MECHANICS. 



19 



if then the weight be increased so as to 

 predominate and tend to raise the power, 

 the resisting forces immediately change 

 their direction and oppose the" effect of 

 the weight. Let us suppose that Pis 

 the power which by any machine would 

 equilibrate with W, independently of 

 friction or any resisting force, according 

 to the principles established in Treatise 

 II. If P be increased, it will have a 

 tendency to raise W; but that tendency 

 will be opposed by the resistances. Let 

 the effect of these resistances on P be 

 R, then it will be necessary to increase 

 P by a quantity greater than R, in order 

 that W should be raised. Again, if P 

 be diminished, W would have a tendency 

 to descend, but this tendency is opposed 

 by the resisting forces ; and, in fact, W 

 cannot descend unless P be diminished 

 by a quantity greater than R. Thus it 

 ajppears, that in order to raise the weight, 

 the power must be greater than P f~R ; 

 and in order to prevent the descent of 

 the weight, the power cannot be less 

 than P R. Hence every power whose 

 value is between the limits P+R and 

 P R, will sustain the weight in equili- 

 brium. The powers P+R andP-R 

 will sustain the weight in equilibrium 

 also, but it will be in a state bordering 

 on motion, the least imaginable increase 

 of the one or diminution of the other 

 necessarily producing the ascent or de- 

 scent of the weight. 



The increase R of the power which 

 balances the resisting forces is not ne- 

 cessarily equal to these forces them- 

 selves, because the increase R generally 

 acts upon the resistances, through the 

 intervention of the machine or some 

 part of it. To explain the methods of 

 determining the quantity R, even in the 

 several simple machines, would require 

 more mathematical investigation than 

 would be suitable to the objects of the 

 present Treatise. We shall, however, 

 explain some of the more simple cases. 



(24.) If a lever rest upon a knife-edge 

 like a balance, the friction will be im- 

 perceptible ; but if it turn upon a cylin- 

 drical axle, this is not the case. Let/ 

 be the absolute quantity of the friction 

 on the axle, determined in the manner 

 explained in the preceding chapters; 

 let r be the radius of the axle, W the 

 weight, and ic its leverage, and let p be 

 the, leverage of the power. In order to 

 raise the weight, the power will have to 

 overcome the friction f acting with the 

 leverage r, and the weight W acting 

 with the leverage w. The moment of 



P'p=Ww+fr.'.P'- 



the power which would exactly balance 

 these would be 



Ww+fr. 



Let the power sought be P' ; hence 

 Ww+fr 



P 



This power will just balance the weight 

 and friction, and any greater power will 

 raise the weight. 



If the weight be supposed to descend, 

 it will be opposed by the friction f acting 

 with the leverage r, which will thus 

 assist the power. Let P" be the power 

 which will just prevent the descent of 

 the weight, and we evidently have 



Any power less than P" will permit the 

 weight to descend, and these powers P' 

 P'', and all intermediate ones, will sus- 

 tain the weight in equilibrium. 



Since P, the counterpoise for W when 



there is no friction, is equal to 

 have 





,we 



which is the limit of the increase or de- 

 crease of the power consistently with 

 equilibrium. 



This investigation Applies also to the 

 \vheel and axle as is evident. In that 

 case, however, the rigidity of the rope 

 must be allowed for, conformably to 

 the principles established in the last 

 Chapter ; and the same may be ob- 

 served with respect to the pulley. 



(25.) Let a body W be placed upon 

 an horizontal plane, and / express 



Fig. 7. 



the proportion of the friction to the 

 pressure. If a force draw it in the 

 direction W A, parallel to the plane, the 

 force which will put it in motion will be 

 equal to the friction, and is, therefore, 

 W/. Let us now suppose that it is 

 drawn along the plane by a force which 

 constantly acts in the direction W B, 

 forming with the horizontal line always 

 the same angle B W A. This force 

 produces a twofold effect. Draw the 

 lines B C and B D so as to form the 

 c2 



