To illustrate the procedure which will be used in the sequel, we will use 

 formula (8) to compute the vertical component of the attraction of a sphere 

 of uniform density and whose center is at a depth h below the surface. 



As formula (8) stands the location and orientation of the coordinate 

 axes with respect to the center of the sphere and the surface of the ground 

 are not specified. The first step then is to chose a set of coordinate axes in such 

 a way as to make the computation as simple as possible. For this purpose, we 

 choose a set of coordinate axes in which the origin coincides with the field 

 point, the positive z axis will be taken vertically downward and the x and y 

 axes turned so that the center of the sphere will be in the (x,z) plane (fig. 27-1) . 



Figure 27-1. 



The coordinates of the source point (u,v,w) now become (u,o,h) and the co- 

 ordinates of the field point (x,y,z) become {o,o,o) . Formula (8) may now 

 be written 



. rtTTTil /f|\ 



* = (u* + wy^ ^ ' 



This formula shows some improvement over formula (8) but as a working 



h 



formula it has one defect, namely, that the value of the factor — 



would have to be recomputed every time we wanted to use a different value of h. 



This may be remedied by writing it in the form — • — ■ — 



h* (w i /h' i + l) 3l ' i or 



587 



