688 SCHULTZ-STEINHEIL, THE ARGUMENT X«. 



where 



W,_ = 'I ( S<» -. Éf) - cos ,f Bf -'Lsf^> + ß,_+f'l-' + ff + 



+ i- {fi •■+«, + if? + i/f ! + :-s:^: {/; ■ ' + «. +/n - 

 -.-iikw''+/n-il^.(/'''+/n (^^) 



From (50), taking (56) into consideration, we have at last 

 for odd m 



'«„.=^0+'''""^"':°"-^''"'— v^S" ^-^ ef 5^^ (68) 



* ° 2 sin /c ^ sin z i sin z ^ 



"VYe may write (44) tor odd m: 



n .1 • , « . 3 , n . 4 



\ sinz+y^ - 



2 sin /t 



^ , ^ . — tp„ ■ sin z + cp, + CP, ■ * cos /, 



q) sin % — q) cos / — q) 



-\ r-^ cos jyiy, + 



2 sin /„ 



>: . 1 , n . 2 , n. 3 ■ 



q cosx + q.^ + q^ sin z 



H ~ sin iivA 



2 sin z 



and for even m: 



, , — (7), Sin -/ + (/;„ -h «), cos X 



''"■ ^^ ~ 2 sin z 



72 . 2 • ?i . 3 w . 4 



(/) Sin z — (5P^ — q, cos z 



H 2r-4 cos ?/2Z + 



2 sin z 



9^r ^ + '/'2 ' *^'°^ ''" + ^'i ' ^ ^^" ''" 



H r — -. sin m-A. . 



2 sin z 



From (61) and (62) it is evident tliat U, V and W are 

 enlarged by small divisors and afterwards these coefficients again 

 are partly diminished by the factor sin wzz a.nd, besides, several 

 great terms disappear in tlie sum. It is however possible to 

 avoid this inconvenience. Km may also be written in the form 



