the Pendulum ice cr Seconds in the Latitude of London. 429 
pression — : for calling «—d, the distance of any particle of 
the body fram its centre of gravity, 7, we have 2?=(d+y)?=d? 
+ 2dy +y?, and 2x?P = =Zad?P + 2dz2 LyP + 2y?P=7Q+04 Ly"P, 
the integral of yP, the product of the distance of each particle 
into its distance from the common centre of gravity always va- 
< Wie SyyP + d2 SyyP 
nishing: consequently 7 = “os tg + d, and l—d 
LyyP 
: . td } ; ¥ > . Y 
=n? which is Huygens’s theorem: the constant quantity 
2yyP being equal to d/—d*. If now we suppose d to be in- 
creased by the small quantity s, the reciprocal, instead of /—d, 
di—dd _ I-d 
— = (1 d)(1—+) =l—d -l- +5, 
Sa ; 
to which bed d eg we have /—/ = + 2s, the increase of the 
will become 
l 
; and making veh equal to — = 7, we have 
—Ir 
S= 577 and pis the pendulum is inverted, substituting /—d 
USO Ret! 
21-2d—1 sey 
the former negative value of the same quantity, must always de- 
stroy it: so that the length of the equivalent pendulum will be 
truly measured by the simple distance of the surfaces of the cy- 
linders, as M. Laplace has demonstrated. 
“¢ There is however another correction, of which it becomes 
necessary to determine the value, when a very sharp edge is used 
for the axis of motion, as in the pendulum which you have em- 
ployed: since it appears very possible, that in this case the tem- 
porary compression of the edge may produce a sensible elongation 
of the pendulum. But it will be found, by calculating the mag- 
nitude of this change, that when the edge is not extremely short, 
and when its bearing is perfectly equable, this correction may be 
safely neglected. 
** Supposing a to be the distance from the edge, in the plane 
bisecting its angle, at which the thickness is such, that the weight 
of the modulus of elasticity corresponding to the section shall 
become equal to the weight of the pendulum, the elasticity at 
any other distance x from the edge will be measured by «x, while 
the weight is represented by a; so that the elementary increment 
for d, the expression becomes , which, added to 
x will be reduced by the pressure of the weight to meet and 
the element of the compression will be —s, and its fluxio™ 
it 
a 
ate 
