.^89 





(-1)" 



06- -1)2"+ 



/— 2 



w/ r 71 + 



so that we find 



u {x) -— 



sin Xjt r f' (I) 



h 



^ J (•«—§) 



1— / 



dt — 



sin ^jr (l-X) 



jr 



mi) 



1 — 



[Gb) 



I /, nl^(-^''-^)l- 



2 7 -('-i) ^" o^'-^r- 



/(§)'-^§' 



5, In a similar way we find by starting from 

 ytv) = r(l-A) (l + ^-^.y^) 2 ^ _J^ ^ v^_l) 



^A 



■(-9 



kv)'^" 



n ! 



1— / 



^( 



sinXjt 1 , , 



(7(^') = .-^-(2-;.)r — 



A-(-l\ sin Xjt 



Jt 



2{-l)n- 



A'2«+^-l 



jr 1 / A+l\ 2ri+;.-2 



nir{n+-^ 



We can again notice here that for :; = the special forms appear 

 as with Abel's problem. 



Substitution in {la) and (16) furnishes the integralequation 



1+/ 

 '1 — /; 



f{w)=r\ 



¥)(j) ■ ƒ■ 



1— / 



v-ê) 2 1 i+A{e(.f-|)|w(§)fZs, (7a) 



with its solution ; 

 Proceedings Royal Acad. Amsterdam. Vol. XVI. 



38 



