448 Influence of Masses of Iron 



ceive the sphere AN'BS' to be subtracted from the sphere 

 ANBS; then the attractions which will act upon p will be — Oi 

 and -iO'. 



But the attraction of the sphere ANBS, minus the attraction 

 of the sphere A N' B S', is the same as the attraction of the solid 

 A N B N' A, together with the repulsion of the solid A S B S'A. 

 Hence, if the space occupied by the first of these solids be filled 

 with a magnetic fluid of either denomination, and that occupied 

 by the second with the opposite ; the effect of their actions upon 

 the magnetism, in any particle p of the sphere, will be as iO 

 and O'i. 



Now let the force with wliich the magnetic action of the earth 

 attracts phe pe; resolve this into the two pk and Ae; of which- 

 pk is in the direction pO, and ke\s at right angles to it: also 

 pk, ke are opposite and proportional to o'i, lO ; it will therefore 

 follow that, if the forces which solicit p destroy each other, they 

 will also destroy each other for every other particle of the sphere : 

 or the fluids will be distributed according to the condition re- 

 quired. 



This result may be readily expressed analytically. Take any 

 point n in the surface of the sphere ; join n O, and let it cut 

 the interior surface of the shell in m. Then if we join mO\\t 

 will be equal to mi; but nO is equal to mO' ; therefore m« is 

 equal to Oi: or putting <|5 for the angle NO?/, T for the polar 

 thickness, and / for the thickness at the point n ; we derive the 

 formula ^ = T. cos <f). 



The law which regulates the equilibrium of the magnetic fluids 

 in spheroids, may also be readily deduced from a similar mode 

 of reasoning to that which we have employed for spheres. 



Thus let N AS B (fig. 7) be the given spheroid, whose major 

 axis NS is in the direction of the dip. Conceive also N'AS'B 

 to be a similar and equal spheroid, whose centre O' is in the hne 

 NS, and at an infinitely little distance from O. Then it is ma- 

 nifest that the magnetic fluids when they are in equilibrio, will 

 be bounded by the surfaces NAN'BN and SAS'B S. For the 

 attraction of any particle p will vary as pm and pv, which are 

 respectively perpendicular to the minor and major axes (Mac- 

 laurin's Fluxions, No. 634 ; or Pliil. Trans. 1S09) : and as this 

 is true for each sjheroid, the difference of their attractions will 

 be zero for the direction pv, and the constant quantity mmf for 

 the direction pm. 



The same solution also extends to spheroids, when the dip is 

 not in the direction of either of the axes. For making the con- 

 struction in fig. 8, where N S is the direction of the dip; the 

 difference of the attractions pm, pm' is mm'; or drawing mh 

 parallel and h p perpendicular to N S, the difference of these 



attractions 



