13 



MECHANICS. 



Thus, if b were known, the effect of the 

 rigidity corresponding to any given 

 weight or pressure would be known. It 

 appears also, that in order to allow for 

 the rigidity of ropes in any machine, it 

 is only necessary to suppose the leverage 

 by which the resistance acts to be 

 greater than it is by a certain quan- 

 tity. 



To complete the theory of rigidity, 

 it will then be necessary to determine this 

 quantity b ; and in explaining how this 

 is done, we shall perhaps be compelled 

 to use more algebraical principles and 

 notation than most of our readers are 

 familiar with. The quantity b evidently 

 depends altogether on the curvature of 

 the rope B'D', Jig. 6. It is easy to 

 perceive the several elements on which 

 this curve depends ; 1 st, on the tension 

 of the rope or the weight B' with which 

 it is loaded ; let. this be called w : 2nd, 

 on the materials of the rope, and the 

 manner in which they have been manu- 

 factured ; let a express the quantity by 

 which this affects the rigidity : 3rd, on 

 the diameter of the rope ; let this be d : 

 4th, on the radius r of the wheel. 



The empirical formula 



d n 

 x = (a+mw) 



is assumed to represent x. By an em- 

 pirical formula is meant one that is 

 conceived or invented without any ana- 

 lysis or demonstration, and the truth 

 (or rather probable truth) of which can 

 only be established by shewing that it 

 is verified by experiment. 



In this formula the letters m and 

 n represent indeterminate numbers, the 

 values of which, as well as that of a, 

 can only be found by experiment. To 

 determine these let four pulleys be taken 

 whose radii are r, r', r", and r". The 

 rope whose rigidity is under examina- 

 tion being successively laid over these, 

 let it be stretched by weights equal to 

 w, w 1 , w", and w'", and let the weights 

 which in each case just give motion to 

 the wheels be x, x', x", and x" 1 . Sub- 

 stituting these in the formula already 

 mentioned, we obtain 



/ , N 

 x = (a+mw) 



x'='~ 

 x"= 



x'"= (a+mw"') 



From any three of these four equa- 

 tions the values of a, m, and n may be 

 deduced. These being known, we have 



*=,.-/. ft=..r: 

 r w 



and hence 



d n 



= (a+mw): 

 to 



thus we obtain the increase of leverage 

 which should be allowed to the resist- 

 ance when the diameter of the rope and 

 its tension are known. 



In order to verify the empirical for- 

 mula just mentioned, or to prove it as 

 far as it is capable of proof, it is only 

 necessary to eliminate the quantities a, 

 m, and n by the four equations (A,) and 

 if the result be an identity, that is, an 

 equation whose members are perfectly 

 the same, the four equations are con- 

 sistent. This species of proof may be 

 strengthened by multiplying the experi- 

 ments, and using different values of x r 

 w, and r ; and if an elimination of a, m y 

 and n, from every combination of four 

 equations, the certainty of the proof is 

 all but equal to that of demonstration.] 



CHAPTER VI. Of the Modification 

 which Friction and other Resistance* 

 produce upsn the Conditions of Equi- 

 librium. 



(23.) IN a machine which is con- 

 ceived to be divested of friction and all 

 other resisting forces, there is one cer- 

 tain and determinate power which will 

 equilibrate with a given weight, the me- 

 thods of determining which have been 

 explained in our second Treatise. Any 

 power greater than this will cause th& 

 weight to ascend, and any less power 

 will allow it to descend ; the equilibrium 

 in such cases being destroyed. If the 

 machine be subject to the effects of 

 friction, rigidity, or any resisting forces,, 

 this will not take place ; and we shall 

 find that any power between two deter- 

 minate limits will sustain equilibrium. 

 This circumstance arises from that pe- 

 culiarity in the nature of resisting or 

 passive forces, that in whatever direc- 

 tion motion, or a tendency to motion, 

 is produced, they assume the direction 

 immediately opposed to that tendency. 

 If the power has a tendency to raise the 

 weight by being increased beyond the 

 value due to equilibrium, by the prin- 

 ciples established in Treatise II., the 

 friction, &c., oppose that tendency; and 



