THEOKEMJTIS FERMATIANL n 



fequente {x i- 2)'^— (x + j)'^ fubtradlus dabit terminum 

 generalem feriei differentiarum fecundarum , qui erit 



— ( .r + 2 j"* — 2 ( .v+ I ) "* + A,' "*. Hinc porro 

 leriei difFerentiarum tertiarum erit terminus geiieraiis 



— (^^+s)'"- 3 (.v+2)'" 4-3 (.v-l-i)'" — A''"; ac tandcm (eriei 

 difftrentiarum ordinis ;« concluditur terminus generalis 

 = (x+/«)'" - "^ (x-\-m- 1 ^^" + pll^) (x-j-m- 2)'" -^'f^'--)(^---) 

 ( .V + /« — 3 ) "* -l^ etc. qui cum fit quantitas conftans, 

 idem erit quicunque numerus pro ;i* fubftituatur , erit 

 ergo 



\el rr /«'"-w(«/-i)'"+':i^\w--2)'"-!:i^;<"'-''(«»- 3 )™ 



+ etc. 



vel =1 («+!)'"-;«. mVj^^-^m-if- ^n^^^Km-zy* 



~\~ etc. 



■vbi in forma priori pofuimus x — o, in pofleriori 



;t = I. 



§. 13. Euoluamus iam cafus huius f lie" fpeciales 

 et a poteftatibus minimis ad altiores afc(P amu : ac po- 

 fito primo w zi: i , feriei i, 2, 3, 4, >, 6, etc. 

 terminus generalis difFerentiarum primiriim erit 

 rr: 1 ' — i .0' rz i ; vel =r 2' — i i ' ^ i . Si w — 2, 

 feriei i ; 2*; 3*; 4*- 5'; etc. diffcreiinue lecundae funt 

 vel a'-2. i', vel 3'- 2.2' + i.i '; ateft^*— 21* 



— 2(2'— i.i'), vnde hae differentiae fecundae funt 

 rza.i. Sit t;^— 3, et feriei i, 2% 3% 4% 5% etc. differen- 

 tiae tertiae erunt \el ~ 3' — 3. 2* + 3 . i^ vel^'— 3. 3* 

 -f-3.2'- 1 .1' ; nt ^' - 5.!z' + 5 .1' = :i {3- 2.z'+ 1.1') 



z=: 3. 2. I, quia ex cafii praecedente cft 3*— 2. 2' +1.1' 



rr 2- I. Simih modo fi m=: 4 feriei i, 2*, 3*, 4*, 5% 



B a etc. 



