146 W, D. Lambert — Mechanical Curiosities 



(2) Tlie smooth lake mth different elevations for its 

 two ends. 



(3) The tendency of large bodies to ^'fall" towards 

 the equator. 



(4) The tendency of a rod suspended horizontally like 

 the Eotvos balance to ^^fall" by twisting about the 

 supporting fiber. 



(5) The existence of great local irregularities in the 

 curvature of the level surfaces and the interesting 

 possibilities that the study of these irregularities 

 seems likely to offer. 



Appendix A. 

 Motion of a Sphere under Gravity on an Equipotential Surface. 



The general problem of a sphere rolling without 

 slipping on a rotating surface of revolution, that surface 

 being one of equilibrium for the attraction of the matter 

 contained within it combined with the centrifugal force 

 of rotation, would be somewhat complicated. In treating 

 the general problem it would be a legitimate procedure, 

 though not necessarily the easiest one, to treat the sur- 

 face as without rotation provided the following additional 

 forces were taken account of: (1) the centrifugal force 

 of rotation; (2) the compound centrifugal acceleration. 

 To obtain the resultant of these additional forces an 

 integration would usually be required. In the restricted 

 problem proposed, where motion is confined to the meri- 

 dian, the procedure suggested is evidently the simpler 

 (1) because the effect of the ordinary centrifugal force is 

 implicitly contained in the gravity potential used; (2) 

 because the compound centrifugal forces do not affect 

 the motion when the latter is confined to the meridian.^^ 



Let h denote the radius of the rolling sphere. Consider 

 the locus of the points at a distance h from the level sur- 

 face on which the sphere rolls, the distance being meas- 

 ured outward along the normals to the level surface. 

 For definiteness, let us call this new surface to which 

 the center of the rolling sphere is evidently confined, 

 the parallel surface and let ds be an element of meri- 

 dional arc of this parallel surface, s being reckoned from 

 the equator toward the pole. 



^ For the equations for motion relative to the earth, see Eouth 's Advanced 

 Rigid Dynamics (5th Ed.), p. 27, or similar works. 



