51 



APPENDIX I 



APPROXIMATE CALCULATION OP THE SURFACE S OP 

 A CLASS OF ELONGATED BODIES OF REVOLUTION 



From Guldin's rule 



S = 2tt [ yds = 2tt f +a y /l + (y') 2 dx 

 S = 2tt f +a yclx + n f +a y y' 2 dx - 7rA m + n f + ° y y' 2 dx [29] 



the main part of the surface is given by n times area of the meridian section 

 A plus a correction term neglecting higher order terms. 











With y 



= bH 



[ 











y* 



_ b_ 

 a 



©H 



the 



correction 



term 



becomes 













IT 



r+a 



1 y y' 2 



J- a 



dx = 7rab 



a 2 



C 



§) 2 d* = 7rab^I [30] 

 ■ ? a 2 



i.e., the correction term is equal to the area of an ellipse with the axes 

 a, b multiplied by the square of the elongation ratio and a numerical value 

 I dependent upon the equation of the curve. To get an idea, with obvious 

 denotations , 



I = 1 6/1 5 H = 1 - | 2 



2 



I = 128/77 H = 1 - £ 4 



T 2n 3 „ _ , t n 



J n ~ (2n-l) (3n-1) H - T - * 



The next term in the expansion of S 



" f C n'*di-f^ J" HH< 4 d| [31 ] 



with H' = dtt/d{ is obviously of the order b 4 /a 4 . However, taking H = 1 - £ n 

 the factor 



1>' 4 d * = K = (4n-3)(5n-3) 



grows with n 3 when n is large. Provided b/a is not too small, say —1/7, the 

 error in neglecting all terms except the first (Equation [29]) is only per- 

 missible as long as n ^ ~5« 



