114 Statics. 



and the velocity of p, namely -^- will be 



mud D 



m D* -f p d a* 



Therefore, in order to know what ratio there must be between 

 -D and d, in order thatp may have the greatest velocity possible, 

 it is necessary to put equal to zero the differential of this express- 

 ion, taken by regarding <5 only as variable ; thus 



muDdS (?nD 2 + pd*} muDS X Zpddd = 0,' 

 (mD* +p$ 2 ) 2 



or 



muDdd (mD 2 + p J 2 ) m uDd X 2p$ dd = 0, 



whence 



or m 

 which gives 



If, for example, the weight /> be lOOOOOlb, and the mass m or 

 moving force be equivalent to a weight of lOlb, we shall have 



= D 1-12 



\ 1000 



!2- = J> 



1 00000 



that is, the radius of the cylinder must be a hundredth part of 

 the arm /E, in order that the effect may be the greatest possible. 

 193. There are many machines which are referrible, either 

 wholly or in part, to the wheel and axle, and consequently to the 

 lever ; such as rack-work, machinery in which wheels are con- 

 nected by bands, tooth and pinion work, and instruments intended 

 for drilling, boring, and screwing, although these last operations 

 Fig.105. often depend in part upon another machine that remains to be 

 described, namely, the inclined plane. In rack-work, the axis 

 FE having a winch FR q, carries a pinion the teeth or leaves 

 of which act upon the toothed bar AB. The leaves of the pin- 

 ion, in turning, raise the bar JIB with a force which is to the 

 force q applied to the winch, as the radius of the winch is to that 

 of the pinion ; and as the radius of the pinion is for the most part 

 small compared with that of the winch, by the aid of such a 



