cal constructions which we have given in the text.

Retaining the notation which we have used in pp 21, 22, the effect of the friction in resisting either the ascent or descent of the weight is

(W cos. e P sin. x) tan. X. This must be added to the element W sin. et (W cos. e-P'

W sin.e of the weight, in the direction of the plane, in order to obtain the force which is to be balanced by the element of P' in the direction of the plane, and must be subtracted from it in order to obtain the element of P" in the direction of the plane. Hence we have

sin. x) tan. X=P' cos. x. W sin. e-(W cos. e-P" sin. x) tan. X=P" cos. x.

Multiplying both members of each equation by cos. X, and observing that tan. X cos. X= sin. X, we find—

W (sin. e cos. X+ sin. X cos. e)-P' sin. x sin. X= P' cos. x cos. X
W (sin. e cos. X-sin. X cos. e)+P" sin. x sin. X=P" cos. x cos. X
..W (sin. e cos. X+sin. X cos. e)=P' (cos. x cos. X+sin. x sin. X)
W (sin. e cos. X-sin. X cos. e)=P" (cos. x cos. X-sin. x sin. X)
sin. e cos. Xsin. X cos. e=sin. (e ± X)
cos. e cos. X ± sin. e sin. X=cos. (e + X)
W sin. (e+X)= P' cos. (x X)
W sin. (e-X) = P" cos. (x + X)

But, by trigonometry,

Hence we obtain,

sin. (e + X)

.. P' = W.

P" = W.

'cos. (x+X)

which formulæ are adapted for computation.

Let us examine under what conditions the two limiting values P', P" of the equilibrating power will become equal.

sin. (e + X)

cos. (x — X)

sin. (e+X) cos. (x + X) ..2 sin. (e + X) cos. (x + X)

But by trigonometry,

If this be the case we must have

sin. (e - X)

cos. (x + X)

= sin. (e — X) cos. (x — X)
2 sin. (e-X) cos. (x — X)

2 sin. (e + X) cos. (x + X) = sin. (e + x + 2 X) + sin. (e
2 sin. (e-X) cos. (x − X) = sin. (e + x

x) 2X) + sin. (e − x)

Omitting the common quantity sin. (ex) in these equals, we have,

sin. (e + x + 2 X) sin. (ex 2X)

Hence the angles within the parentheses must be either equal or supplemental. 1st. Suppose them equal,

e + x + 2 X = e + x 2 X
.. X = 0

the case in which there is no friction, and therefore but one value of P.

Hence the angle x, which the direction of the power makes with the plane, is equal to the complement of the elevation e. This is the same result as was obtained in (34.) geometrically.

It is very easy to shew that the geometrical construction in fig. 12, exhibiting the value of P' and P", might be derived from the formulæ for these quantities which we have just found, or, vice versa, that the formulæ may be derived from the construction.

In fig. 12, WG is equal to FI, or to W; the angle W GH is equal to K FI,

or e; and the angle HGO or HG O'is equal to X. Hence WG B' is equal to WGH-HG O′, or (e — X); and W G B is equal to W GH+H GO, or e+X. Also, G W B is equal to G W H BWH, or GWH

Hence,

G WH 90°

=

WGH 90°

GW B 90° (e + x).
GB'B = GW B' + WGB'
= 90° (e + x) + e-X
=90°~ (x+X).

Also GBW

but B'G B

Hence,

GB'W - B' GB;

2 X.

GBW = 90° + (x + X) — 2 X,

or G B W 90° (x − X).

By trigonometry we have

x. But

WB: WG:: sin. WGB: sin. WB G

or P': W :: sin. (e + X): cos. (x - X)

WB: WG :: sin. W GB': sin. W B'G = sin, G B'B
or P": W:: sin. (e

Hence we find,

which are the formulæ already determined analytically, and by reversing this process, the construction may be deduced from the formulæ.

If the power be parallel to the plane x=o, and the formula become

(36.) It is evident that all the preceding reasoning will be applicable, whether the body slide or roll, or be moved on wheels. The only difference will be that the proportion of the friction to the pressure, or the value of ƒ or WA will be different in each case.

CHAPTER VII.-Tables of the Results of Experiments on Friction and Ri gidity of Cordage.

(37.) SINCE no theory of friction and the rigidity of cordage has been yet established on perfectly satisfactory principles, and all our knowledge respecting it must be derived immediately

from experiment, we shall lay before the student some tables containing the results of experiments instituted by Coulomb, and by comparing these results with the principles which have been deduced from them, the degree of validity to be allowed to these principles will be apparent.

The following tables are extracted from Dr. Gregory's Treatise on Mechanics. TABLE I.-Friction of Woods, the Directions of the Fibres being the same, and the pressure being Unity.

Friction of Woods, the Directions of the Fibres being at Right Angles.

TABLE II.-Friction of Rollers of Lignum-vitæ of six inches and two inches diameter. Pressure 1.