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!.
The number of products such as x r, xr-3y3, x r-2 z 2,.
(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).
Thus if the plane is normal to Or, the resultant thrust R =f fpdxdy, (r) and the co-ordinates x, y of the C.P. are given by xR = f f xpdxdy, yR = f f ypdxdy.
To construct circles coaxal with the two given circles, draw the tangent, say XR, from X, the point where the radical axis intersects the line of centres, to one of the given circles, and with centre X and radius XR describe a circle.
If P, be the force at (Xr, Yr, Zr), acting,in the direction (1, m, n), the formulae (6) and (~) reduce to X = Z(P).