It is convenient to retain x, to denote x r /r!, so that we have the consistent notation xr =x r /r!, n (r) =n(r)/r!, n[r] =n[r]/r!.
(ii.) Repeated divisions of (24) by x+x, r being replaced by rd I before each division, will give (I +xy 2 = I -25+3x2-4x3-F...+(- )r(r (I)xr + (-) r+l x r+1 1(r+ I) (I +5)- 1 + (1 + 5)-21, (I-Fx)-3=I - (3x-6x2 - IOx3+...+(-)rï¿½ 2l(r+I)(r+2)xr +(-) r+l x r+1 12 (r+I) (r+2) (' +x)-1+(r+I)(I - Fx) - 2 +(I +x)-3},&c.
Comparison with the table of binomial coefficients in ï¿½ 43 suggests that, if m is any positive integer, (I +x)-m =Sr+Rr (25), where Sr=I -mx+mx2...+(-)rm[r]xr (26), Rr_(_)r+1xr+11m[r] (1Fx) - 1+(m - I[r](I+x) m) (27).
+(plq)(r)xr} 4 =1 +p(1)x+p(2)x2+...+p(r)xr +terms in xr+1, xr+2, x4r.
(vii.) The comparison of the numerical value of I-Fn(1)x +n(242+...+n(r)xr, when n is fractional, with that of (i+x)n, involves advanced methods (ï¿½ 64).