MAGNETISM. 



39 



of Magnetism, already referred to, M. 

 Poisson deduces, from the theory above 

 stated, the analytical equations which 

 express, for all possible cases, the laws 

 of the distribution of magnetism within 

 bodies that are rendered magnet ical by 

 induction, and those of the actions, whe- 

 ther attractive or repulsive, which they 

 exert on points given in position. The 

 first problem to be resolved is to reduce 

 the resultants of all the attractions and 

 repulsions of the magnetic elements of 

 a magnetized body, of any imaginable 

 form, "on such points, situated either 

 within or without the surface, to three 

 directions at right angles to one another. 

 By adding to the resultants which relate 

 to any interior point those of the ex- 

 ternal magnetic forces which act upon 

 the body, he obtains the whole forces 

 which tend to separate the two fluids 

 that are united at that particular point. 

 Were the matter of the body to oppose 

 no sensible degree of resistance to the 

 displacement of the fluids in each mag- 

 netic element, or, in other words, if there 

 were no coercive force, it would be ne- 

 cessary, in order that there might be an 

 equilibrium, that the attractions and re- 

 pulsions should destroy one another ; or, 

 speaking algebraically, that their sum 

 should be equal to zero ; since, if any of 

 them were uncompensated, they would 

 effect a new decomposition of the neutral 

 fluid, which may be regarded as inex- 

 haustible, and the magnetic state of 

 the body would be altered. The sum of 

 the resultants must, therefore, be made 

 equal to zero, with respect to each of the 

 three rectangular directions to which 

 they are referred. The equations of 

 equilibrium, thus formed, will always be 

 possible, and they will serve to deter- 

 mine, for each point of a magnetized 

 body, the three unknown quantities 

 which they comprehend; namely, the 

 intensity of the action of a magnetic ele- 

 ment on a given point, and the two an- 

 gles which determine the corresponding 

 direction of the line of polarity. At the 

 extremities of each element, these joint 

 resultants will not vanish ; they will 

 give rise to pressures from within each 

 element, tending outwards, and counter- 

 balanced by the obstacle, of which the 

 nature is unknown, but which opposes 

 the passage of the fluid from one ele- 

 ment to another, and also its escape from 

 the surface. 



(160.) When the coercive force of 

 the magnetized body is also taken into 



account, it will then be sufficient for the 

 magnetic equilibrium that the resultant 

 of all the exterior and interior forces, 

 acting upon any point of the body, no- 

 where exceeds the given magnitude of 

 the coercive force : so that, in this case, 

 the equilibrium may take place in an 

 infinitude of different ways, and the 

 problem is, in this respect, wholly inde- 

 terminate. This indeterminateness is a 

 source of considerable difficulty in the 

 resolution of questions of this nature. 

 The following general consequence, 

 however, may be deduced from the 

 equations of magnetic equilibrium 

 formed in the manner above described ; 

 namely, that although in a solid body, 

 magnetized by induction, the austral 

 and boreal fluids are distributed in an 

 active state throughout the whole mass 

 of that body, yet the attractions and re- 

 pulsions which it exerts externally are 

 precisely the same as if they proceeded 

 from a very thin stratum of each fluid, 

 occupying the surface only, both fluids 

 being in equal quantities, and distributed 

 in such a manner as that their total ac- 

 tion upon all the points in the interior 

 of the body is equal to nothing. If the 

 body be hollow, or contain an empty 

 space within it, and if the centres from 

 which magnetic forces proceed be situ- 

 ated within this space, the body must 

 be considered as terminated by two thin 

 strata of fluid, situated, the one on the 

 external, and the other on the internal 

 surface; and the action of these two 

 strata on any point, within the substance 

 of the body, joined to that of all the 

 given centres of magnetic action, must 

 produce a perfect equilibrium; and, in 

 this case, the two fluids may be in diffe- 

 rent quantities in each of the thin strata, 

 provided that they be always in equal 

 quantities in the two surfaces taken 

 together. 



(161.) Thus it appears, that the 

 theory of magnetic attractions and re- 

 pulsions is reduced to the same princi- 

 ples, and leads to the same formulae, as 

 the theory of electric forces in conduct- 

 ing bodies ; and the perfect correspond- 

 ence between the two may be illustrated 

 in the following manner. We may sup- 

 pose an aggregate mass composed of 

 minute grains of metal, or other con- 

 ductor of electricity, each grain being of 

 so small a size that its dimensions may 

 be neglected in comparison with the 

 whole mass, and each being surrounded 

 by a substance impermeable to^electri- 



