304 «or«,W^,«» «v L. EULERI OPERA POSTHUMA. Amiym. 



1 /. n — 2 n(n-*-3) — < n(n-i-4) (n-»-5) —6 n 



Hae scilicet binae series conjunctae verum valorem ipsius cos ncp^ existente cos </? = aj , exprimunt, 

 idque sine ulla restrictione, ita ut pro n omnes numeros tam negativos quam positivos, tam iategros 

 quam fractos assumere liceat. Ubi quidem per se est perspicuum , sive ipsi n tribuatur valor nega- 

 tivus quicunque, sive idem positivus, easdem binas series ordine mutato resultare. 



15. Jam binae hae series conjunctae pro quovis numero integro n eosdem cosinus per for- 



mulas iinitas exhibent, quos initio recensui. Pro casu quidem /i = res est manifesta, cum inde 



11 

 fiat cosO^J = - H-— = i. Reliquos igitur casus simpliciores evolvamus : 



I. Sit /1=1 eritque 



—2 1.2 —i 1.3.4 —6 1.4.5.6 



/, 1 —2 1.2 —i 1.3.4 —6 1.4.5.6 —8 ^ \ 



1 /. 1 -2 1.4 —^ 1.5.6 -6 1.6.7.8 —8 ^ \ 



4x\ 4 4.8 4.8.12 4.8.12.16 / 



ubi potestates negativae ipsius x sponte se destruunt, uti ex sequente repraesentatione fit perspicuum: 

 £iuu<ji^ 1» N^uai^, % .^Uf^qii^i»- uoa ?.«uiafi*> niih» jhfj-imjn. 



1 — * 1.2 —3 1.3.4 —5 1.4.5.6 —7 

 ^os^=x-^~x -^j-^X -^^^x -^X12X6* -^^' 



/ 1—1 1.1 —3 1.1.4 —5 1.1.5.6 —7 



4 4.4 4,4.8 4.4.8.12 



ita ut sit cos 90 = oj. 



: ;!iii');Hh';'ji>i) ti«'<! ihIiujuihi/ Hii >ijpd;i 'ii)J '.iifnfino 



II. Sit /1 = 2 eritque 



f,o^2<f^^xx(^i^\^" ---^-^^^' ^^ 



1 /. 2—2 2.5 —4 2.6.7 —6 2.7.8.9 — » ^ \ 



Sxx \ 4 4.8 4.8.12 4.8.12.16 / 



quae binae series ita ordinate exhibeantur: 



o n i n 2.1 ~2 2.2.3 — * ^ 2.3.4.5 —6 



0082^ = 20^-1-2. — a. -2.^—0. -2.^-^^^^ -etc, 



M ■ U:-i ..ui J - ■ 



• • ~* l -^2 1.2 "r-f ' -1.2.5 -^s - 



ubi potestates negativae omnes tolluntur, ita ut prodeat cos 290 = 2cca? — 1. 

 -•>Im.jjj gj^ w = 3 eritaue ™*"V* '^^ tMiaiiT^loq. oi9t)fjl«flfi2 oJi«inp*»i iisllB 

 t9 l^yilB^i n 2iiHfU).(/M inifiM •uiJi^imi''! ..^ivnUilj.') 'UJV'^»'''— 4' «» o 1 ^ — s •'«'♦bsJitni 

 cos 3g) = i^s?» j( 1 — -?-a; r- OaP — ^^«? -- . l IL^ — -etc.) 



^ ..■T<TT \',-,i.-*.,4 ;•■, '.■■:■, .yT' !■,■■■- 4.8.12 T^ — r.4:8.t2il6 <0-) ~ ( , / 

 / ,1 /- . 3 —2 3.6 —* 3.7.8 —6 3.8.9.10 — » , N 



^^1-^ * -^16?^X?-*"T^ -^478^ -^4X12^ -*"4XT2Ti6^ rt-eM^.-). 

 qui termini hoc modo in ordinem redie-antur: . . ,1 * i 1 j i • 



