348 Quinby on the maximum effect of Machines. 
Pn? 
plane, the force which accelerates A willbe = Pat gWo 
And either of these values may be readily introduced into the 
investigation. 
Cor. 4. The work done in the time t, if we retain 
rP—r? iW = 
R?® P+r aw s 
the original notation, will be = 
RrPW—?W? 
RPyrPw eo 
Cor. 5. When the work done is to be a maximum, and 
we wish to know the weight when P is given, we must make 
the fluxion of the last expression = 0. ‘Then we shall have 
P? —2r?R?PW—r4*W?2=0 5x and W=Px 
Cor. 6. IfR:r+::n:1, the preceding expression wil 
become W=P x ( ,/n* +n? —n?). 
€or. 7. When the arms of the lever are equal in length, 
that is, when n=1, then is W=Px (/2—1)= .4142 P, or 
nearly 3°; of the moving force. 
: Scholium. ._ 
If we compare the values of S and » in this proposition, and 
the first corollary with those in the fourth example, art. 267, 
oe relates to motion on the axis in peritrochio, it will be 
the expressions correspond exactly. Hence it fol- 
rig that when it is required to proportion the power and 
weight, so as to obtain a maximum effect on the wheel and 
axle, (the weight of the machinery not being considered,) We 
may adopt the conclusiens of cos. 5 and 6 of this proposi- 
tion. And in the extreme case, where the wheel and axle 
Se pulley, the expression in cor. 7. may be adopted. 
ns may be applied to machines in gene- 
ral, if R and r represent the distances of the impelled and 
working points from the axis of motion; and if various 
kinds of resistance arising from friction, stiffness of ropes, &c. 
be. properly reduced to their equivalents at the working 
‘points, so SO as = = comprehended in the character W for re- 
a aie demonsttelieg. Dr. Gregory has proved his 
+ but in the corollaries which he has hae anni and 
