Propositions in the Theory of Magnetic Force. 245 



PP', there must clearly be an augmentation (a " component " 

 augmentation) from P towards P'; and therefore (Prop. IV.) 

 the line through P must be curved, with its concavity towards P', 

 and also a "component" augmentation from N towards S, and 

 therefore the end S must experience a greater force than the 

 end N. It follows that the magnet will experience a resultant 

 force along some line in the angle SNP', that is, on the whole 

 from places of weaker towards places of stronger force, obliquely 

 across the lines of force. 



Prop V. {Mechanical Lemma.) Two forces infinitely nearly 

 equal to one another, acting tangentially in opposed directions 

 on the extremities of an infinitely small chord of a circle, are 

 equivalent to two forces respectively along the chord and per- 

 pendicular to it through its point of bisection, of which the 

 former is equal to the difference between the two given forces 

 and acts on the side of the greater; and the latter, acting 

 towards the centre of the circle, bears to either of the given 

 forces the ratio of the length of the arc to the radius. 



The truth of this proposition is so obvious a consequence of 

 " the parallelogram of forces," that it is not necessary to give a 

 formal demonstration of it here. 



Prop. VI. A very short, infinitely thin, uniformly and longi- 

 tudinally magnetized needle, placed with its two ends in one line 

 of force in any part of a magnetic field, experiences a force which 

 is the resultant of a longitudinal force equal to the difference of 

 the forces experienced by its ends, and another force perpendi- 

 cular to it through its middle point equal to the difference be- 

 tween the force actually experienced by either end, and that 

 which it would experience if removed, in the plane of curvature 

 of the line of force, to a distance equal to the length of the 

 needle, on one side or the other of its given position. 



NS being the bar as before, let I denote the intensity of the 

 force in the field at the point occupied by N, I' the intensity at S, 

 J the intensity at P on the line of force midway between S 

 and N, and J' the intensity 

 at a point P', at a distance 

 PP' equal to the length of 

 the bar, in a direction per- 

 pendicular tothelinc of force. 

 Then ifm denote the strength 

 of magnetism of the bar, ml 

 and ml' will be the forces on 

 its two extremities respect- 

 ively. Hence by the mecha- 

 nical lemma, the resultant of these forces will be the same as 

 the resultant of a force »«(I — I') acting along the bar in the 



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