792 PROF. W. H. MILLER ON THE CONSTRUCTION OF THE NEW STANDARD POUND. 
Let t be the temperature of the air in centesimal degrees, b its pressure in millimetres 
of mercury at 0°, v the pressure of the vapour contained in it, also in millimetres 
of mercury, 0•378^’, AP, AQ ratios of the densities of P and Q at 0° to the 
maximum density of water ; eP^, eQ^ the expansions in volume of P and Q. Then 
log weight in grains of the air displaced by P 
= log A 4- log A^-f- log ( 14 -eP^)+log weight of P in grains — log AP. 
If vP be taken to denote the volume of P at 0°, the unit of volume being the 
volume of a grain of water at its maximum density, 
log vP=: log weight of P in grains — log AP. 
^die expression for the weight of air displaced by Q differs from the above only in 
the substitution of Q for P. 
The value of A is deduced froin h by means of Table II., assuming that the amount 
of vapour in the air is two-thirds of the quantity in saturated air. Table I. gives the 
second term for the expression for the weight of the air displaced, and Tables III., 
IV., V. give the third term according as the weight is of brass, bronze or platinum. 
Calculation of Densities. 
Let P in water at f appear to weigh as much as Q in air. Then weight of water 
at t° displaced by P= weight of P— weight of Q4- vveight of air displaced by Q, 
log ^;P= log weight in grains of the water displaced by P4- log Wj— log (1 -peP/), 
where VP^ is the ratio of the maximum density of water to its density at f obtained 
from Table VI. Log AP= log weight of P in grains — logt>P. 
An approximate value of vV having been found by assuming the weight of P equal 
to its apparent weight in air, this value of «;P may be used in reducing the weight of 
P, and thus a more accurate value of vV obtained, by means of which a closer 
approximation to the values of the absolute weight of P and of AP may be found. 
This process is to be repeated when greater exactness is required. 
Densities of the Troy Pounds constructed in 1758. 
Though it appears that only two of the five weights with which U was compared 
are in a state of unexceptionable preservation, and the number of trustworthy com- 
parisons is reduced frotn 608 to 440, these are amply sufficient for the purpose of 
ascertaining the apparent weight of U in air (^=65‘66 F, h=29’75 inches). But, in 
order to find the absolute weight of U, or indeed its apparent weight in air of a 
density different from that which it has when ^=65 66, l) = 29'7^, a knowledge of the 
volume of the lost standard is requisite. It is not probable that U was ever weighed 
in water, and certainly no record of any such weigliing is known to exist. There is 
therefore no direct method of finding its volume. An indirect way of arriving at it 
was suggested to Professor Schumacher by an examination of three Parliamentary 
Reports, the first presented May 26, 1758, the second April 11, 1759, and a third 
oidered to be printed March 2, 1824. 
