764 PROF. w. H. MILLER ON THE CONSTRUCTION OF THE NEW STANDARD POUND. 
when the balance is in its position of equilibfium, with Q in the left-hand pan and P 
in the rig-ht-hand pan. Then 2Q = 2P-l-m[(P, Q) — (Q, P)]. 
In making- a large number of comparisons, the weights are exposed to the risk of 
being injured by wear. In order to obviate this danger, two light pans were pro- 
cured of very nearly equal weight, each of which has a loop of wire forming an arch 
the ends of which are attached to the pan at opposite extremities of a diameter of the 
pan, by which the pan could be lifted with the hook at the end of a long handle, into 
or out of either of the pans of the balance. Calling the pans X and Y, and the 
weights to be compared P and Q, P was placed in X and Q in Y, and P-j-X com- 
pared with Q-j-Y n times ; then P was placed in Y and Q in X, and P-f-Y compared 
with Q+X n times. The weights were thus exposed to the wear of two ordinary 
comparisons only in the course of 2n comparisons. The mean of the 2n compari- 
sons gives the difference between P and Q, unaffected by the very small but unknown 
difference between the weights of the pans X and Y. This contrivance was found to 
be especially useful when either of the weights to be compared consisted of several 
parts. 
An improvement upon this was made by using the pans of one of the balances 
employed by the French Pharmaciens, which resemble those above described, with 
the addition of an iron hook at the highest point of the wire loop. Either pan is 
suspended by a wire of suitable length bent into a hook at each end, from the ring 
attached to the agate-plane. In using the method of double weighing, the original 
left-hand pan of the balance was suffered to remain with the counterpoise in it, and 
the pan X containing the weight P, and the pan Y containing the weight Q, alter- 
nately suspended from the right-hand end of the beam, and the positions of equi- 
librium observed (usually about twenty times). The weights were then interchanged, 
and pan Y containing the weight P, and pan X containing the weight Q, suspended 
from the right-hand end of the beam, and the positions of equilibrium observed the 
same number of times. Tlie w'eights of X and Y were frequently reduced in order 
to make them as nearly equal as possible, and sometimes in order to remove rust 
from the iron hooks. 
In using Gauss’s method, it was desirable to be able to transfer the pans, and the 
weights contained in them, from one end of the beam to the other without opening the 
doors of the balance-case, and thus avoid sudden changes of temperature of air within 
the balance-case, and consequent production of currents of air. In order to effect this, 
various contrivances were tried. Of these the following proved the most successful. 
A slender brass tube, 38 inches long', passes freely through two holes in tlie ends of 
the balance-case, which is 22‘75 inches long, near the top of the case, and half-way 
between the balance and the front of the case. To the middle of the tube is attached 
a descending loop of wire. Suppose that by sliding the rod, the loop is brought near 
to the right-hand end of the beam, and a pan with a weight in it transferred from 
the end of the beam to the wire loop by a brass rod having a deep groove filed round 
