288 
PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 
the operation of the moving power, 2 u the whole number of such units absorbed 
in overcoming the prejudicial resistances opposed to the working of the machine, 
2 U 2 the whole useful work of the machine (or that done in producing its useful 
effect), and (t\ 2 2 — v x 2 ) one half the aggregate difference of the vires vivse 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, 
2 Uj = 2 U 2 + 2 u + Yg 2 w ( v 2 2 ~~ v i 2 )> (1-) 
in which v x and v 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 V 2 that at its termination, and v x and v 2 
the velocities of any other element at the commencement and termination of the same 
cycle or period ; then X . V 1 = v x , X . V 2 = v 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 
2 w v x 2 = V x 2 . 2 w X 2 , l,wv 2 2 = V 2 2 2 w X 2 ; 
(V - *¥*) = \ (V - V 2 2 ) .2 
* 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.’ 
