Vertical Cylinders 



A method much like the one just described for infinite horizontal cylinders 

 may also be used to compute the gravity effect of vertical cylinders or prisms 

 of finite length. In this case we start with the problem of computing A z for 

 a body with vertical sides, whose top end is plane and at a depth h below the 

 surface of the ground and whose lower end is at infinity (fig. 27-6). 



Figure 27-6. 



As before, we take the field point at the origin of coordinates so that 

 equation (16) reduces to 



A z = 



k >sss; 



wdudvdw 



h (u 2 + v z + w 2 ) s/2 

 In the integral 



C C dudv „ 



kp J J (u* + v* + h*)*l* W 



J J (u 



dudv 



( U 2 + V 2 + h*yi* 



we set u = ha, v = Kb, so that 



A z = kphjj- 



dadb 



(a* + b® + iy> 



(22) 



Evaluating (22) over the narrow rectangle shown in Figure 27-7 we get 

 approximately 



A' z = k P h [log (x 2 + V--^ 2 + 7s* + 1 ) (23) 



- log ( Xl + V-^i* + ys 2 + 1) ] (y, - yi) 



A contour map of log (x + \/x 2 + y 2 + 1 ) for positive values of x is 

 shown in Figure 27-8. The images of the contours on the positive side of the y 



593 



