92 APPLICATION OF THE PRELIMINARY RESULTS 



But dy and dV being perfect differentials, X'dx'+ Y'dy' + Z'dz 

 must be so likewise, making therefore 



d$ = X'dx + Tdy + Z'dz', 

 the above, by integration, becomes 



const. = (1 - ITT&) fi + fop + kV. 



Although the value of k depends wholly on the nature of the 

 body under consideration, and is to be determined for each by 

 experiment, we may yet assign the limits between which it must 

 fall. For we have, in this theory, supposed the body composed 

 of conducting particles, separated by intervals absolutely im- 

 pervious to the magnetic fluid ; it is therefore clear the magnetic 

 state induced in the infinitely small sphere dv', cannot be greater 

 than that which would be induced, supposing it one continuous 

 conducting mass, but may be made less in any proportion, at 

 will, by augmenting the non-conducting intervals. 



When dv is a continuous conductor, it is easy to see the 

 value of the potential function at the point p" 9 arising from the 

 magnetic state induced in it by the action of the forces X, Y, Z, 

 will be 



Bdv X (x - x'} + Y (y" - y) + Z (z" - z) 



seeing that - = a 3 - a representing, as before, the radius of the 



sphere dv. By comparing this expression with that before 

 found, when dv' was not a continuous conductor, it is evident k 

 must be between the limits and f TT, or, which is the same thing, 



g being any positive quantity less than 1. 



The value of k, just found, being substituted in the equation 

 serving to determine <', there arises 



