146 
Proceedings of the Royal Society 
for the right arm, the balance being adjusted so that, when both 
riders are at their zero-points, it is in equilibrium, and p being 
chosen so, that, supposing the large rider to be shifted to the 
n mark, and the small one to the m mark, this virtually amounts 
to the addition of lOn + ra milligrammes to the charge in the 
right pan. 
There is no need of my explaining how the balance is meant to 
be used; I will rather avail myself of this opportunity for drawipg 
the attention of readers interested in the subject to a few inferences 
from the theory of the balance, which, obvious as they are, have 
hitherto not been sufficiently appreciated by either the authors of 
our physical handbooks or by practical balance-makers. 
I. Given a balance in which everything is constant except the 
distance s of the centre of gravity of the empty instrument from 
the axis of rotation, and it is easily shown that (for a constant 
charge) the deviation a of the needle for a given over-weight A, and 
consequently the 
“sensibility” a — is the greater the less 
s. This, of course, is duly stated by all authors; but what is 
always forgotten to be pointed out are two things, viz. — 1st, That 
the “ sensibility ” has nothing to do with the inherent precision of 
the instrument ; and 2dly, That supposing the sensibility to be 
increased, all the other good qualities of the balance get less ; we 
diminish the rate of vibration (this rate being proportional to 
we diminish the range of differences of weight deter- 
minable by the method of vibration ; we diminish the relative 
constancy (in opposition to variations in the charge) of the 
sensibility and the time of vibration. Considering now, — 
II. The case of a balance to be constructed , the arm-length l and 
weight w of the empty beam also become variables, related to 
each other, according to some equation like w = const. l 7 and 
(assuming each of the pans to bear a certain medium charge p) we 
have 
t i— - 
— t== const. J l v const. + const. I , 
sja 
i.e ., by diminishing l we can increase the sensibility without 
diminishing the rate of vibration (or vice versa)) but the other 
