154 - 10 - 



Considering any haK-perlod, and changing the independent variaole in (l«) to a, we 

 obtain 



-.S^? + \^ * 2{ !l!£i2 ♦ (i_n)-l, b„^ = (m) 



■utting 



(«5) 



and using (l7), this equation Becomes 



i£, * i { tia) ♦ I } — - (n - 1) Jj { t(a) -1)0 M 



•here the suffix (n,s) is omitted from/3 for the present, and 



,(a) - -^^ ^.3 _ ^-3y ^ ^ (^ , ^) (^-3 . ^, 



(■17) 



When y = iJ/3, the equation (46) reduces to 



+ (n- 1) J (|* j^a) a-2 I /3 ■= (») 



The general solution of this eqiation is required for the range l<a ^ ot • It may be noted 

 that when O" -• the eqiation reduces to a Lani4 eqiBtion, and can De transformed to Legendre's equation 

 for P "^ (t)) by the subst itut ion tj = / ( 1 - cl), but m, r and T) ire all complex or imaginary. 



The equation has regular singularities at the origin and at infinity, and also at the four 

 zeros of 



a = t. V -I ("n* "± ' '%'*')• 

 At the four latter singularities the indicial eqiation is the same, viz. 



2 M (m - 1) * M = 0. 

 so that the exponents relative to any one of these singularities is or ^. 

 Hence two fundamental solutions of (19) can be obtained in the forms 

 = A^g.(z) - A^* A^2* a/ ♦ 



where 2 =a-at, anda. is any one of the above four singularit ies. 



By considering the positions of the singularities in the complex plane, it can be seen that the 

 series expressions (50) relativetoa= 1 (i.e,withz = a-l)are convergent for z < 1, i.e. for 

 a < J, and the series with z = a - a are convergent for z <a - t, i.e. for the whole range (except 

 for a = l) over which the solution is required. 



The 



(50) 



