144 A. TANAKADATE. 



The Vertical Force Z is 



Maximum at 6=0 i.e. epicenter 

 and Minimum at 6 = - i.e. antipode. 



When the source is above the level, the sense of the force is 

 IT versed near the place directly below it, which we may now 

 call subcenter ; while on the antipode side the sign remains 

 unchanged. The surface is thus divided into two regions of 

 positive and negative vertical forces by the nodal circle 



6 = cos-^A (.■>) 



within the limit ll<r<co ; 



evidently this is the circle along which a pencil of rays from 

 the source touches the sphere. A new maxinuun occurs in the 

 pc>sitive region along the circle 



<*=cos-(2R-^j-) (0) 



within the limit Pi< r <2E . 



It beo-ins with at the lower limit and ends with - at the 

 higher. 



The Horizonlal Force H 'vanishes always 



at = and t: 



whether the source is inside or outside the sphere, being 



Maximum at ^=cos-4(y(^ J. + J^y_.^-|.) (7) 



0<r<co ; 



the value of ^ remains the same by replacing r by 1/?', so that 

 there are two values of r corresponding to the same circle of 

 maxinmm horizontal force ; its position changes from ^ ^^ as 



