84 APPLICATION OF THE PRELIMINARY RESULTS 



from an infinite quantity Q of magnetic fluid, concentrated in a 

 point p on this line, at an infinite distance from A. Then the 

 origin of the rectangular co-ordinates being anywhere within 

 A, if x, y, z, be those of the point p, and x, y, z', those of any 

 other exterior point p, to which the potential function V arising 

 from A belongs, we shall have (vide Mc. Cel. Liv. 3) 



r ^(x 1 + y* + z'*) being the distance Op. 



Moreover, since the total quantity of magnetic fluid in A is 

 equal to zero, U (0} = 0. Supposing now r' very great compared 



jj() 

 with the dimensions of the body, all the terms after ^- in the 



expression just given will be exceedingly small compared with 

 this, by neglecting them, therefore, and substituting for Z7 (1) its 

 most general value, we obtain 



-4, B, C, being quantities independent of a?', ?/', z', but which 

 may contain x, y, z. 



Now (Art. 6) the value of Fwill remain unaltered, when we 

 change x, y, z, into x', y ', z', and reciprocally. Therefore 



Ax + By' + Cz _ Ax + B'y + G'z 



r' B r* ~' 



A, B', C'j being the same functions of a?', y t z, as A, B, C 7 

 are of a?, y, z. Hence it is easy to see that V must be of the 

 form 



v _ d'xx+l"yy'+ c"zz'+e"(xy'+yx) +f"(xz'+zx'}+g"(yz'+zy) 



rV' 3 



a", b"j c", e",f", g", being constant quantities. 



If X, Yj Z, represent the forces arising from the magnetism 

 concentrated in p, in the directions of x, y, z, positive, we shall 

 have 



Y -Q x . Y -Qy . 7 -Qz. 



=-- = ~~ = ~- 



