SECT. VIII.] 
OF THE REGULATION, &c. 
257 
engine, the fly is used. Its heavy mass of matter must be so shaped, as to balance 
itself in any position on an axis connected with the machinery, and turning round 
with a part of it. 
The proportions of the fly wheel must be derived from the laws of rotary motion. 
They are not often stated very clearly, and rather in too comparative a form for 
the purpose of application : Dr. Jackson’s equation 1 is derived most in unison with 
my own methods, and adding the time, the radius corresponding to the angular 
velocity of the exterior ring of the wheel, and comparing with the force of gravity 
to obtain the coefficient, it is 
32 P art 
= n V. 
b ¿c 2 
In this equation 
P = the mean quantity the moving force varies in its intensity in excess 
above the resistance, 
t — the time in which that variation takes place; 
v = the velocity, 
n v — the greatest variation of velocity, 
a = the leverage the force P acts with, 
r = the radius corresponding to the velocity v, and 
w = the weight of the fly acting at the distance x from the axis. 
It is obvious that the mass of the fly must be sufficient to receive the excess of 
force during the time it acts, and afford it again to the machine in an equal lapse 
of time, and so that the velocity shall not vary more than the nth part. The only 
point, therefore, which depends on practical experience, is what variation of velocity 
may be allowed. On this point however there is no difficulty, as the practice of 
different makers is so different, as to show that it may be taken with considerable 
latitude. 
The weight of the rim may always be considered to be collected at the extremity 
of the radius; and then x = r, and the equation becomes, 
32 P at 
= n v. 
w r 
The effect of the arms of the wheel may be neglected, as it is a problem which 
neither requires nor admits of a very refined solution, in consequence of the un 
certainty regarding the precise variation of the intensity of the moving force; 
hence it ought not to be rendered complicated. 
537. From this equation it appears, that when the weight or the diameter of 
1 Theoretical Mechanics, art. 400—403. 
2 k