GRAVITATION AS AFFECTED BY TEMPERATURE. 
355 
(2) Trouble attends the use, as in these experiments, of a vacuum. The beam of a 
torsion balance has five degrees of freedom, and each mass suspended from the 
beam end has two effective degrees of freedom. In a vacuum, where damping is small, 
the beam system often gets beyond control. 
To come to details, the large attracting masses outside the vacuum will be denoted 
as before by letters M, M, and the small ones inside the vacuum by letters m, m. The 
masses used are of various forms : (a) spheres ; ( b) cylinders, with axes vertical; (c) the 
approximately cylindrical form of vertical chains. The spherical form is the most 
sensitive, yet the cylinder has two advantages. It is compact for some heating 
purposes ; also the law of force for cylinders involves distance to the first power 
approximately, so that error in position is not so serious as for spheres where the 
square of distance is involved. The law for spheres being well known, that for 
cylinders only will be proved :— 
2. The Law for Cylinders. —-The law of force for infinite parallel cylinders, one 
having much greater sectional area than the other, is found thus :— 
Let the small cylinder cut the paper normally at O. Its attraction on unit mass 
at P = 2Gm-j/R, where It = distance from P normal to the cylinder, and <m x = mass 
of unit length of the cylinder. Let the second parallel cylinder of large section have 
axis at O' and have mass per unit length m 2 . Consider element of cross-section ds 
at P. Its mass = m ' 2 - > where a = radius of cylinder. 
7 -a 
The total attraction per unit length is 
cos \[s . ds 
the integral to be taken over the whole cross-section of the cylinder. Using the 
symbols in the figure 
P cos f = d—r cos 6 
R 2 = d 2 + r 2 —2d . r cos 0 
ds — r. dr . dd, 
