58 ME. A. B. BASSET ON THE MOTION OF 



Now, 



d /v\ 1 dv v 

 dr \r/ a a dr a* 



Hence, if <u were uniform we should have 



(<lr\ 2a> [" > T cat 



, ) = 2u + T- exp. ( 2w </vt/a tr) du 

 (If/a \/7r J o 



Putting u + Y/ (vt)a = )8, the definite integral 



.^/a.fvA[ ^A"0 , W 1 



1 2 a 3a ' J 



y 



= 2 a 



if i>< be small ; whence 



/dv\ 2co f . 



[ -7-) = o --- 7- v^ 



\drja av^7T\ v 



ao> 



2a 

 Changing t into T, and w into F' ( T) dt, (47) becomes 



Putting 



(48) becomes 



10p . 7 



- A 1 , fe = X, 



<ra- 



" + x " + ; vb i >' <' - ' - " * 



((-r) i = 4^N. (49) 



Now we have supposed the motion to have commenced from rest under the action 

 of the couple N' ; but if the sphere had initially been set in rotation with angular 

 velocity fl, and then left to itself, it can be shown in the same manner as in 7 that 

 the equation of motion would be 



