MECHANICS, 



2.5 



ferences of the thicker and thinner parts 

 of the axle. 



To determine the conditions of equi- 

 librium in this engine, we shall refer to 

 Fig. 37. 



vrl 



Jig. 37, which is a section of the appa- 

 ratus taken perpendicular to the cylin- 

 der, of which C is the centre, C B the 

 radius of the thinner and C A of the 

 thicker part, and C D the lever by 

 which the power P acts. As the weight 

 W hangs from the rope, which passes 

 round the wheel E F, the parts E B and 

 F A of the string are equally concerned 

 in supporting the weight ; and, conse- 

 quently, half the weight is the force 

 which hangs from each part of the 

 string. If, then, the lever by which 

 . the power acts be R, and that r be the 

 radius C A, and/ be C B, the condition 

 of equilibrium, conformably to the prin- 

 ciple laid down in (1 0), is 



P x R + 4 W x r 1 = J W x r; 

 that is, if the power be multiplied by its 

 leverage, and half the weight by the ra- 

 dius of the thinner part of the cylinder, 

 and these two products be added toge- 

 ther, their sum will be equal to the pro- 

 duct found by multiplying half the 

 weight by the radius of the thicker part 

 of the cylinder. Or, what amounts to 

 the same thing, we have, 



P x R=~4W xr JW x /, 

 or P x R = W x i (r r') ; 

 that is, the power multiplied by the 

 lever by which it works is equal to the 

 weight multiplied by half the difference 

 of the radii of the thicker and thinner 

 parts of the cylinder. 



This condition may also be expressed 

 thus : the weight is to the power, as the 

 lever by which the power works is to 

 half the difference of the radii of the cy- 

 linder ; i. e. 



. W : P : : R : J (r - r'). 



Fi cm considering this condition, it is 

 plain that, where the mechanical effi- 

 cacy depends on the proportion of the 

 lover of the power to half the differ- 

 ence of the radii of the cylinder, this 

 efficacy may be increased, without any 

 limit, by merely diminishing the diffe- 

 rence of the radii of the cylinder, with- 

 out either increasing the leverage of 

 the power or diminishing the thickness 

 of the cylinder. Thus, by this contri- 

 vance, the power will act through a 

 convenient space ; there will be nothing 

 unwieldy in the construction of the 

 apparatus ; and all the requisite thick- 

 ness and strength may be given to the 

 cylinder. 



A capstan, constructed upon the 

 same principle, is represented in/g". 38. 



Fig. 38. 



A great advantage attending this ma- 

 chine is, that there is no recoil upon the 

 weight upon the remission of the power, 

 and, therefore, the use of the ratchet- 

 wheel and catch (52) is superfluous. 



Almost the only disadvantage attend- 

 ing it is, that a considerable quantity of 

 rope is requisite to raise the weight 

 through a very small height ; but still 

 much less than what is requisite in any 

 machine of the same power in which 

 rope is employed. 



Dr. Gregory states, that although 

 this invention is generally ascribed to 

 George Eckhardt,v*ho probably invented 

 it without being aware that it had been 

 previously used; yet that there is a 

 figure of it in some Chinese drawings, 

 more than a century old, from which 

 Dr. Gregory took the sketch, Jig. 36. 

 See Gregory's Mechanics, vol. ii. p. 3. 



(60.) The principle of varying the 

 diameter of the cylinder while the le- 

 verage of the power remains unvaried, 

 is sometimes used to accommodate an 

 uniform power to a varying resistance. 

 We may state this principle still more 

 generally. If a power, varying under 

 any given conditions, be required to 

 overcome a resistance which varies ac- 

 cording to some other given conditions, 

 the one may always be accommodated 

 to the other by producing a variation in 

 the leverage by which one or both acts, 



