TO THE THEORY OF MAGNETISM. 85 



and therefore V is of the form 



a'Xx'+V Yy'+c'Zz'+e'(Xy'+ Yx'}+f(Xz'+Zx}+g( Yz'+Zy'} 



r' 3 ~> 



a, lj ', &c. being other constant quantities. But it will always 

 be possible to determine the situation of three rectangular axes, 

 so that e, f, and g may each be equal to zero, and consequently 

 V be reduced to the following simple form 



a, b, and c being three constant quantities. 



When A is a sphere, and its magnetic particles are either 

 spherical, or, like the integrant particles of non-crystallized 

 bodies, arranged in a confused manner; it is evident the con- 

 stant quantities a', &', c', &c. in the general value of F, must be 

 the same for every system of rectangular co-ordinates, and con- 

 sequently we must have a =b' = c, e = o, f = o, and g = o, 

 therefore in this case 



a'^+Yy' + Zz') 



/3 W'| 



a being a constant quantity dependant on the magnitude and 

 nature of A. 



The formula (a) will give the value of the forces acting on 

 any point p' t arising from a* mass A of soft iron or other similar 

 matter, whose magnetic state is induced by the influence of the 

 earth's action ; supposing the distance Ap' to be great compared 

 with the dimensions of A, and if it be a solid of revolution, one 

 of the rectangular axes, say X, must coincide with the axis of 

 revolution, and the value of V reduce itself to 



a and V being two constant quantities dependant on the form 

 and nature of the body. Moreover the forces acting in the 

 directions of x, y', z, positive, are expressed by 



_ fdV\ _ (dV\ _ / dV 



