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 -m[1]x+m[2]x2...+(-)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).