458 Aloysii Casithelm 



A6=m-}-|m(m— 1 )4-ljj;M(m— 1 )(m~2)-\rjini...{m—l)-+- - m. . .(ni— 4)-l-.-y?i . ..(m— 5) 

 A7=:m+3m(m — 'i)-]--m{m — 1 )(m — 2)-+— j-m...(m — 3)4-— m...(m— 4)-4-— ^m...(m — 5)-f-j^m...(m— 6) 



etc. etc. clc. etc. 



Quae formulae (acillime ad sequentes reduci possuut. 



A.2=/i»-H;-m(m — 1) 



A3=m -4-^ m(m— 1 )•+--/» (w—1 )(m— 2) 



A4=^4-|m(m— 1)4-|m(m_1);^m— 2)-4-im...(m— 3) 



A5=m-4-jm(m— 1)-+-^m(m— 1)(m— 2)+j;m...(m— 3)+ji^w,..(m— 4) 



A6=ni4-?m(m— 1)+-m(m— 1)(m— 2)4--OT...(,;j_3)+jj^ru...(m— 4)4-^-|jm..^ 



A7=m4-|m(m-.1)4-%,(m— 1)(m— 2)4-|^m...(»»_3)+j^w...(n»— 4)+r^:»t...(TO— 5)4-j^^ 



etc. etc. etc. 



Atque ita lex barum expressionnm manifesla est. In qiia- 

 vis eaiiim nam era to res coelbcienlinni numericornm sunt coef- 

 ficientes ipsi seriei in qnam evolviiur potentia cujuscumque 

 binomii cujns exponens aequat inJicem coeOicientis quem 

 aeqiiat exprcssio ipsa, unitate miuiiliun. Sic in exprcssione 

 aequivalcnii A, nnrneralorcs coefficientiiira sunt 1,6,15,20,15, 

 6, 1 scilicet coeflicieotes potentiae sextae cujuscumque bino- 

 mii. 



Ilaque si Ap indicat coeiEcientem quemcumque seriei in 

 quam evolvitur potentia m esima illius polynomii, erit 



m(m-l ) (A^-l )(p-2) m (>n-1)(m-2) , (;._1 )(,.-2X>-3) m...{m- y) 

 Ap =m+(^-1)— 2— -^ 2 2:1 "" n 27371" 



m(m—1)....(m-p~\-1) 



•4-ec. -+- jr-^ — 



2 . i . 4 p 



Sit nunc trinomium i-+-x-<-x' ad poiestatem m esiraam e- 

 vehendum. Erit fl,= i,c,= i,«j=o,<74:=o,a5=o,fl«=oec, 

 Ideoque 



Ai=m 



A.i=:fn-4-;;m(m — 1) 



\ 



