126 = Examination of Mr. Quinby's Crank Motiou. 
tation when the crank is at a, is equal to the tension of the 
shackle bar, at that time, multiplied by the distance Ce ; 7. e. 
= (_F xrad. ) x Ce: and the effect produced at the point 
Cos< ASa 
d, is equal to (Peles = Cc” : these expressions are not 
equal to the tendency to rotation, but they are proportion- 
ai to it. To be equal, each should be divided by the radius 
of the crank: that is of no consequence just now, since they 
are to be put in the form of a proportion: ‘ and now, if the 
inference drawn by Mr. Ward were true, then would. rx ra 
: Cos< ASa 
x rad oe 
rape ag er xCc:: am: dn; or (by dividing the first 
and second terms by P. rad, and substituting in place of am 
> z e Ce 
x Ce: 
sé 
Cos< ASa Cos< Atd 
>: Ce: Ce.” But CeD and Cc are not proportionals of am 
and dn, foram:aS :Cc: , and dn: dt (or aS) :: Ce: 
Ct; in these two proportions, the third terms are the same, and 
in order that the terms of the first couplet in one proportion, 
should be proportionals to the terms of the first couplet of the 
other, the fourth terms must be equal; but Cz is evidently 
less than CS, hence Ce and Ce are not proportionals to am 
and dn, and the remaining part of Mr. Quinby’s demonstra- 
tion founded upon this assumption, can be of no avail. 
In showing the actual loss of power in the application of 
the crank, we will consider as proved, the fact shown by 
Mr. Quinby, that in the actual case in practice, the tendency 
to rotation is the same as it would be if the shackle bar re- 
mained constantly vertical; a refutation of his demonstra- 
tion is unnecessary. 
Let Sa (fig. 3) represent the position of the shackle bar, 
at the point a of the circle AEBD, in which the lower ex- 
tremity of the bar moves, and Sa the value of the constant 
power P, applied to the upper extremity of the bar ; by re- 
solving the force Sa into the two forces ab and Sh, the first 
in the direction of the radius, the second parallel to the tan- 
gent at the point a; Sb is the component tending to produce 
rotation ; the SJ multiplied by the arm of its lever, Ca, must 
be equal to P, multiplied by the lever am, or (calling Ca, R; 
and dx their proportionals Ce and Cc) 
