IMPERIAL STANDARD TROY POUND WEIGHT. 
483 
Q . . . the specific gravity of the water * at the temperature which it 
has when the body is immersed in it, and by 
m!, r\ q', R' . . . the values of m, r, q , R at the weighing in water 
the following two equations, viz. 
/ R 3 q\ ( r 3 q\ 
for weighing in air, M ^1 — = m ^1 §- ) 
( R , 3 Q\ / r' 3 q'\ 
for weighing in water, M ^1 — ~~ £ ~ ) — \ t j-j 
whence, by eliminating M, we obtain 
R ' 3 Q (i -^) -»/R 3 ? (i ~ r ^f) 
A = 
m 
(i) 
(5) 
( 6 ) 
r 3 q 
or, if for brevity’s sake we put = 
jiSqi 
= a 
A = 
m R /s Q ( 1 — a) — m' R 3 q ( 1 — a! 
(7)X 
m ( 1 — a) — in' ( 1 — a!) 
If only the first power of a is taken into consideration, which (with the exception 
of elastic fluids) can cause no perceptible error, we have the approximate formula 
A = — ^-,R' 3 Q--^— R 3 ? + -m 
m — m m — m ( m — mf 
or, because R and R' are nearly equal to 1, and q so small that it may be neglected, 
we may put R' 3 Q — R 3 q = Q, and obtain 
m m 
R 3 q) (a — a 1 ) 
( 8 ) 
A = 
m 
7 R ' 3 Q 
m 
m — m' m — m' * ' ( m — in f 
R 3 q -j~ 
in m 
(a a') 
(9)t 
29. It remains now to determine the numeric values to be used for these reduc- 
tions, and to give Tables that make the application of the formulae more easy. 
* The unity adopted for specific gravities being pure water at its maximum of density, Q is of course the 
density of pure water at the temperature T, divided by its greatest density, or 
q density of pure water at the temperature T. 
density of pure water at tbe temperature of nearly 39° F. 
T is the common temperature of the water in which the body is immersed, and of the body immersed in it. 
f The weight of the body immersed in water ( m ') is evidently different from its weight in air (m), and the 
temperatures of the water and of the air, at the moment when the body is weighed in water, will generally be 
different from the temperature of the air for the moment when the body is weighed in air. The values of 
r, q, R depend on these temperatures, and will consequently generally be different in both cases, so that 
they ought to be distinguished by a particular notation with accents : r', q' depends on t 1 (— common tempe- 
rature of the air and the weights when the body is weighed in water), and It' depends on T (— common 
temperature of the water and the body immersed in it). 
I When the atmospheric circumstances are the same at the weighing in water as they were at the weighing 
in air, or, in other words, if b' — b, and t' = t, it is not necessary to know the specific gravity of the weights em- 
3 q 2 
