Thickness of the Earth's Crust. 11 



had suddenly become one solid body and thereby suddenly re- 

 tarded the rotation. This, before proceeding, I will illustrate 

 by an example for the use of your mathematical readers. 



3. Suppose a spherical shell or crust of mass C to have within 

 it a solid spherical nucleus, of radius b and mass N, fitting it 

 exactly ; and the crust to receive an angular velocity of rotation 

 around an axis fixed in the crust, the nucleus at that moment 

 having no angular velocity ; but suppose that a slight force of 

 friction between the surfaces gradually generates a rotary mo- 

 tion in the nucleus; and suppose this force to vary as the differ- 

 ence between the angular velocities of the crust and nucleus — 

 that is, of the surfaces in contact. Let co and ool be the angular 

 velocities at the time t, k and k 1 the radii of gyration of the two 

 bodies, F(a> — o 1 ) the force at the equator of the nucleus which 

 represents the friction between it and the crust. Then the equa- 

 tions of motion are 



dco ¥b , ,, dco' ¥b , ,. /1N 



di = -Ck^ (0 -° ) ^ Ht=WF^ (0 - 0>] - ' ' (1) 



Suppose also that /3 would have been the angular velocity, when 

 the primitive impulse was given, on the hypothesis of the crust 

 and nucleus being rigidly connected so as to be one mass. Then 



/3(C£ 2 + JW 2 )=«C£ 2 (2) 



Subtracting the second of equations (1) from the first, putting 



F <i + i) =c > ( 3 ) 



and integrating, we have 



G)—ft)'= const, x e~ c K 

 When £ = 0, co = u and co' = 0; 



,\ cD — (o l = ae~ ct . 

 Hence, by the first of equations (1), 



dv_ ~Fbu _ ~Nk ,2 cu 



dt = 



C& 2 * " C^ + M' 2 * Ct> by W' 9 



a(l — e~ ct )', 



and also 



C£ 2 + 1W 2 

 N£' 2 



=0(l+|^«-*),by(2). 



The first of these expressions shows that the angular velocity 

 of the crust begins with ol ; and when ct becomes very large in- 

 deed, it is reduced to jS. Hence the effect of the constant friction 

 of the nucleus against the inner surface of the crust is at last to 

 reduce the velocity of the crust to what it would have been at 

 first if the crust and nucleus had been one solid mass. 



