( 287 ) 



as (1+^^) (1-2,?) = (1— 2/?+2.t-,i-3.#^) - ..i-?(l-/i). 

 Now we get fiirtlier tbr the factor between [ ] : 



J ,7^ + 1 



2(.c + l)' .7;+l 2(.«+l) 



+ ^r4T^ ^(1-^) (H 2(,.-l).'i-3.#)(.v;4^.^)'- — 477, (?^(l-/?r(.r + y,)'. 

 2(c'?; + l)' 2(a- + 1)' 



In this we have : 

 - — - ^{i-i3){.>:^^<py = —4- .i(i-.?)(^-+<r) + :~-—8{i-i3Mx+rp), 



2 A'-f-l 2(;r-|-l) J(a'-|-1) 



80 that the factor mentioned becomes : 



^ + ^^./?(l-/?)(.^ + ^/')+— i77.|:?(l-i^)(l+2(..-l)/i-3.^^^ 



in which the siipplementaiy piece S is represented by 



2(.r + l) .t-+l 2(.r + l)- 



The first two terms give ^^(1 — /^) /(-'■ + ^/>) : the two following 



2(.f+l) 



1 



ones 



2(.r + l) 

 represented b}' 



1 



/5'(1 — /^)''/('<'- +</')' ; so that the first four terms can be 



^(1— ^)^(.7; + g)) 





2(.^+l) 



and this is evidently the fifth term apart from (he sign. So the 



supplement S is := 0, and we finally get : 



y_^, 2 1 r 3 X 



Z X -\- 1 



1 



3(l + 2/r 



+ 77-TT^ ii(l-/?)(l + 2(..— l)/?-3,./?')(^-4-rp)^ 

 If we henceforth put : 



20 



Proceedings Royal Acad. Amsterdam. Vol. XIV. 



