294 APPLIED SCIENCE 



The conception of a complete machine has not been recognised 

 by any writer on mechanics as necessary for the statement 

 of problems connected with the lever. If, however, these 

 problems are to be practical, and not confined to abstrac- 

 tions, such as ' forces applied to points,' they do require the 

 consideration of a complete machine, as here drawn. The fric- 

 tion at ab is usually taken into account ; that at ae and af is 

 more generally neglected, and the friction at be and bf has 

 perhaps never been thought of as an essential part of the 

 problem. The reason of the neglect is clear. The forces re- 

 presented by links 1 and 4 are in many problems due to at- 

 traction between some parts of elements a and b, as, for instance, 

 when these forces are due to weights actually forming part of 

 the element a, and attracted by the earth which supports, and 

 is indeed part of, the element b. In this case the joints ae, of, 

 be, and bf are frictionless, or may be said to disappear as joints. 

 When the weights are hung by pins at ae 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 take a 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. 12a. When 

 the wheel and axle are circular, there is no friction at joints eb 

 and/6 ; moreover, the pins and eyes which form the joints at ea 

 and fa are replaced by the flexible rope. There is no friction at 

 the joint eb, since e does not rotate relatively to &, and we may 

 therefore assume that the force in the tie e is uniformly dis- 

 tributed relatively to its cross section : the resultant force, there- 

 fore, will pass through the centre of the pin at eb, 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 

 fib and/&, tangent to the dotted circles drawn with the effective 



