306 mmi^\^m m L. EULERl OPERA POSTHUMA. Anaiym. 



Q«~/3-l » (^-*-0 i^-^^) (^-*-3) ' • • («-*-^ ~ *> = Q/3^a-i , » ( ^-*-*) (^-^^) ("-*-3) • • » («-*-/^ - i) 

 ■^ * 4.8.i2.16...4« * 4.8.12. 16... 4/3 ' 



seu utrinque per 2'^'*'^'*'* multiplicando 



o2a n(^^-4-t)(|3-H2)(/3-4-3)...(«-i-i3-i) __ga^ n (aH-1) (a-t-2) («-h3) . . . (a-n^- 1) ^ 

 4.8.12.16.. .4« * 4. 8. 12. 16... 4/3 



Cum jam ia priori forma factorum denominatoris numerus sit =«, singulique per quaternarium sint 

 divisibiles, hos factores ita repraesentare licet 



; , , ! 4". 1.2.3 a = 22\ i . 2 . 3 . . . a 



simiii modo denominator alterius formae ita exprimi poterit 



k^,i .2.3. ..,/3 = 2^^.1. 2. 3.. ../3 



unde haec aequalitas ostendenda superest 



n (/3-H 1) (/3-1- 2) ( /3-i-.3)...(aH-/3-l) n (an- 1) (a-f-2) («h-3) . . . (a-H/3 - 1) 



17273.4...« "~ 1.2. 3. 4... /3 



quae per crucem multiplicata manifesto utrinque praebet idem productum 



/i.i.2.3.^ («-^-^— i). 



i9. Paradoxon ergo initio propositum satis distincte explicatum videtur, simulque ratio patet, 

 cur haec aequatio: 



«U'imi*m> _^ „/ n — 2 n(n-.3) -•* n(n-4)(n-5) — « . \ 



cos nq>=z2 ^ x" { i x ~+- -~~ x . '' — - x -h etc. ) 



tum demum sit veritati consentanea, quando n denotat numerum integrum positivum, simulque 

 omnes potestates ipsius x exponentes negativos habiturae expungantur, et cur his restrictionibus 

 non observatis, haec expressio in errorem praecipitet. 



20. IVunc autem pro casibus, quibus n est numerus fractus, veras series exhibere possumus, 

 quae cosinus ang-ulorum submultiplorum exprimant. Quod ut ostendam, sit primo n = — > eritque 



.1 Vx f. 1 —2 1.5 — * 1.7.9 —6 1.9.11.13 — « » \ i 



C0S-^^= ^2 (i— ^O) -^^X -sT^a. -8716724:32^ - «'^^•) 



1/7 1 —2 1.7 — * 1.9.11 — « 1.11.13.15 —8 ^ \ 

 -*-272^C^-^T^ -^8716^ -^8716724^ -^ 8.16.24.32 ^ H-etC.j,, 



quae in ordinem secundum potestates redacta dat 



J^ Vx / 1 l^ _1 l^ 1.7 1.7.9 1.9.11 \ 



*^*^* Y ^ "~ yS V 2a. 8^~*"2.8*3 8. I^ar^ "^ 2.8. 16a:5 8. 16.24a;« "*" 2.8. 16.24«' V 



ubi, si quiiibet coi^fficiens per praecedentem dividatur, haec resultat series: 

 •»?. i^tiTi ?!<vti(> 2^' T' 2^* 8* To' 12' n' **^* 



l^ J \_ 5 ^ 9^ 11 



V 4' ¥' 8* 10' 12' 14 



mww 1 yx/, 1 —1 1.1 —2 1.1.3 —3 1.1.3.5 — * 



, p- rntJvi? 1 yx/ 1 —1 1.1 —2 1.1.3 —3 1.1.3.5 — * ^ N 



ideoque manifesto habebilur cos-r-^=^*M h — j = ^-?~' ^*^' constat. 



