3152 .mvjtoW^st» $« L. EULERI OPERA POSTHUMA. m otMsbo Amiym. 



dv^ dz^ ndxdz nndx^ 



— = 1-— — , ergo 



vv zz xz Axx ° 



ddv ddz ndxdz n(n-*-2)da?* 



uDde facta substitutione : 



.<: 



4-020; (1 — x) knx (1 — x) 1- n (n -i- 2) (1 — a;) dx^ 



«g. .,..„.< „,„:;..,.,„-, -f-2x(2-3x)^-«(2-3x)dx' [=0. «" 



— nndx^ 



kx (1 — x) ddz — k (n — 1) dxdz h- 2 (2/i — 3) xdxdz — n{n — 1 ) zdx^ = 0. 

 Gum hic yariabilis x partim unicam, partim duas dimensiones obtineat, distinguendo hos terminos 



-f- kxddz — k (n — i) dxdz 

 ft^<^ >. — kxxddz -+- 2 (2/t — 3) xdxdz — n{n — 1 ) zdx^ = 



statuamus BJfioHqr z — A '-^- Bx -+- Cxx h- Dx^ -i- . . . . -h Mx'-+- Nx'~*' * -+- etc. 



et potestatis a?' coefficiens erit ^ibinoTfr 



'^ -4- N{M {i-t-i)'— k (n— 1) (t-+-l)) -H M(— 4-1 (1—1) -+- 2 (2n— 3) t — /i (/i — 1)), 



qui cum evanescere debeat, habebitur 



(2<-n)(2<-n-l) 

 4(i-Hl)(»--nH- 1) 



IVunc autem novimus esse y^ = 2" a , quare sequentes coefficientes eruut 



.-'J «M'31)l ' > 



4 (n — 1) 4 



^ (n-2)(n-.3) p . » (n - 3) ^ 

 ^= 8(n-2) ^ = -* 4:8-^ 



(n-4)(n-5) ^_ n(n-4)(n-5) . 



^ ~7 - .__ 12^»— .3) 4.8.12 



(a-^^ijVai^ i- 

 etc. 



(mlRtilBfloif 



Sumatur a = 2~", ut fiat ^=1, eritque 



/l-t -y(l-a;) V_ , n n(n-3) n(n-4 )(n-5) 3 n(n-5)(n-6)(n-7) , 



^ 2 J — ' — 4^"*"~4:8~^^ 4:8.12 "^ . 4.8.12.16 ^ ®^^' 



quae est series quaesita. 



30. CoroU. 1. Sumto x negativo, sequentis seriei 



n n(n-3) n(n — 4)(n— 5) 3 n(n — 5) (n — 6) (n — 7) , 



summa erit = ( "^ 1 "*" ) ? ex cujus combinatione cum praecedente, alternis tantum terminis 

 sumendis, summa assignari poterit. 



