TO THE THEORY OF MAGNETISM. 91 



By the equation (I 1 ) of the preceding article, we see that 

 when dv is a perfect conductor of magnetism, and its particles 

 are not regularly arranged, the value of the potential function 

 at any point p", arising from the magnetic state induced in dv 

 by the action of the forces X, Y, Z, is of the form 



a (Xcos n. + Fcos /3 + ^cos 7) 



r being the distance p", dv', and a, /3, 7 the angles which r' 

 forms with the axes of the rectangular co-ordinates. If then 

 x", y ', z" be the co-ordinates of p", this becomes, by observing 

 that here a = Jcdv', 



kdv' \X(x" - a?') + Y(y" -y] + ~3F (*"-*)} 



k being a constant quantity dependant on the nature of the body. 

 The same potential function will evidently be obtained from the 

 expression (a) of this article, by changing dv, p ', and their 

 co-ordinates, into dv'j p", and their co-ordinates; thus we 

 have 



dv'{X'(x"-x'}+Y(y"-y'}+Z(z"-z'}} 

 r' 3 



Equating these two forms of the same quantity, there results the 

 three following equations : 



dx ' 



dy dy ' 



d^f 



Zj == K^i == -k'TrlC^J rC 5 7~ 



dz 



since the quantities x", y", z" are perfectly arbitrary. Multiply- 

 ing the first of these equations by dx, the second by dy, the 

 third by dz, and taking their sum, we obtain 



= (1 - frrk) (X'dx + Y'dy' + Z'dz') + k'd+' + UV . 



