194 Methods for the Solution of Special Problems [OH. vm 



By addition we eliminate a,, and obtain 



e, | e a = 



or, if we put - = u 8 



.(128), 



and from symmetry it is obvious that the same difference equation must be satisfied by a 



quantity u' 8 = . 

 e a 



The solution of the difference equation (128) may be taken to be 



U 8 = A 



where a, /3 are the roots of 



ab 



The product of these roots is unity, so that if a is the root which is less than unity, we 

 can suppose 



so that 



and similarly 



We now have 



To determine A, , we have 



so that 



f a + ba 

 where = 



Thus 



(1 

 rqp + F^3 + r= 



To determine .4', ^', we have 



a ab 



I "A 



