422 Archdeacon Pratt on the Method of determining 



surfaces gradually generates a rotary motion in the nucleus; and 

 suppose this force to vary as the difference between the angular 

 velocities of the crust and nucleus — that is, of the surfaces in con- 

 tact. Let ft) and w' be the angular velocities at the time t, h and y 

 the radii of gyration of the two bodies, F(ft) — o)') the force at the 

 equator of the nucleus which represents the friction between it and 

 the crust. Then the equations of motion are 



Suppose also that ^ 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 



/S(C7c3 + m'2)^^Cyt2 (2) 



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



^Kc^+w^)=^' (3) 



and integrating, we have 



ft)— Gj'= const. X e""'. 

 "When ^ = 0, 0) = a and ft)' = ; 



.*. ft)— ft)' = ae~'''. 

 Hence, by the first of equations (1), 



It- QW' - c/t^+m'^" ,t^yc^;, 



and also =^(l+^.-), by (2). 



The first of these expressions shows that the angular velocity of 

 the crust begins with a ; and when ct becomes very large indeed, it 

 is reduced to /3. 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. 



We may conclude perhaps that the same effect would be produced, 

 though in a much longer time, if the interior were not a solid sphere, 

 but a fluid mass. 



The above reasoning shows that if the disturbing force producing 

 precession and nutation did not exist, and the interior of the earth 

 were fluid (whatever the thickness of the crust), it may be fairly 

 assumed that the motion of rotation of the crust would now, the earth 

 having existed so many ages, be exactly what it would have been 

 had the earth been one solid mass, all difference of motion having 

 been long ago annihilated by the internal friction and viscidity. 



