144 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



in comparison with the distance from it of the nearest portion of 

 the original boundary. 



Let be the velocity-potential of the motion when the sphere is absent, 

 and KJ, K 2 ,... the circulations in the various circuits. The kinetic energy of 

 the original motion is therefore given by Art. 55 (5), viz. 



where the integrations extend over the various barriers, drawn as in Art. 48. 

 If we denote by + </>' the velocity-potential in presence of the sphere, and 

 by T the energy of the actual motion, we have 



the cyclic constants of <' being zero. The integration in the first term may 

 be confined to the surface of the sphere, since we have d$/dn = and d(f>'/dn = 

 over the original boundary. Now, by Art. 54 (4), 



so that (ii) reduces to 



dS ......... (iii). 



Let us now take the centre of the sphere as origin. Let a be the radius of 

 the sphere, and u, V, W the components of its velocity in the directions of the 

 coordinate axes ; further, let w , v , W Q be the component velocities of the fluid 

 at the position of the centre, when the sphere is absent. Hence, in the 

 neighbourhood of the sphere, we have, approximately 



where the coefficients A, B, C are to be determined by the condition that 



- 



for r=a. This gives 



4=ia(o-u), 



Again, -*-- j r 



when r=a. Hence, substituting from (iv), (v), and (vi), in (iii), and re- 

 membering that J/A- 2 ^=^a 2 . 47r 2 , jjyzdS=O t &c., &c., 

 we find 



