inside a Hollow Unlimited Boundary. 

 we get on integration 

 i/^ = C\ — fxa cos 6 — sin 



235 



«r-l <*-*>}£*' + 



^.n+i 



1 , , N r 3 „ , " f a n r 



P. 



...(51), 



where P n ' stands for dPjdO. (48) and (51) belong to points 

 inside the sphere r = a. For points outside this sphere (38) 

 becomes 



^ = Vrcos0- b f-l^-b o )^P 1 + l(f,-b o )^P 2 + 



+ 2 



(^P.)...(52). 



Putting in (49) for the value of <£ 2 from (52) and integrating we 

 get 



r Yr 2 

 yjr 2 = G 2 — b o acos0 — sin ^-sin^ + 



+ §fc- W |jY-j<*-w£p.'-S( , ^iv)] (53). 



10. If in (51) i^! = is to be the equation of that particular 

 stream line which cuts 00' (fig. 1) orthogonally, it must be satis- 

 fied by = ir and r = OB, so that we must have 



G 1 = — fia. 



.(54). 



OB is most easily got from (36) by putting in it dfa/dr = and 

 = 7T. We thus get from (36), (45) and (46) for finding OB 



0-^-5{«^-|0.-w}+^.0»-W-fc...(««). 



Remembering (T.L.F. Art. 14) that P n ' vanishes when n is even 

 and = 7r/2 we get from (51) for finding OE 



1 - -,».-\- a F-|(^-6 )|£ 



(56) 



accurately. We see from this that if V = we must for OE to be 

 real have 6 > /a. We have next to get the value of 2 in (53). 

 This may be done in two ways. We can either suppose V to be 

 finite and remembering that from (35) 4//,o.7r is the quantity of 

 liquid given out by the source in the unit of time, and making the 



