A PENDULUM FOR THE REDUCTION TO A VACUUM. 
401 
other. And until this is practically accomplished, and can be practically re¬ 
peated, I do not think that the true length of the seconds pendulum can be 
considered as satisfactorily determined. 
Reduction to a vacuum. 
M. Bessel has shown, in his very interesting work on the pendulum *, that 
the usual formula for the reduction to a vacuum, as far as'the specific gravity 
of the moving body is concerned, is very defective; and by no means expresses 
the whole of the correction which ought to be applied : in fact, that a quan¬ 
tity of air is also set in motion by, and adheres to, the pendulum (varying 
according to its form and construction), and thus a compound pendulum is in 
all cases produced, the specific gravity of which will be much less than that of 
the metal itself. He states (page 32) that “ if we denote by m the mass of a 
“ body moving through a fluid, and by m' the mass of the fluid displaced 
“ thereby, the accelerating force acting on the body has, since the time of 
TYi — jyi} 
“ Newton, been considered equal to ———. This formula is founded on the 
ce presumption that the moving force, which the body undergoes, and which is 
“ denoted by m — m', is confined to the mass m. But, it must be distributed not 
“ only over the moving body, but on all the particles of the fluid set in motion 
by that body; and consequently the denominator of the expression, denoting 
“ the accelerating force, must necessarily be greater than mT M. Bessel then 
enters into a mathematical investigation of the principles from which the 
results of his experiments are deduced: and at length comes to the following 
important conclusion : viz. “ that a fluid of very small density, surrounding a 
w pendulum, has no other influence on the duration of the vibrations than that 
^ it diminishes its gravity and increases the moment of inertia. When the in- 
“ crease in the motion of the fluid is proportional to the arc of vibration of the 
“ pendulum, this increase of the moment of inertia is very nearly constant: in 
“ all other cases it will depend on the magnitude of that arc.” 
The obvious inference from those experiments and researches is, that the 
amount of the correction will not only vary according to the length, magni¬ 
tude, weight, density and figure of the pendulum ; but also that in the case of 
* Untersucliungen iiber die Lange des einfachen Secundenpendels, von F. W. Bessel. Berlin 
1828, 4to. This work forms part of the Memoirs of the Royal Academy of Sciences of Berlin for 
1826. 
