152 A. TANAKADATE. 



the same direction becoming more and more pronounced until it 

 touches the source at the surface where the variation will be 

 indefinitely great. 



When the source is above the level, the variation changes 

 sign on the subcenter side, and the force increases upward, while 

 on the antipode side it decreases as before : the circle of no 

 variation being the same as the nodal circle of the vertical force. 

 When the height of the source is small, the maximum is close 

 to the subcenter and the minimum to the antipode from where 

 they expand, as the source rises, approaching the assymptotic 

 positions 54° f from either of the extreme points ; the circle 

 of no variation begins at the subcenter and tends to bisect the 

 sphere in the limit. 



If the horizontal force is resolved along any given directions 

 in the horizontal plane, the magnitude of the variation will change 

 in the same ratio as its respective components. Taking com- 

 ponents along the cardinal directions as before we have 



9X 

 dz 



- .3 Wir < R — ?^[siri ^ sin ^^ + cos^ cof^o cos(^ — XJ] i 



] R- + r^ — 2rR[sin^ sin^o + c(^^<P cos^y cos(^ — \^] i ^ 



[cop^ siij^o— sin^ coF^„ cos(^ "■^u)]V(20) 



^ y, 3 wi?'< R — r[sin^ sin^o+ cos^ cos^o cos(A — ^)] [ 



-^ = - 1^ ^coP^o sin(; - ^o). 



]R- + r^— 2rR[sinç7 sin^o + cosçj cos^o cos^A — ^)]v ^ 



The Variation of the North Component always vanishes on the 

 nodal ellipse. When the source is below the level, the space inside 

 both of those ellipses is the region of upward decrease of the 

 north component, and the outside zone that of upward increase. 



When the source is above, the variation vanishes besides on 



