150 



Dr. T. J. I'a. Bromwich 



solutions would not be valid in all nngular space. Thus, on 

 making n-^-^0 and ?? 2 — ^tt, it is evident that both terms in 

 equation (2-32) will tend to zero: and accordingly the value of I 

 is zero. 



Returning now to the form (2-3), we see that since B and C 

 are positive 



S=°> S=° (2 - 4 > 



at all points : and accordingly 17 is independent of 77, £ (or 



It now follows from equations (1'6) that Y = 0, Z = 0, /3 = 0, 

 y = 0: and accordingly ail the components of force are zero, 

 as already stated. 



It may be noted that if the solutions do not refer to all 

 angular space (so that the arguments used above may no 

 longer apply), we can often draw the same conclusion, by 

 using the boundary-conditions laid down in the special problem, 

 to prove that (2*31) is zero. 



§ 3. Special form of solution in terms 0/ spherical 

 polar coordinates. 



Taking 



S=r, 



■0=0, 



e=#, 



and 



A = l, 



B = r, 



C = rs'mO, 



the equations (1*5)-(1*7) yield the general solutions in 

 spherical polar coordinates. It will be convenient to state 

 the results here in a complete form, which wall be useful 

 for subsequent reference : 



X = 



?3 2 U _/*Kyu 

 1 yu _n /^ 7 \ 



r'dr'dO p'dQKc ~dt J' 



_i a»u ,1 a_/A*av\ 



where p = r sin 0, 



m " Br 2 c» 3**' 



1 3 2 V ljd /KBU\ 



c? 



1 d 2 V _1B_/K3U\ 



(3-1) 



(3-2) 



