98 Capt. Kater’s experiments for determining the 
have been inferred at once, from the principle of virtual velo- 
cities, 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 follow the mechanical steps by which 
the operation of the law takes place. If we make r— o, we 
have —jy = /, for the length of the equivalent pendulum 
when the surface of the cylinder is supposed to be the centre 
of suspension ; and it follows from the well known properties 
of the centre of gravity, that P the sum of the product of 
all the particles into their distances, is equal to Q d, the product 
of the whole weight Q into the distance of the centre of gravity 
from the point of suspension ; and = / = dQl, 
so that the equivalent length for the rolling pendulum 
, dl Q_ dl dl l 7 r\ 
becomes ? ( X+r ) p 2a:P+2rP <*Q^+rQ_ , r ‘'\ l ~~~d' > 
1 +d 
r being supposed very small ; which, for a simple pendulum, 
when d—l, becomes/ — r, as it ought to do. We must how- 
ever 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 be easily 
deduced from the expression 
■z xx P 
: for calling x — d, the 
distance of any particle of the body from its centre of gravity, 
y, we have x % =(r/-f.y)*= d 2 -\- 9 ,dy -f -y 2 , and £.r 2 P = £d a P 
+ zd%yV + %??=d*Q + o + ^y 2 P, the integral of SyP, the 
product of the distance of each particle into its distance from 
the common centre of gravity always vanishing : conse- 
quently / = == ^ + d, and l — - d = ^ ; which 
is Huygens’s theorem : the constant quantity being equal 
