302 L. EULERI OPERA POSTHIJMA. Anaiym. 



{xx — 1 ) dds -+- xdxds = nnsdx^ , 

 quac jam formam habet deslderatam, ita ut quantitas s nusquam plus una dimensione habeat, et 

 quantitas x ab omni irrationalitate sit immunis. 



10. Quia hic quantitas x in aliis terminis duas, in uno vero nullam tenet dimenslonem, facta 

 hujusmodi distinctione, ut slt , ,- , ;• i i -:i:> 



-.::i ^.-^vyi-f ,.'v,^s/, - im . xxdds -{- xdxds — nnsdx*—'dds = 



ponamus * ^ " 's = ax'"X- /3 x"' -''-*- yx'"-\ . . ..-^jux'"-'-*- rx'"-'-^-^^^' 



et facta substitutlone, potestas x'" — ' — ^ talem acclpiet coefflcientem 



QunVi .:..:;;... 



-fli mm ,iu^j(!^--r-4T^Mi^T^^-ir-.^}c+^^ (m — {m — i— 1), 



qiH Cum etanldscbre debeat, qtiantttas p"ex ^ ita definitur, ut sit 



(m — i) (m — i — 1) 



V = -^^ — ; — II, 



(m — t — 2)'' — nn ^ 



^ - '..••- e^' ^ o ^ i f\ i»i (m-t-2) (mH-i) ^ 



Statuatur jam pro mitio t = — i-2, ut iiat i^ = « et /« = 0, proditque «= — 0, quae 



littera ut mancat indefinita, esse oportet mm=^nn, ideoque vel m = nf vel m = — /i. 



11. ISostro autem casu est, ut supra vidimus, m = n, atque cc==2"J, quare posito 



5 = «a;"-f-/5£c"-*2_^_^^«-4 -^-jLix''-'-t-yx"~'-^--t-ctc, 



'•' *^ .. (n — «)(n — t — 1) (n — «)(n — «'— 1) 



■ (n — « — 2)'— nn "^ (t-»-2) (2n — » — 2) '^ 



unde sequentes prodeunt coefficientium determlnatlones: 



^ — n(n— 1) — n 



' 4(n — 1) 4 



-(n-2)(n — 3) , -^-n(n— .3) 



y— 8(n-2) '^~~ 4.8 " 



t, — (n — 4) (n — 5) — n (n — 4) (n — 5) 



o = — ^^ — — — y = -^- -' u 



12 (n- 3) ^ 4.8.12 



— (n-6)(n — 7)^ -»-n(n — 5) (n — 6) (n — 7) 



b 



16(n-4) 4.8.12.16 



etc. 



12. Posito ergo s = A{x-^y{xx — 1))", ob a = ^Ay habebimus hanc serlem, qua quan- 

 titas s exprimltur: 



S—^irAx^^d—'' -^ , n(n-3) -^ n(n-4)(n-5) -« n(n-5)(n-6)(n-7) "« \ 



s—^Ax\\ — jX ^—^x — ^g^2 ^ -» 4T8.12.I6 * ®^^V 



Quare si pro A capiatur -j orietur ipsa ilia forma, quam initio pro cos ncp assignavimus, existeute 

 a;=cos9J, atque nunc quidem patet ilJius expressionis in Infinitum conlinuatae verum valorem esse 



