356 
DR. P. E. SHAW ON THE NEWTONIAN CONSTANT OF 
therefore 
therefore 
cos . ds 
~R~ 
F = 
( d—r cos 0) r . dr dO 
d 2 -t-r 2 —2d . r cos 6 
J_ / d 2 -r 2 
2 d \ d 2 + r 2 — 2d . r cos 0 
0 2Gm,m 2 f a r dr 
2 —5 s — 2 —r 
a Jo d 
2Gm 1 m 2 /d. 
I 
for finite cylinders (M andm) whose length (l) is great cf. with the distance between 
the axes, we have approximately 
F = 2 G^-y -Ifd 
L L 
= 2 GM m/ld .II. 
hence increase in length of the cylinder, for constant mass, decreases sensitiveness, 
since l increases faster than d decreases. 
The exact solution for finite cylinders would be exceedingly difficult and for present 
purposes is not required. 
3. The Sensitiveness Attainable for Spherical Masses. —Many factors are involved. 
Let a — arm of the torsion balance, n = rigidity of fibre, r and l equal respectively 
radius and length of fibre, 6 — angular twist of fibre ( cc sensitiveness), K = moment 
of inertia, and T = period of the torsion system. Using other letters as before, we 
have 
2GM ma _ irnr' l 6 
d 2 ~~2T’ 
couple = 
therefore 
but T 2 = 8 tt 2 K —— ’ therefore 
7 rnr 
4GM?na l 
~d 2 ' 
4GMmaT 2 
d 2 . 8 tt 2 K ' 
I. 
II. 
* 
When M, m are close together, on being small, we have 
therefore 
where 
but Kima 2 , therefore 
cT 
4GMmaT 2 _ GW’onaV 
(6 M/tt) 2/s . 8 tt 2 K Iv 
C = constant, 
6 = CM 1/s T 2 /a. . 
. III. 
