24 



MECHANICS. 



Fig. 11, 



those lines which are merely introduced 

 to supply the links of the analysis, the 

 two limits of the power may be thus 

 determined. Let WB be the direction 

 of the power. Take WA and WA', (fg. 

 12.) each equal to the friction due to the 

 pressure of the weight upon the plane, 

 the weight being supposed not to be 

 affected by any power. Take A O and 



Fig. 12. 



A'O', each equal to IK, and through O 

 and O' draw lines each inclined to OO' 

 at an angle equal to the complement of 

 X, and produce the direction *bf the 

 power to intersect these lines at B' and 

 B. Then WB' will be the least power 

 which can prevent the descent of the 

 weight, and WB will be the greatest 

 power which can be applied without 

 causing its ascent. All intermediate 

 powers will produce equilibrium. 



The construction which we have 

 given not only exhibits in every case 

 the two limiting values of the equili- 

 brating power, but also shows what the 

 single value of this would be if there 

 were no friction. Let D be the point 

 where AVB'B intersects th'e perpendicu- 

 lar GH from G on OO'; WD is the 

 value of the power which would sustain 



the weight were there no friction. For 

 the element of this, which is in the di- 

 rection of the plane, is WH ; but since 

 O'GO is isosceles, H must be the mid- 

 dle point of OO', but A O' is equal to 

 AO: take AO' from both, and the 

 remainders, AA' and OO', are equal ; 

 and therefore WA, which is half of the 

 one, is equal to HO, which is half of 

 the other ; add to both AH, and WH 

 is equal to AO, which by construction 

 is equal to KI, or the force down the 

 plane. Hence, the element of WH in 

 the direction of the plane would be 

 equal to the force down the plane, and 

 W G is, therefore, the equilibrating 

 power. 



(34.) From considering this construc- 

 tion it appears that if the direction of 

 the power be that of the line WG pass- 

 ing through the intersection of the lines 

 drawn through O' and O, the two limits 

 of the power become the same, the 

 points B' and B coincide, and there is 

 but one power, WG, which will keep the 

 weight in equilibrium; every greater 

 power will cause its ascent, and every 

 lesser one will permit its descent. This 

 may be easily accounted for, and is in 

 fact what might be expected. Since WA 

 = HO and WA is the friction due to the 

 weight, HO is equal to this friction ; 

 and since HGO is the complement of 

 HOG, it is equal to the angle X, and 

 therefore HO is to HG as the friction 

 to the pressure ; but HO is the friction 

 due to the weight, and, therefore, GH is 

 the pressure. But since WG is the 

 power, HG is its element perpendicular 

 to the plane. Hence the part of the 

 power which tends to diminish the 

 pressure is equal to the entire pressure. 

 The pressure being thus destroyed, there 

 is no friction ; and hence it is that the 

 two limits of the power become equal, 

 their difference, which is always twice 

 the effect of the friction, having va- 

 nished. 



Since GH is equal to FK, and WH 

 to IK, and the angles at H and K are 

 right, it follows that WG is equal to 

 FI, and that the angle GWH is equal 

 to FIK, and, therefore, that WG is pa- 

 rallel to IK. Thus it appears that this 

 is the case, in which the direction of 

 the power is vertical, and is, therefore, 

 equal to the w r eight, and sustains it in- 

 dependently of the plane. 



(35.) [The preceding results may 

 very easily be obtained analytically, 

 and the formulae thus found are better 

 fitted for calculation than the geometri- 



