97 
length of the pendulum vibrating seconds. 
to the force at an equal distance in the path of A, as a to dElL, 
or as fl-l+.r) to 1 . Now the point of the rolling pendulum 
confined to the vertical line is not the centre of curvature, 
but initially the surface of the cylinder : so that this must be 
considered as the point of intersection with the vertical line, 
and as the fulcrum of the lever; consequently the distance of P 
from the vertical line will be, to that of the pendulum A, as 
x to a, and its immediate force will be ■ . -.P=^-tfP; 
but this force, acting only at the end of a lever x, will have 
its effect at A again reduced in the ratio 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 character 3, 
whether found by a fluxional calculation or otherwise, we 
have the whole force, at A, 2 - P. The reduced or rota- 
tory inertia of the body, sometimes very improperly called 
the “ momentum ” of inertia, will also be expressed by 
2 — P, being reduced in the ratio of the squares of the dis- 
tances from the fulcrum ; consequently the accelerative force 
£~P 
will be to that of the pendulum A as ■ — a -—~ to i, or as 
2^+Ip 
a 
— 2 xx — -- to i ; since it is indifferent whether the integral 
aZ(x+r) p ’ o 
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 as the force is 
weakened, so that we have for this length — TjftT, It is 
obvious that the denominator of this fraction is the same that 
would express the force of the body with regard to the 
centre of the cylinder as a fixed point ; and it might indeed 
mdcccxviii. O 
