200 ilir Robb, Discussion of a difference equation relating to 



afterwards justify the assumption by investigating the coavergency 

 of the series. Let us assume then, if possible, that 



r,. = r(i-.,|.-,,|;.-a,|;-,..). 



where a^, a«, a-^, ... G and K are quantities to be determined. 

 We have then 



^ »-i — -* l^ /i'* K'^"' K'-''^ 



i„+,-lll ^^^^„ K^K-n K^K^n 



-2hT,. = -2bT{l-a,^^-a,j^^-a,j^^--), 



thus 



T„-, + Tn+,-2bTn-a + y^^ 



Thus we get 



K-^\-2{b-\-^)l = 0, 



an equation which determines K. 



The calculation of the successive coefficients (as well as the 



value of K itself) may be considerably facilitated by the help of a 



table of hyperbolic functions*. In order to do this let us write 



c s 



cosh &) = 6 + ^„ = l+ »^^jis (^'^^ + Si'iu-E) 



and ff = 



2T' esT' • 



* Good tables of these functions by C. Burrau have recently been published; 

 Berlin. 1907. Georg Eeimer. 



