A PENDULUM FOR THE REDUCTION TO A VACUUM. 
405 
But, here it will be proper to remark that S does not denote the specific 
gravity of the pendulum, as determined in the usual manner when at rest, 
unless the mass be homogeneous : for, in all other cases, where the pendulum 
consists of several parts, whose specific gravities are different, we must com- 
pute the vibrating specific gravity of the mass in the following manner. Let 
cP, d", d"', &c. denote the distance of the centre of gravity of each body re- 
spectively from the axis of suspension*: w', w" , w'", & c. the weight (in air) of 
each body: s', s", s'", &c. the specific gravity of each body, determined in the 
usual manner. Then will the required vibrating specific gravity of the pen- 
dulum bef 
c , ’id d 1 + ’vo" d" + id" d" 1 + &c. 
^ — idd' WdP WdJ" 7 ( 2 ) 
-T + - 7 t - + - t »- + & c - 
And, it is in this manner that I have deduced what may be called the vibrating 
specific gravity, for all those pendulums, which, in the following experiments, 
consist of substances of different specific gravities. 
With respect to the other quantities involved in the above formula (1) there 
are two modes which have been pursued for expressing them numerically : 
viz. one by assuming Sir George Shuckburgh’s determination of the relative 
weights of air and water, as stated in the Philosophical Transactions for 1 777 ; 
that is, a = (3 = 29-27* and t — 53° : and the other by assuming the more 
recent determination of MM. Arago and Biot ; that is, a = (3 = 29-9218, 
and t = 32°. The former has been adopted I believe by most English experi- 
mentalists ; but, as I conceive the latter to be the more accurate determina- 
tion, I shall adopt it in all the present reductions. They differ from each other 
about gVth part of the whole correction : the French result being the greatest 
in amount. 
The expansion of mercury is generally assumed equal to *0001 for each de- 
gree of Fahrenheit’s thermometer : but the expansion of air is not quite so 
* When the body is below the axis, d is plus : when above, it is minus. 
f I am indebted to Professor Airy for this formula : which, although of considerable importance in 
all investigations relative to the pendulum, has not, as far as I am aware, been alluded to by any 
writer on the subject, except Bessel. 
3 G 
MDCCCXXXII. 
