244 Prof. Thomson's Elementary Demonstrations of 



point whose coordinates are (x,y, z), the expression Xdx + Ydy + Zdz 

 must be the differential of a function of three independent vari- 

 ables.] 



Examination of the Action experienced by an infinitely thin uni- 

 formly and longitudinally magnetized bar, placed in anon-uniform 

 Field of Force, with its length direct along a line of force. 

 Let SN be the magnetized bar, and ST, NT' straight lines 

 touching the line of force in which, by hypothesis, its extremi- 

 ties lie, and P a point on it, midway between them. The result- 

 ant force on the bar will be the resultant of two forces pulling its 

 ends in the lines ST, NT'. If these two forces were equal (as 

 they would be if the intensity of the field did not vary at all 

 along a line of force, as for instance when the lines of force 

 are concentric circles, as they are when simply due to a cur- 

 rent of electricity passing along a straight conductor; or if 

 P were in a situation between two dissimilar poles symmetri- 

 cally placed on each side of it), the resultant force would clearly 

 bisect the angle between the lines TS, T'N, and would therefore 

 be perpendicular to the bar and to the lines of force in the direc- 

 tion towards which they are curved ; that is (Prop. IV.), would 

 be from places of weaker to places of stronger force, perpendi- 

 cularly across the lines of force. On the other hand, if the line 

 of force through P has no curvature at this point, or no sensible 

 curvature as far from it as N and S, the lines NT and ST' will 

 be in the same straight line, and the resultant force on the bar 

 will be simply the excess of the force on one end above that on 

 the other acting in the direction of the greater ; and since in 

 this case (Prop. IV.) there is no variation of the intensity of the 



force in the field in a direction perpendicular to the lines of 

 force, the resultant force experienced by the bar is still simply 

 in the direction in which the intensity of the field increases, 

 although this is now a direction coincident with a line of force. 

 Lastly, if the intensity increases most rapidly in an oblique 

 direction in the field, from P in some direction between PS and 



