inside a Hollow Unlimited Boundary. 233 



Outside the sphere or for r > a we will take for velocity-potential 



fc-Froosfl-^ + Sf^P,.) (38). 



In case the boundary is tubular at infinity, the 1st term gives us 

 V the constant velocity at infinity. If we want the boundary to 

 resemble a hyperboloid or paraboloid at infinity, F'will be 0. The 

 2nd term in (38) represents the fictitious source at 0, which will 

 ultimately be seen to represent the effect of the boundary on the 

 velocity at a distance. In the 3rd term in (38) n is supposed to 

 have the same values as in (35) excepting zero. Then 



d(f> 2 

 dr 



= FcoS + ^_£j M% + i)^> w j (39)) 



l^-Fsintf+if 6 ^ 1 ^ (40) 



r d6 Sm ° + 7 V r»+ 2 dd J ( U) ' 



It will be seen either from (35) and (38) or from (37) and (40) 

 that the motion is symmetrical with respect to Ox. 

 On the surface of the sphere r = a we must have 



£ + -f(^)-K»i + M^(- + i)S)..^ 



*&%)--V*ne + lfe<*-) (42). 



a dd J x \a dd 



These must be true for all values of 6 from up to a certain 

 definite angle, which may be 7r/2 or any less angle. 



In (42) we must have generally a n = b n except when n = 1, in 

 which case remembering that - sin 6 is dPJdd, we must have 

 b T = a x — aV. Since dP /dd is zero, a is arbitrary. Putting these 

 values in (41) it will become 



^ + a ^l + li(na n P n ) 

 a a a 2 



h 2 i * 



= VP 1 +-?- n (a 1 -aV)P 1 -±X(n+l)a n P n 



Uj \h (Jo 2 



CO 



or (ji - b ) + 3P X {a 1 -aV) = -X (2n+l)a n P n (43). 



2 



Whether V is finite or zero (43) can be satisfied in an infinite 

 number of ways, so that we get an infinite number of solutions of 

 the problem proposed. The method is as follows. 



9. By Todhunter's Laplace's Functions (T.L.F.) Arts. 28 and 

 62 it can be shewn that between the limits = and tt/2 any 



