41 



than the first in order to obtain the same accuracy, but, since all graphical 

 operations are eliminated, the second method is suitable for processing on an 

 automatic-sequence calculating machine. 



The result of the polar transformation has effectively been given 

 in [95]. We have 



pEULziU) dx . J" EM^^yU) „ [1W] 



where 



x - t = y(x) cot 6 [70; 



It is desired to evaluate this integral for a series of values of t . In gen- 

 eral this can be done with sufficient accuracy by means of the Gauss 7- (or 

 11-) ordinate quadrature formulas. This gives 7 (or 11) values of 9 at which 

 the integrand needs to be determined for a given t. The value of x occurring 

 in the integrand is determined implicitly, for given values of t and B, by the 

 polar transformation [70]. In practice the 7 (or 11) x's can be obtained 

 graphically from the intersections with a graph of the given profile of the 7 

 (or 11 ) rays from the point x = t on the axis at the angles required by the 

 Gauss quadrature formula. If greater accuracy is desired, these graphically 

 determined values of x can be corrected by means of the formula 



t-x + y(x ) cot 6 



x = x + S S [105] 



g 1-y' (x ) cot 6 



in which x is the graphically determined value and y' denotes the derivative 

 of y with respect to x. 



Now let us derive an alternate, completely arithmetical procedure 

 for evaluating the integrals. Put 



, , , V f (x) 



k(x ' t] = [(x-t) y + f(x)] a ' 2 



k .(x, t) = — g (x ' t] 



[(x-t) 2 + g(x)] 3 ' 2 



where y 2 = f(x) is the equation of the given profile and y 2 = g(x, t) is 



the equation of the prolate spheroid whose ends coincide with the ends of the 



given body, and which intersects the given body at x = t . i.e., 



