264 Prof. Challis on Hydrodynamics, 



lows that 



dcr //+F f + W 



and <?= -jq, it follows that 



av \ r 2 r ) 



Consequently by integration, 



As the sphere is supposed to be very small and to disturb the 

 fluid through a very small space compared to the magnitude of 

 the waves, it will be assumed that wherever r is very large cr is 

 equal to the condensation cr } of the waves at incidence. Con- 

 sequently yjr(r, t) is equal to a v and does not contain r, cr i being 

 supposed to be a function of the time. Again, from the equa- 

 tions 



Ka *dr + dt - U ' * a 'rdd + dt ~ U> 

 it will be found, assuming U and W to be wholly periodic, that 



/ca~ \ r s r 2 ) * 



Ka \ ir r z r ) 



/j and F x being respectively put for \fdr and J Fdr. It will 

 now be supposed that the incident waves are denned by the 



. Sir/cat , , . . i , 



equations w l =/cacr 1 =m sin — r- — , w x and <r 1 being considered 



uniform at each instant through the small extent of the influence 

 of the sphere's reaction. Also since by hypothesis the functions 

 / and F are periodic, and must have their period determined by 

 that of the incident waves, it will be further assumed that 



f=m 1 sm — (r— mt+cj and F=m 2 sin ~ (r+/cat± c 2 ). 



Now if c be the radius of the sphere, the condition must be 

 satisfied that U = 0, where r=c for all values of 6 and at all 

 times. Hence, substituting c for r in the expression for U, we 

 shall have, independently of the time, 



c 



which equation furnishes two relations between the arbitrary 

 constants m„ m 2 , c v c 2 . 



I shall now introduce a condition which has special reference 



