TO THE DETERMINATION OF THE EFFICIENCY OF MACHINERY. it 
machine, ashere drawn, The friction at ad is usually taken into account ; that at 
ae and af is more generally neglected, and the friction at be and 47 has perhaps 
never been thought of as an essential part of the problem. The reason of the 
neglect is clear. The forces represented by links 1 and 4 are in many problems 
due to attraction between some parts of element @ and element 0, as, for 
instance, when these forces are due to weights actually forming part of the 
element @, and attracted by the earth which supports, and is indeed part of, 
the element 0. In this case the joints ae, af, be, and bf are frictionless, or may 
be said to disappear as joints. When the weights are hung by pins at aé and 
af, the friction at those pins must be taken into account, and whenever the 
- forces represented by links 1 and 4 are due to another machine, to springs, or 
any other material element, the problem requires all the circumstances to be 
taken into account which are indicated in the dynamic frame as shown. 
§14. Wheel and Axle.—The wheel and axle, with its driving element, resisting 
element, and bearing, forms a complete machine when the parts are connected, 
as shown in fig. 12. The wheel and axle constitute the element a, and 
the other elements have names given to them, corresponding to those for the 
lever. The dynamic frame is shown in fig. 124. When the wheel and axle 
are circular, there is no friction at joints eb and /d; moreover the pins and 
eyes which form the: joints at ea and ja, are replaced by the flexible 
rope. ‘There is no friction at the joint eb, since e does not rotate rela- 
tively to 6,and we may therefore assume that the force in the tie ¢ is 
uniformly distributed relatively to its cross section: the resultant force, 
therefore, will pass through the centre of the pin at ¢, and similarly the 
resultant of the resistance will pass through the centre of the pin at fb. If the 
ropes were perfectly flexible, we might, in fig. 12a, draw links 1 and 4 from the 
centres of the pins at eb and /b, tangent to the dotted circles drawn with the 
effective 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 brotvight’ nearer the centre of a, and the 
resisting link removed further from this centre. The distances d and d,, by which 
each half link is shifted, are given by the expressions. 4 =a 4=55 where m or m7, 
is the moment of the couple required to bend the given rope to the given radius, 
and P or P, is the stress in the lmk. 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 ¢ we must proceed, by trial and error or simultaneous equations, to 
find the stress in / before we can determine exactly the distance by which the 
VOL. XXVIII. PART I. E 
