Enodatio insignis cujusdam paradoan circa mulliplicaltonem angulorum. 303^ 



sicquc ralio aberralioiiis a valore cos /19?. est mauifesta, atque nunc quidem evidens est, cur sumto 



\ 

 n = 0, prodeat summa nostrae seriei =-^h reliquis vero casibus summa fiat imag;inaria, si quidem 



I 



sit a;<l. At si sumatur x=i, quicunque numerus pro n accipiatur, summa semper est = a^> 



eritque propterea .k.^j 



4 _ O" fi'^l^^.t- "^""-^^ n(n-4)(n-5) n(n-5) (n-6) (n-7) \ 



~ ^ .^:t^.. M^nh, 4.8^^-^ 4.8:12.16 '''-r 



quod certe est tbeorema non inelcgaus. 



13. Alio modo concinnius valor ipsius s exprimi potest; cum enim sit 

 a3 = cos9?, erit y{xx — l)=y — l.sm^p, 



" ^'0 80f) ^fiil 



ct ex notis sinuum proprietatibus ; ^ 



(cos (p -+- y — 1 .sin (py= cos ncp -+- y — 1 . sin n^. J>li'i^ 1 ~ j\ li<^ .1 

 Quare posito cos cp = x, erit , I , _ ^ g^^ 



-./ 4 • cMi n t A n — 2 n(n-3) " ^ n(n— 4)(n — 5) —6 ' \ 



cosncp-*-y — l.sm/ia) = 2"aj"(l — ~x -+- — ,—5— »; — ^ , ^ ,^ — ^o; -i-etc.lj 



^ \ 4 .4.0 4.8. 12 / 



« 



unde patet summam hujus seriei in infinitum continuatae esse imagiuariam, nisi sit a:=l, se.u 

 ^p = 0. Realis quidem semper erit dum sit a? > 1 ; sed his casibus non amphus ad sinus et cosinus 

 referri potest. Vehiti si xx = 2, ob 5 = ^^(1-1-^2)" erit 



/-./.^ . i\n^^-^fA ■'■■'»■. n(n-3) «(n-4)(n-5) . n(n -6) (n-6) (n- 7) \ 



(y2-4-I) _2^(^1-.--*--^;^. ^__^ _________ etc.J. 



At si ponamus x-*~y{xx — i)=y, fit a;= — — , unde obtinetur sequens summatio non contemnenda: 



--• -x .;:-: .-i 

 fjyy= 1 — - yy n(n-3) y* n(n-4)(n-5) _^y^ .-+. etc 



W-H,iy_ 1 •(«/-*- 1)*"*" 1.2 •(j/y-Hn*^ 1.2.3 * (w-^-i)'."^^^ 



quac cum etiam vera sit sumlo n negativo, erit 



/yy-^-iy _ I . ^ yy . »(»-*- 3) y* n (n-H4) (n-t-5) y' ^^ 



V yy y ~ 1 '(yy-*-!)' i-2 •(yy-Hi)*"^ 1.2.3 •(w-^i)«. . * ., 



' ii .' . '»h;i:>K» iJ)l 29JTW OBOld OfiiJp 



Sit porro ^^^ — = -, ct habebitur 



n z-l „(„H-3r '(?-!)* n(nH?^4^(n-4-5) («-<)' . ,^^ 



r = 1 -f--^ . — -f- -j-^ . --^^^ 1 ^-^. . -^ -4- etc. 



■ ■ ' , i 



ubi pro n omnes numeros assumere licet. 



xS = ry^' «< ,;toi «snrao «vile^gaa aotsJasJoq idji 



H. Hinc etiam alteri requisito satisfacere poterimus, quo ejusmodi expressio infinita deside- 



ratur, quantitatem cos nrp sine uUa rcstrictione exhibens. Sumatur enim exponens n negative, et 



cum sit cos ( — ncp) = cos ng> et sin ( — n<p) =; — sin lup , erit ex superiori forma 



/ . . 1 A n • — 2 n(n-^3) — * n(n-H4)(n-+-5) — ^^ .\ 



cosn^) — y— l.sm/i9> = ^(^l-4--.a> -^ -^^ x ■-+- > ^^^\^, 'x -t- etc.j 



,,,.,.- ,. .... „., ^ in. i' . v^iJi.M ii ohom r)ori JTilmTJt iiip 



addendis his formuhs pars imaginana toUitur, et summac scmissis daDit ^ 



