APPLICATION OF GRAPHIC METHODS 295 



radii of the wheel and of the axle ; from their intersection link 

 3 would be drawn tangent to the friction circle for the joint ab. 

 The stiffness of the rope must, however, be taken into account, 

 and this can be done by drawing links 1 and 4 as broken links, 

 of which the lower halves are drawn as above described, while 

 the upper halves represent the lines of action of the forces shifted 

 sideways. The driving link is brought nearer the centre of a, and 

 the resisting link removed further from this centre. The dis- 

 tances d and d v by which each half link is shifted, are given by 



the expressions d="~. d { = 1, where m or m^ is the moment of 



* i 

 the couple required to bend the given rope to the given radius, 



and P or P, is the stress in the link. It must be remembered 

 that this stress is not the same in the driving and resisting 

 links, and that if we are given the stress in e we must proceed, 

 by trial and error or simultaneous equations, to find the stress 

 in f before we can determine exactly the distance by which the 

 link 4 is shifted. When the ropes are long, their efficiency 

 must also be taken into account, when our object is to com- 

 pare energy exerted with work done. When we simply wish to 

 compare effort and resistance, the loss of energy due to the 

 stretching of the rope may be neglected. Inasmuch as the 

 axes of elements e and / are assumed to lie in parallel planes, 

 perpendicular to the axis of a, the forces in the elements e and 

 / (unless parallel) give rise to an injurious couple on the bear- 

 ings, which, except when these are very far apart relatively to 

 the distance between the planes of e and /, sensibly diminishes 

 the efficiency of the machine. 



15. Inclined Plane. The idea involved in problems on the 

 ' inclined plane ' is that one element, sliding on another with 

 a plane joint between them, shall be employed to maintain 

 equilibrium between forces applied to the sliding element in a 

 plane perpendicular to the joint. We may embody this idea 

 in a simple complete machine, as shown in Fig. 13, where b is a 

 fixed element, e a driving element jointed to b and a, the sliding 

 piece having a plane joint with b; / the resisting element 

 jointed with b and a ; the axes of e and /are in a plane perpen- 

 dicular to the joint ab. We have here a self-strained combination 

 fulfilling all the required conditions. The dynamic frame with 



