452 

 I make in (4) the substitutions 



p = 3 , z = «*«'«' , 

 p = b , z =\. 



These substiliitions give the four equations 



^. + ^.. = ^^.. 



u.+ ü,+ u,,^.-L^^u,,, 



2^ 

 1 



3^ 



and eliminating i\, U^,, U^^ we get 



Now, taking s=i2h-\-l, we ma}' expand (/,, U,, U„ U^ by 

 applying equation (5) and we will get ['^^ = ^{2h -\- 1) expressed in 

 terms of the (piantities u(3), C(5), . . , C(2/i — 1) and of four power- 

 series in lu, the indices of which aie respectively 9^, -5-!^, rè-ö. tIt- 

 Since the formulae become somewhat complicated, I will consider 

 only the simplest case /i = l. Then we have 



336 4(::!) = — 900 U^ -{- 225 C', + 800 i\ + 900 U, 

 and hence 



689g(3) = 450u.'(36-241og».-331og2-81og3)-900 V — . h 



^ ^ ^ ^ ^—'(271 + 2)/ 2n 



;(:=:oo 



+225 y _J^_}^^^}^ + 800 y -^^ <i!!»!l! + 



^-; (2w + 2)/ 2n ii-'(2n + 2)!' 2n 



+ 900 V 



U (2n + 2)/ 2w 



»i=i 



I will end with remarking that the calculation of the sum of 

 the series 



o, 1 1 1 1 



' ^ ' I2h ^2h ^ 52;. 72A ^ 



may be conducted along similar lines. Eqnation (3) gives, if we take 

 k = 2h,y = ill, 



