mi^ v^mi' L. EULERI OPERA POSTHUMA. Anaiym. 



18. Quoniam irrationalitas negfotium turbat, ad ejus speciem saltem tollcndam pono 

 /w tang; ^ = tang 6J, ut sit m m sin^ = cos^ lang^ co^ 

 hincque V(cos^ 6 -\-mm sin^ 6) = cos 6? "]/(l -\- tang^ w) = ^-^» 

 Hac substitutione facta nostrae coordinatae erunt 



/dOcoiOcoim . /«de cos sin ffl ' 

 et r= / -1 

 COS W ^" J cos o 



ubi notandum est angulos <9 et « ita a se invicem pendere, ut sit 



^ . .1 mdd du 



m tang 6* = tang w, ideoque - — j— = — g— 



COS (/ COS CJ 



Statuatur jam 



g)z=n0 — co, et ob cos9p = cosn^ cosGJ-i-sin/id^sin « et sin go = sin /i^ cos « — cosn^sinw 

 coordinatae ita exprimentur, ut sit ob tang « = /w tang ^ 



X = /dO cos 6 cos nd -f- /dO cos <9 sin nO tang w = fd6{cos 6 cos /i<9 -i- /w sin 6 sin /i^), 

 y = fdO cos 6^ sin /i^ — fdO cos ^ cosnO tang w = fdO{cos sin /i<9 — /w siu cos /i<9), 



quas formulas, quicunque numerus pro /i assumatur praeter unitatem, manifestum est semper esse 

 integrabiles. 



fflWT 19. Cum igitur sit cos cosnO = -^ cos (n — i)0 h- ^ cos (/i-f- 1 )^, 



sin^ sin nO = -^ cos (n — i)0 — -^-cos (/n-1)^, 

 V . iiii loi^ -^ ^ 



1 1 



ivrwii ziisi nstioantit) (joncos sin nO = ^ sin (/i — 1)^ -*- ^ sin (/i-i-l)^, 



2 ^ ^ 2 



2 



1 1 TJJ'" 



— sin ^ cos /i<9 = -^ sin (/i — i)0 — — sin (/i-i-l)^, 



substituendis his valoribus habebimus 



a? = y yd^ ((/?» -I- 1) cos {/i — 1)^ — (/w — 1) cos (/i -I- 1)<9), 



> ^^-^/^^((/w-f-l) sin (/i-— D^ — (/w— 1) sin (/i-Hl)^), 



unde valores integrales sponte fluunt: 



X = -t-(m-t-l) sin (n — 1) 5 (m — 1) sin (n -i- 1) 

 ai^60iblOOt> gijfllfiuti 2(n— 1) ~ 2(n-*-l) 



.... y= — (nt-f- 1) cos(n — 1)0 (ni — 1) cos (n -t- 1) 



2(n-l) ' 2(n-t-l) 



Hincque cum pro n numeros quoscunque rationales praeter unitatem accipere liceat, innumcrabiles 

 lineae algebraicae exhiberi possunt. , .. 



20. Gum igitur unitas pro /i substitui nequeat, casus simplicissimus prodibit, si ponatur n=0, 

 quo ergo habebitur 



(c = — (/n -H 1) sin ^ — -^ (m — 1) sin ^ = sin ^, 



1 . ^ 



j = — (/WH- l)cos^-t- -x-(tn — 1) cos<9 = mcosO, 



