302 MR. S. H. BUKBURY ON THE INDUCTION OF ELECTRIC 



or 



Treating G< and H< in the same way, and arranging, we obtain 



iff^J;/ rfH dG \ I** ^\ f dG rf 



T = i < <M3 -- ^H~ w l -- T~) + n (~j -- ^ 

 ~JJ I \rt# (/ \02 a#/ \ dx dy 



But 



_ j _ 



dy dz' dz dx' dx dy 



are respectively the x, y, and z components of magnetic force, or which is here the 

 same thing, magnetic induction. That is, if fl be the magnetic potential of the system, 



rfH rfp- _ dn 



dy dz d$ ' 



Hence 



This value of T is unambiguous ; because, as is well known, dfl/dv is not discon- 

 tinuous at the sheet, even when the superficial currents are finite. To show this, it 

 is sufficient to take the tangent plane at any point for the plane of x, y. Then 



<m _ dfl _ rfF dQ 

 dv dz dy dx 



Now, F and G, or JJ (hu/r) dS and JJ (hv/r) dS, are the potentials of imaginary matter 

 distributed over of surface densities hu and hv respectively. Therefore, dF/dy and 

 dG/dx, corresponding to tangential components of force, are continuous, although, if 

 hu be finite, dF/dz may be discontinuous, at the sheet. 

 Since 



dn_/dH_dG 

 dv \dy dz 



and 



