288 PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 



the operation of the moving power, 2 m the whole number of such units absorbed 

 in overcoming the prejudicial resistances opposed to the working of the machine, 

 2 Ug the whole useful work of the machine (or that done in producing its useful 



effect), and ^^^tv {v^ — v-^) one half the aggregate difference of the vires vivae of 



the various moving parts of the machine at the commencement and termination of 

 the period during which the work is estimated, then, by the principle of vis viva, 



2Ui = 2U2 + 22. + ~SM;(r22-V), (L) 



in which v^ and ^2 represent the velocities, at the commencement and termination of 

 the period, during which the work is estimated, of that moving element of the 

 machine whose weight is w. But one-half the aggregate difference of the vires vivae 

 of the moving elements, represents the work accumulated in them during the period 

 in respect to which the work is estimated. 



3. At every period of the motion of a machine, there obtains a relation between the 

 motion of each one of its elements, and that of every other element, so that the 

 velocity of every other moving element of the machine may at any time be expressed 

 by an algebraical function of the velocity of that one element, and the space traversed 

 by it from a given period of the motion, the constants entering into which function 

 are determined by the forms, dimensions, and combination of the elements of the 

 machine*. If any one such element be made to move uniformly, the other elements 

 will either move uniformly or with a periodical motion, or some of them uniformly, 

 and others with a periodical motion. In the first case it is evident that the motion of 

 every element will bear a given constant ratio to that of every other. In the second 

 case, that it will bear to it a ratio which will become the same at the expiration of 

 each given period ; it is evident moreover that this given ratio between the velocities 

 of the moving elements, will obtain constantly or periodically under a variable as well 

 as a constant motion of the first element of the machine. Suppose the work to be 

 estimated during a period which is a common multiple of the periods or cycles of the 

 different moving elements. Let Vj represent the velocity of the moving point, or first 

 element of the machine at the commencement of this cycle or period, which is a com- 

 mon multiple of all the other periods, and Vg that at its termination, and v-^ and v^ 

 the velocities of any other element at the commencement and termination of the same 

 cycle or period ; then X . V^ = i^j, X . Vg = ^2? where X represents a constant quantity 

 given in terms of the forms, dimensions, and combination of the intervening ele- 

 ments of the machine. The same being true of every other element, it follows that 



.•.y2«)(V-«;22)=l(Vi2-V22).2M)X2. 



* Professor Willis has determined the form of this function in respect to each of the principal elements of 

 complex machinery, in his work recently published, entitled ' The Principles of Mechanism.' 



