486 MK. II. A. AVEBfe ON THE CONVEKGENCE Otf 



Let 



x = +iri, t = & + ir) l . 

 The condition is that 



sin cosh 17 < sin , cosh 77, ........ (10). 



5. The equation 



= ;/. (11) 



is satisfied by LEUENDRE'S associated functions 



Q+r"-'(*) and Q-.- 

 We find that when n is a large positive integer 



Q--fi(~\ _. -(n + BUr. II (/?) TT 2 -,.- tt -i +-!/, i 



Z 

 and 



_._ a _ _ 7a 



approximately. 

 Whence 



Accordingly the series 



J^Q^...^).^.-*-'^) ........ (12) 



H=l 



converges if 



\\-*\<\i-e\ 



and diverges if 



\l-x>\>\l-t*\; 



x 



the c's being arbitrary except that the power series 2 C H Z* has unit radius of 



x=l 



convergence. 



: |i_^| = |i_^|, ^ :,,:_:-,-; 



the convergence depends on the values of the c's ; we shall exclude this case. 



Let 



x = %+ii), t = 



|1-^|<|1- 

 if 



(r+T ,y_2(f_^)<( 



Now let us choose the c's according to the following law. Unless n is an integral 

 power of 3, c = ; ii 



"= 8 " c - 



