Nova methodds etc. 461 



m(m_1) . . . (m—1n) 

 '2.6.1 . r.C2/;-j-l) 



Clar. Eulcrus potentias irinomii 1 -♦-jr-f-x' pluries exami- 

 nl suhjiecit, ejusque de hoc argumento dissertationes variae 

 insertac sunt in Actis Acadcmiae Petropolitanae. Ipse, dedu- 

 clis prioriljiis potcnliis, putabat, ope mnltiplicaiionis succes- 

 sivae trinoinii in se ipsum , invenire legem seriei in quam c- 

 volviiur poteniia qnaevis (i-f-or-i-x')'" mi inveniuir polenlia 

 qnaevis, regula Newtoni, binomii cujuscunique. Seriebiis prio- 

 ribus hoc mode deductis erit. 



(1_l-,r-|-x2)2= l4-2;c4.3,r2-4-2x^4-x' 



(1+a:.-f-x2)^= 1-(-3x4-6^'2-4-7jc3+(Jx'+3r54-xC 



(1_|_x-|-x2) >= 1 + 1x4-1 Ox*+1(5j:3-f-1 9x^+1 6x5+1 0x6+4x'4-r8 



(H-x-f-x2)s= l~t-JX-Hl5x2+30x3-J-4ox'+51x3-+-4:a;C-f-50x'+15x'< 



-4-5xi+x'» 

 (1-f-x-|-x2)6= 1-i-6,r4-21x2+:0x''-f-90xf+12Gx'-f-1 11x«4-126x' 



+90x8+50x9-1-2 1 x' ''+6x< '+xi2 

 (1+x+x2)7=1+7x+28x2+77x''+161x<+266x^+357xfi+393x' 



+357x8+266x9+161x<''+77x"+28x'-+7x'3+x" 

 (\-{.x+x'i)»= 1+8x+36x2+1 12x3+266x»+504x5+784x6+loi9x'' 



+1110x8+1019x''+784x'»+501x<'+26Gx<2 



+11 2x"+5Cxi <+8x'5+x'8. 

 (l+x+x2)f— 1+9x+43x2+1j-6x3+ll4xi+882x^+1554x«+2304«' 



+2907x8+31 39x9+2907x'«+2304x«'+1 554x>» 



+882x'3+414x'<+156x'5+45x<6+9x"+x'8. 



(1+x+x2)in— i4.i0x+5Jx2+210x3+615x'+l452x5+285OxG+4740x' 

 +67G5x8+8350x3+8953x'«+83J0x"+676JxH 

 +4740x'3+28:0x' '+1 452x'5+6 1 5x'6+21 0«<' 

 +55a;'8+10x«9+a;20. 



Ex hisce forniuhs Eulerus deducit terminum generalem se- 

 rierum, quae conslituunlur ex coeflicienlibus potenliarura jt, 

 x'-,x^,x\jc^, sed pro sericbus successivis fatelur ordinem sea 

 legem reconditam esse. Verum cum omnes hujusmodi series 

 sint arithmeticae , quarum unaquaque consiantes habet diflfe- 



