction, the other that of least squares. The first method is the more simple and requires but little calculation; though neatness and care are necessary in constructing the diagrams. The second method will be described for the benefit of those who possess the requisite mathematical knowledge. The formulæ used in the preparation of the tables have been generally deduced from the method of least squares, as the results are to a slight, though insignificant, extent more accurate than those of the method of graphical construction. This remark will explain why the figures in some of the formulæ are carried to a greater number of places of decimals than could be obtained by the other method.
We shall confine the numerical examples to [Tables III]. and [IV]., and show how the formulæ of these tables have been deduced by the two different methods.
Tables [V]., [XIV]., [XVI]., [XXI]., are to be found in the same manner as [Table III].; and [Tables VI]., [IX]., [X]., [XI]., [XV]., [XVII]., [XVIII]., [XIX]., [XX]., [XXI]., [XXII]., in the same manner as [Table IV].
THE METHOD OF GRAPHICAL CONSTRUCTION.
[Table III].
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.
