312 



H. XAGAOKA. 



lions îiinkc tlic foniiiihic jiist Lj'ivcri less r-onvcuioiit for (■.•ilciihilion 

 tliîiii llic i(<i'iniil:i(' ü'i\('ii lu'low. 



As :ili(':i(lv iiUMilioiicd, tlic Inrin of tlio intOL'Tal /sliows \]\:\t wlioii 

 llie liniirs lie iVoiii o to -, il Ik'couk,'.-! ('(|ii;il lo - J °(ç). Tiiiis J 

 iiirliidc:-! IV'ssel's i'iinctioii oi'ilic llr.u kind willi index :im :i |):ii'1 i('id:ir 

 case. l>v a s])f'(:'i:d transiocnialion, J can Ix' inadc to d('|)('nd on 

 J° ()iz) as will now 1k' shown. 

 Tuttinii' Il ^ cos if , wc \v.\\v 



nu 

 — I dii. 



J 



Expanding' 



V^ I — \C- 



:zir m a r Olivier series. 



1 



1 "^ 



^ [ — li- 

 ]\Inlti])l_vinii' tin's l)y ^' ^ ", and intea'i'atin^- 



After a simple reduction, "we liave 



/: 



ju - — 



?' ./?«/ 1 



Ivjiiatino- the real and imaginary ])arts of both sides of tlie 



equation 



(0 /:^^*,, = -Lj 



V 



z h - ~ sin ^ M > 



1 



Ç- — n- -- 



2 r ro.s ç ?/, V 



^nJ° [n ~) sin [n - ii] 



Ç- — n- -- 



I 



