428 An Account of Experiments for determining the Length of 
of x to a, and will then become =<"P: : and if we express the 
sum of all the similar forces belonging to the body by the cha- 
racter =, whether found by a fluxional calculation, or otherwise, 
c+r 
we have the whole force at A, = P. The reduced or rota- 
tory inertia of the body, sametiibas very improperly called the 
«* momentum ” of inertia, will also be expressed by == P; being 
a 
reduced in the ratio of the squares of the distances from the ful- 
crum ; consequently the accelerative force will be to that of the 
Sr2P 
pendulum A as oes Fl 
rt+rp az(x+r)P 
to 15 since it isin- 
a 
different whether the integral or the differential be divided by 
the constant quantity a: and in order to express the length of the 
equivalent pendulum, we must suppose a to be as much lengthened 
Y2aP 
=(a@+r)P° 
It is obvious that the denominator of this fraction is the same 
that would express the force of the body with regard to the cen- 
tre of the cylinder asa fixed point; and it might indeed have been 
inferred at once, from the principle of virtual velocities, that the 
force must be the same in either case, however irregular the form 
of the body may be: but it is somewhat more satisfactory to fol- 
low the mechanical steps by which the operation of the law takes 
as the force is weakened, so that we have for this length 
place. If we make r=0, we have 222° =I, for the length of the 
equivalent pendulum when the surface of the cylinder is supposed 
to be the ecntre of suspension; and it follows from the well- 
known properties of the centre of gravity, that =xP, the sum of 
the product of all the particles into their distances, is equal to 
Qd, the product of the whole weight Q into the distance of the 
centre of gravity from the point of suspension; and 2a*P=LaP/ 
=d Ql, so that the equivalent length for the rolling pendulum 
11 1 ll 
aceiee ero oy ies dkO = =U 1—-), r 
Ss Gas) Pi eteaPSrP dQ+,Q 1+" 
being supposed very small; which, for a cae pendulum, when 
d=/, becomes /—r, as it ought to do. We must however find 
the displacement of the centre of suspension which is capable of 
producing an equal alteration in the length of the equivalent 
pendulum; and for this purpose we must have recourse to the 
theorem of Huygens, which may he easily deduced frnm the ex- 
pression 
e 
