Lcmim 



J. 3. Si X zz: x\, et x fuaejjivc augeatur incremen* 



tis 7, a, 3. etc. erit ^Xn. . •• ; (n — \)n f 



feu nta dijferentia poteftatis n mriabilis x eji conjians. 



Poteftatum ac differentiarum .altiornm relationem ha^c 

 fatis quidem cognitam , fequente modo fuccir.Qe demonftrare 

 non inutile erit. Cum differen-tia prima quarrtitatis x 9 

 h. e. (i+i) 8 - x n figno A x n denotari foleat, illa abeat in 

 ^Af, pofito 1 + 1 loco x, in /y Ax n , pofito x ■+- 2 loco x, 

 etc. ita ut flt 



'a x 1 — (x •+- 2 y — (x -f- 1 )\, 



"A x n ~(x-^^) n - (x-t- c)\ 



et fic porro: tumque erit 'A x n A f =rA ! x , 1 fecunda 

 differentia quantitatis x n , quae fubftituto iterum x+i lo 

 co x ahit in 'A~ x\ unde porro fit fertia differentia quan- 

 titaiis x l feu A 3 x n ~''^ 2 x n ^ 2 x', et fic pono: ut igi- 

 tur in genere fit A r x n zz: 'A^ 1 x n — A r " ] x\ 



lam vero cafu n ~ 1 eft A n X - A x zz (x±i) - x~t^ 

 et cafu n ~ 1 9 A n X — A 2 x 2 zr ' x — A x 2 : unde ob 

 Ax 2 r(i+i) 2 — x 2 — 2 x -f- 1 , ideoque ln hac expreflio- 

 ne fubftituto x -f- 1 loco x, 7 A x 2 zz x-f- 3, fit zi 71 X zz 

 2~ 1.2. Cafti denique n = 3 eft A" X ~ A 3 x 3 . Eft au- 

 tem A x 3 ~ (x -f- i) 3 — x 3 ~ 3 x x -t- 3 x H- 1 , proinde 



7 A x 3 zr 3 (x -+- 1) 2 ■+• 3 (x +- 1 ) +• 1 — 3 xx +- 9 x -+- 7 , 



et A 2 x 3 = 'A x 3 — A x 3 ~ 6 (x -+- 1 )« ideoque 7 A 2 x 3 — 

 6(x+- r ), unde fit A 3 x 3 zz ^ 2 x 1 A 2 X 3 zz: 6 zzr 1. ?. 3. 

 Vnde veritas Lemmatis cafu xz. 1, aszzs, x:3, patet. 



Simul* 



