60 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



enclosing the similarly formed body of the earth at rest within it, 

 while the physical forces at work (force of attraction) are perfectly 

 independent of the condition of motion of the mass of the earth 

 It therefore appears profitable to approach the above stated prob- 

 lem of the absolute motion under the influence of the force r u? 

 sin <p directed towards the pole and to treat it at least approximately 

 just as the problem of the oscillations of a pendulum is solved par- 

 ticularly in the case of infinitely small amplitudes. 



In the vicinity of the pole sin <p changes only very slowly, but r 

 very rapidly; no great error will then be committed if we give to 

 sin <p the limiting value i at the pole and neglect the corresponding 

 component of the motion parallel to the earth's axis, that is to say, 

 the motion is to be considered as taking place in a plane; the error 

 is thus purely geometrical and easily estimated inasmuch as the 

 forces arising from the special form of the surface are already taken 

 into account. Adopting the coordinates x and y in a tangent plane 

 and the origin at the pole the differential equations of motion are 

 therefore as follows: 



d 2 x x , 



= — rur - = — xor 

 dt 2 r 



d2 y 2 y 2 



= — rur- = — yar 



dt 



0) 



(Strictly speaking these equations apply to the absolute motion of 

 a liquid particle parallel to the level bottom of a circular vessel 

 revolving with the velocity co, in which the liquid is subjected to a 

 fi irce perpendicular to the bottom surface — in so far as this abso- 

 lute motion can be considered as entirely unimpeded.) 



Equations (i) agree perfectly with those on which is founded the 

 theory of oscillations in an elastic medium; they can (for example 

 by the substitution of x or y = e xt ) be integrated separately and 

 lead to the final equations* 



x = a sin tot ) 



(2) 



y = b cos wt j 



U r = cud cos cot 



(3) 



U y = — bco sin tot J 



5 Compare, for example, the following treatises on physics: Wullner: 3d 

 Edition, I, pp. 443 and 450; Mousson: 2d Edition, II, p. 531; Miiller: 7th 

 Edition, I, pp. 278 and 281. There will also be found given in these places 

 the geometrical representation to be spoken of presently. 



