dudw (44) 



- 2In \ i v ~ ^r— — p ""■' dwdw 



J J [ (x — w) 2 + (z — u> ) 2 \ 2 



At the point (0,0,0) equation (44) reduces to 



H * = 2/ ' XX (,/+% i *""' " 2/ " XX ("'7^) r *-*»' <*) 



which may be written 



ff. = - 211 (f -^— i-^r-j) dudw - 21nff-£— ( ™ ) dudw 



J J ^u u 2 + w i •* J qw u 2, + w £ (46) 



Following the method used in computing the attraction of horizontal 

 cylinders we will first evaluate the integrals in equation (46) over narrow 

 rectangles, the first integral over a rectangle such as that shown in Figure 27-12a 

 and the second over one such as that shown in Figure 27-12b. If (z 2 — Zj) and 

 (x Jf — x 3 ) are small, a good approximation to the first integral is given by 



211 ( v 2 T 7 2 - X 2 Z i tt ) ( z * - Zi ~) and to the second b y 



The function to be mapped is — - — ■ 7- . The contours are circles passing 



x 2, + z~ 



through the origin with centers on the z axis. The radii of the circles cor- 

 responding to constant values of — : — - — -r are given in Table 27-VII. The contour 



r x z + z 4 



map is shown in Figure 27-13. 



There is one difference between the use of this map and that of the maps 

 described previously. In the present case two counts must be made around 

 the boundary of the body, one for the terms containing the factor // and one 

 for the terms containing the factor In. For the terms containing the factor II 

 the cross section of the body is divided into horizontal rectangles so that on 



the left hand side of the body the values of — ; — ■ - read off the map will be 



X" + z 2 



602 



