.374 BELI. SYSTEM TF.CllNICAE JOURN.IL 



Coiuinuin,i; this process it is easy to show that, for positive indices 

 {m positive), 



A,n = -h,„C,„A,n^x (306) 



where C,„ designates the continued fraction 



1 



\ — hmhm-\-l 



1 



1-// 



m+I"m+2 



l—JlN-lhl 



The procedure for the coefificient ^_,„ is precisely similar. For 

 con\enience we write A-,„ = A,„, Z_„, = Z,„, and r/Z-,„ = hm- In 

 this notation we get 1)>- precisely similar procedure 



^;« = -/4cu;„-i (307) 



where C,„ designates the continued fraction 

 1 



1 — //„,// ,„4-l 



1 



1 —flU + l}l»i+2 • 



1 — hN-ihN 



We now put the index N equal to infinity and the continued fractions 

 Cm and C'„, become infinite instead of terminating fractions. 

 Collecting formulas we now have 



A' - —Ji' r' A' , 

 and 



Ao^Jo-{hcA, + h:A\) 

 whence 



Jo 



Ao = 



1 — hohiCi — ho'hi'Ci 



'h I r I ' 



The coefficients are thus all determined in terms of Jo- 



The practical value of this method of solution will depend, of course, 

 on the rate of convergence of the continued fractions. While no rigor- 

 ous proof has been obtained, it is believed that they are absolutely 

 convergent for all ph\-sically possible systems, but this question cer- 

 tainly requires fuller inxestigation. Ne\ertheless any doubt regard- 



