Coefficient of End- Correction. 251 



Again, since 



Y 2=0 = E + 2«r JoC^rOT") 

 r 



= A + B{l-v 2 ) + C{l-*7 2 ) 2 + T)(l-^)\ 

 if we multiply by ot and integrate from to 1, 



E=A+iB+JG + iP (ii.) 



In the region I. we have to make V have a definite value 



over z = 0, tzr^l and V = when ~=+co and -^-=0 when 

 5=0, <*>!. ° 2 



Introduce * new variables f f which satisfy the relations 



•«ee OT =v / (i-rjii+n- 



Then the solution of Laplace's equation which makes V = 

 at infinity and finite when w = is 



V=2AJ»„($)Q»('f). 



n 



where P n is the first, Q n the second Legendre function, and 

 i is the imaginary v / — 1. 



But when ^ = 0, 



«r>l, |=0, |^=0 



and therefore ^ =0 when f =0. 



Then only even integers n may be taken, n — 2r say. 

 When c = 0, w< 1, f =0, so that 



Vz=o=2A 2r P 8r (f) 0^(0). 



r 



Let A 2r Q 2r ( v 0) = t 2 r- 

 Then, since when f=0, 1 — -& 2 = i; 2 it follows that 



S&2rP2r(f ) = V_. =0 = A + B£ 2 + C£ 4 + Df . 

 r 



This equation will yield the coefficients b 2r - 

 When 5=0, ot<1 or J = 0, 



^~ far 



=-^A 2r p 2r (i)[^Q 2r (/r)]_ o . 



* Jeans, • Electricity and Magnetism/ p. 252. 



