140 W. D. Lambert — Mechanical Curiosities 



for the moment the torsion of the suspending fiber. 

 Under the influence of the earth's field of force the rod 

 tends to turn into the plane of the prime vertical, because 

 in so doing it falls. How may it be said to fall when the 

 height at which the center is suspended remains 

 unchanged? Let us consider the curvature of the sec- 

 tions of the level surfaces in the meridian and in the 

 prime vertical. It is easy to calculate that (in the normal 

 case) the curvature of the meridian section is a maximum 

 and the curvature of the prime vertical section a mini- 

 mum. The two sets of traces of the level surfaces on 

 the planes of meridian and prime vertical are shown in 

 ^g. 5, the prime-vertical traces being represented by full 

 lines. The level surfaces are arbitrarily numbered with 

 the numbers increasing outward, which corresponds to 

 values of the potential increasing with height. The line 

 PP^ represents the suspended rod of the Eotvos balance. 

 We see that its ends (where the weight is concentrated) 

 are in the surface numbered ^'3'' when the rod is in the 

 meridian but are in the lower-lying surface marked ^ ^ 1 " 

 when the rod is in the prime vertical. Thus there must 

 be a force tending to swing the rod from high to low, or 

 towards the prime vertical. 



The same thing may be seen from a direct consideration 

 of the forces acting. The moment of these forces comes 

 out 



m r sin 26 



^ \R N/ 



In this expression 2m is the mass of the rod, % its length, 

 B the angle between the plane of the rod and the meridian, 

 / the intensity of gravity, which acts vertically at the 

 center of the rod, and R and N are the radii of curvature 

 of the meridian and prime vertical sections respectively. 

 If we substitute for R and N their values in latitude </>, 

 there results for the turning moment very nearly^ ^ 



— ^ efcos'cf, sin 20. 



In this expression e is the ellipticity of the earth and a 

 its equatorial radius. For Eotvos 's apparatus we may 



^^ The turning moment is zero for either (9 = 0° or ^ = 90°. The former 

 value, corresponding to the rod in the meridian, gives unstable equilibrium. 

 In the prime vertical equilibrium is stable. 



