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).

Now 2 area 17r=2Xr; so that, in order to reconcile the amplitude of the primary wave (taken as unity) with the half effect of the first zone, the amplitude, at distance r, of the secondary wave emitted from the element of area dS must be taken to be dS/**Xr** (1) By this expression, in conjunction with the quarter-period acceleration of phase, the law of the secondary wave is determined.

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).