Deinde, suStrahenda hic utrinque primum terminuni _H_ et di* 

 videndo per ^^^ , oritur ista: 



' i — _i_ -4- ^ iq — i\ ^ ^ (?— 3j (?-4T I 6(^-31 (»7'— 4) (?-5) I . gj^^ 

 8 ^-t-a (5+*)(5+4) (2+3)(i+4)(^-Hj) (5-f-3)(e-!-4) (^ 1-J) (5-r6) 



Porro subtrahendo utrinque terminum primum — ^ et dividenda 

 per ?^^^ « nanciscimur 



I __ _j»_ I f (r7 — 4T r_ 6 (? — 4T {q — n t^ l (»7—4) f^ — 51 (^-6T t q^q^ 

 ^ ^ + 4 t?-4-4)(2+5) ('I'+'4)(<jM) (^'+6) (g + 4) (5+-S-)(^+6) (<i+7) 



onde iam facile intelligitur, has operationes quousq^uc lubuerit 

 continuari posse, atque in genere fore \ 



I __ _n__ _y ( n+i) (q — nj . ■ , (n+2j ig — n) {g — n — l) i^ ^^^^ 

 * q-^n. {q + n) \ii-rn-rl). {q-hn) (^ + Ji + I) (? + n-h2) ~ 



quicun^ue numeri pro litteris n et q accipiantur. 



J. 21, Tsta consideratio simuf viam nobis aperit ad de» 

 monstrationem hujus summationis memorabilis perveniendL Sit 

 enim summa adhuc incognita, ponaturque; 



S ~ "^ -4- (n + I) {q-n) . (n^2) {q-n) {q-n-l)' _ _^ gf^^ 



q-Hn. (3 + n) (.j + n-t-I). (f+n; (f-i-TH-x) (ri + n+2) 



T ZZ: -^L±JL -4- (^-^2.) (<7— n— I) _, ( n-i-3T (q—n — J.){q—n~ 2) ^ ^^q^ 

 q •hn-hx'^ {q-hn-hX) (q-f>-n + 2)'^ [q^-i-n-hl) [q+n-^2) (^+n+S) 



onde si has duas series addamus invicem et a se invicem sub- 

 trahamus, relicto scilicet primo termino seriei S^ et combinan- 

 do sequentium quemque cunx termina seriei T^ qui subscriptum 

 praecedit, habebimus; 



S i.Tz— H-Bl- r "-t-^ _i_ Ct+2y (?-n-i T _^_ fn+3,) f^-n-D {q-n-Z) _^ qIqJ 



5 + n 5-1-n '■(2 + n + ll) (5-r-n+l)(5-hn+2) (^+n+i) (5+n-i-2)(^+n-(-3) 

 ■_n_ __^ _2n_ r n-t-l _^ (iz-4-2T (-7— n— i ) 

 3+» 9+n 4+a+I {«-^n^iJlg-i-n-i-i;) 



S — Tr-l^ — ^ f t-^-I _^ (g-4-2T (-7-^-1 ) _^ (n -4-3T {q-n-X) [q~n-a) ^.^.etc.l 



n {3+n+I)(3+»H-2j (5-hTi-i-i)" -' 



hoc 



