442 ELEMENTARY ACTIONS. 



VI. Principle of symmetry. The application of the principle of 

 symmetry will determine the direction of the force. 



We see at first that : 



i st. The action of a pole on an element of current perpendicular 

 to the right line which joins it to the pole, is perpendicular to the 



ds 



p 



Fig. 104. 



ds 





 Fig. 105. 



Fig. 106. : 



plane passing through the pole and the element. Let us join the 

 pole P to the centre of the element ds (Fig. 104). We already 

 know that the action is perpendicular to the element. It is also 

 perpendicular to the right line PO, for if the figure is turned 

 through 1 80 about the right line, the force must change its sign 

 without changing direction (II). 



2nd. The action of a pole on a current element, the prolongation 

 of which passes through the pole, is zero. This action must be 

 perpendicular to the element ds (Fig. 105); on the other hand, it 

 should not change in direction when the element is made to turn by 

 any quantity about the right line PO ; it is therefore null. 



Let there now be an element ds (Fig. 106) which makes an 

 angle a with the right line joining it to the pole; the element of 

 current ds may be replaced by its two projections ds cos a and 

 ds sin a, the one along the right line PO, the other in a perpendicular 

 direction. 



The action of the pole on the former is null ; there only then 

 remains the action of the pole on ds sin a. This latter is propor- 

 tional, as we have seen, to the mass m of the pole, to the intensity i 

 of the current ; it is also proportional to the length ds sin a of the 

 element, and lastly to a certain function of the distance f (r). 

 Hence, if d<}> is this force, and k a coefficient to be determined by 

 experiment, 



d<j> = mkids sin of (r) . 



