a^ + a^ + Hg + ... + a^ = [4] 



a^^m + a^m^ + . . . + a^m" = -q- [5] 



a^ + aa^m + . . . + naj^ra'^''' = [6] 

 The radius of curvature R may be evaluated from the formula 



-±ih©7 



dY^ 

 which may be written In the dlmensionless form 



^-F^hi^ar 





But differentiating Equation [2] successively with respect to y gives 



2y = (a^ + 2a2X + . . . + na^x^'^ ^ [8] 



and 



2 = (a, + 2a2X + ... + na^x""^ ) ^ + [2a^ + ... + n(n-l la^x^'^] (||f [9] 



dx 

 If aj^ / 0, it is seen from Equation [8] that when x = 0, -t— = 0, 



and hence, from Equation [9], that ^-|- = ■— . Consequently, substituting these 



dy 1 

 values into Equation [7], we obtain 



a = 2r^ [10] 



10 ■■ ■' 



If, on the other hand, a = 0, the body has a pointed nose and r =0. Hence 



Equation [TO] is valid for both cases. 



dx 



Similarly when x = 1 , y = and from Equation [8] -g- = 0, unless 



a^ + 2a2 + . . . + na^ = [11] 



Hence Equations [7] and [9] give 



a, + 2a^ + . . . + na = -2r, [12] 



X 2 n -^ 



