33 
“ On the Estimation of Small Excesses of Weight by the 
Balance from the time of Vibration and the angular Deflec- 
tion of the Beam/’ by J. H. Poynting, B.A. ; B.Sc. 
While working last year on an experiment to determine 
the mean Density of the Earth by the balance, I had to 
measure such an exceedingly small difference of weight, 
that I could not at that time estimate it by means of a rider 
but was obliged to adopt the method described in this paper. 
Stated generally it consists in treating the balance as a 
pendulum. Knowing the nature of the pendulum (that is 
its moment of inertia), and its time of vibration, we can 
calculate what force acting at the end of one arm of the 
beam will produce a given angular deflection. It is in. fact 
an application to the common balance of the method which 
has always been used with the torsion balance when it has 
been necessary to calculate the forces measured in absolute 
measure. I cannot find any record of a previous applica- 
tion of the method, and as it might be of use in very delicate 
weighings, or in verifying the small weights in a laboratory. 
I have thought it worth while to give a full account of it. 
When small quantities of the second order are neglected 
and the oscillations are of the first order, it will easily be 
found that the equation of motion of the beam of the balance 
is 
(MP + 2 Pa 2 ) Q + (2P& + M glc)Q — ap (1) 
9 
where MI 2 = moment of inertia of beam about central knife edge 
M = mass of beam 
a — half length of beam. 
P = weight of either pan and the mass in it. 
h — distance of line joining terminal knife edges below the cen- 
tral knife edge. 
h - distance of centre of gravity of beam below central knife 
edge. 
p = small excess in one pan. 
