

Propositions in the Theory of Magnetic Force. 243 



shorter of the two paths, must exceed the mean intensity of the 

 force along the other ; and therefore the intensity of the force 

 increases from P in the perpendicular direction towards which 

 the concavity of the line through it is turned. 



Prop. II. The augmentation of the component force in any 

 plane at an infinitely small distance from any point, towards the 

 centre of curvature of the line of the component force through 

 it, bears to the whole intensity at this point the ratio of the infi- 

 nitely small distance considered, to the radius of curvature. 



If, in the diagram for the preceding proposition, we suppose 

 AB and CD to be infinitely near one another, and each infinitely 

 short, they will be infinitely nearly arcs of circles with infinitely 

 neai'ly equal radii. Hence the difference of their lengths must 

 bear to either of them the ratio of the distance between them to 

 the radius of curvature. But the mean intensities along these 

 lines must, according to the preceding demonstration, be in- 

 versely as their lengths, and hence the excess of the mean inten- 

 sity in CD above the mean intensity in AB must bear to the 

 latter the ratio of the excess of the length of AB above that of 

 CD to the latter length ; that is, as has been shown, the ratio of 

 the distance between AB and CD to the radius of curvature. 



Prop. III. The total intensity does not vary from any point 

 in a magnetic field to a point infinitely near it in a direction 

 perpendicular to the plane of curvature of the line of force 

 through it. 



Prop. IV. The total intensity increases from any point to a 

 point infinitely near it in a direction towards the centre of cur- 

 vature of the line of force through it, by an amount which bears 

 to the total intensity itself, the ratio of the distance between these 

 two points to the radius of curvature. 



These two propositions follow from the two that precede them 

 by obvious geometrical considerations. 



[They are equivalent to asserting, that if X, Y, Z denote the 

 components, parallel to fixed rectangular axes, of the force at any 



other, would be the supposition that a thermo-dynamic engine could absorb 

 heat from matter in its neighbourhood, and either convert it wholly into 

 mechanical effect, or convert a part into mechanical effect and emit the 

 remainder into a body at a higher temperature than that from which the 

 supply is drawn. The investigation of a new branch of thermo-dynamics, 

 which I intend shortly to communicate to the Royal Society of Edinburgh, 

 shows that the magnet (if of magnetized steel) does really experience a 

 cooling effect when its pole is carried from A to B, and would experience a 

 heating cH'ect if carried in the reverse direction. But the same investiga- 

 tion also shows that the magnet; must absorb just as much heat to keep up 

 its temperature during the motion of its pole with the force along AB, as it 

 must emit to keep from rising in temperature when its pole is carried 

 against the force, along DC. 



B2 



