156 W. D. Lambert — Mechanical Curiosities 



To form a numerical estimate of the equatorward force 

 let us take o- = 2.7 and p = 3.2 ; these figures correspond 

 roughly to the case of a mass of rock floating in a heavier 

 Ibasic magma. By neglecting the curvature of the earth 

 and the vertical variation of gra^dty, it is easily seen that 

 the spherical mass of rock will float with one-half of its 

 radius above the liquid,^^ that is, with 5/32 of its volume 



X^U.r°Xg.7>X/ ^°^;g'^'^ ' 



above the liquid and 27/32 below, that is, with C = - - 

 i\^e find also z^— —~, v — %Tcr\ and v^ = f 7rr'. Therefore 

 by (4) and (11) 



^f Trr^ X 2.i\ X /• 



The value r = 3 kilometers corresponds with an aver- 

 age elevation above the liquid of 833 meters, which is 

 about the average elevation of the land surface of the 

 earth; on the floating-crust theory the elevation of the 

 land above the sustaining magma would exceed this 

 amount. For this case the force X is 1/3000000 part 

 of gravity for </> c= 45°. This force X, though small, would, 

 if acting continuously without resistance, bring the sphere 

 from latitude 45° to the equator in about three weeks.^- 



If the preceding discussion be examined, it will be 

 noticed how little use has been made of the spherical form 

 of the floating body. A symmetrical body, like a paral- 

 lelepiped, if placed symmetrically with respect to the 

 meridian, could be substituted in the discussion in place 

 of the sphere. The parallelopiped would not be attracted 

 by the earth exactly as if the former were concentrated 

 at its center of gravity, but the error in assuming that it 

 would be is very small. The chief difficulty arises from 

 the fact that the resultant pressures have moments about 

 the center of gravity of the parallelopiped ; but it can be 

 shown that these do not affect the general validity of the 



^^ This is simply a convenient coincidence ; the problem of the floating 

 sphere requires the solution of a cubic equation. 



^- If the sphere were not constrained to float along a meridian, the deflect- 

 ing force of the earth's rotation would cause its path to take a curious 

 wavy or looped form resembling a trochoid; if the sphere were to start from 

 rest, the general direction of its advance, apart from the loops or undulations, 

 would at first be at right angles to the meridian. The discussion of the 

 exact form of the path is not necessary for the matter in hand, since the 

 time needed to reach the equator is of the same order of magnitude, whether 

 the sphere be confined to a meridian or not. Eesistance would make the 

 sphere move more nearly in the original meridian. 



