],S0 R. FUJISAWA. 



niid if// be ;i ratioiml iiite^Tal fnnotioii of the tliird degree, -y^ and 

 all the higher differential coefficients vanish, so tliat 



: coef. of h- i„ 4 j Ä 4^^ + ^ ;,= i!* + I /,' 4^ }' , 



r ! ( d.r 'J d.r (j d.r ) 



Ji-r 



X„_, - coef. of //' m --— -r -y;. + TT ^' -xj- + IT '^' 



or 



UT — r>r>of (^f 7)' 111 I ,, _ ,„ 



(/i — r)! ( dx '2 c?u^'- 6 djf 



wlienee follows that A",,., vanislies for all values of r for which 



2» 

 8 



2» 

 r > :lu — '2t\ that is, r)/- ^ i^//. Denoting the integral ]>art of—- by /, 



we have 



+ (--lrl.a...(2;/-2r-3)(2/yyX„_,+ ... + 1.8....(2«--3)X„| , 

 wliere 



"-'- ■ |(» - 2;-)! r! V f7.r / V '2 r/.r"-^ / 



■^C»-2r+l}!(r-2)ll! V7/7y/ \2d.r'0 \(]d.r) 



{n - 2r + 2;! (r - 4)! 2! V dx ) \ 2 dx'') \ G f/./'V "^ ' ' ' ) ' 



+ 



the last term beins" 



1 (ävS:~'u\ d'iiy 



(,,_.;j;.)!(J_). \dx) V«', ./W 



