94 APPLICATION OF THE PRELIMINARY RESULTS 



</>, i/r and V belonging to a point p, within the body, whose co- 

 ordinates are x, y, z. It is moreover evident from what precedes 

 that the functions $, ty and V satisfy the equations = &/>, 

 = &Jr and = BV, and have no singular values in the interior 

 of A. 



The equations (b) and (c) serve to determine < and -^, com- 

 pletely, when the value of V arising from the exterior bodies is 

 known, and therefore they enable us to assign the magnetic state 

 of every part of the body A, seeing that it depends on X, Y, Z, 

 the diiferential co-efficients of <j>. It is also evident that -v/r', when 

 calculated for any point p', not contained within the body A, 

 is the value of the potential function at this point arising from 

 the magnetic state induced in A, and therefore this function is 

 always given by the equation (&). 



The constant quantity #, which, enters into our formulas, 

 depends on the nature of the body solely, and, in a subsequent 

 article, its value is determined for a cylindric wire used by 

 Coulomb. This value differs very little from unity : supposing 

 therefore g = 1, the equations (Z) and (c) become 



const. =^r+ V ...................... (c'), 



evidently the same, in effect, as would be obtained by consider- 

 ing the magnetic fluid at liberty to move from one part of the 

 conducting body to another ; the density p being here replaced 



by ( -] , and since the value of the potential function for any 



point exterior to the body is, on either supposition, given by the 

 formula (), the exterior actions will be precisely the same in 

 both cases. Hence, when we employ iron, nickel, or similar 

 bodies, in which the value of g is nearly equal to 1, the observed 

 phenomena will differ little from those produced on the latter 

 hypothesis, except when one of their dimensions is very small 

 compared with the others, in which case the results of the two 

 hypotheses differ widely, as will be seen in some of the applica- 

 tions which follow. 



