6 EESEARCHES ON II. 



in which, h is a co-eflEicient depending on the intensity of the current in the curve, 

 and the magnetism in the needle. 



For the other pole of the needle, marking with a dash the quantities analogous 

 to those which compose formula (1), we shall have 



(2) ^'^ + ^1^}^. 



The sign is opposite, because, if the force be attractive in the first case, it will be 

 repulsive in the second. 



The experimental confirmation of the preceding law was first established by 

 Blot, and afterwards by Pouillet, in a series of very nice experiments, some of 

 which are given by the latter in his Traite de Physique} Notwithstanding this, 

 we shall take this law as a mere hypothesis, upon which to base our calculations ; 

 the confirmation of theory by observation will show how far it extends, and whether 

 it can be pushed to indefinite limits. 



In order to transform the expression of the force into another, more suitable for 

 analytical calculations, the following consideration will be useful. If we imagine 

 a straight line having one of its extremities fixed at the point P, around which it 

 may revolve, always touching the perimeter of the current MNQ, it will describe a 

 conic surface, which will be generally oblique, having its vertex at P, and for its 

 base the directrix MNQ, and the apothemes or sides of the cone will be the distances 

 already indicated by r. If now we pass a plane tangent to the conical surface, it 

 is evident that it will pass through the element ds, and the direction of the plane 

 will be that of ds itself Therefore the resultant of the forces of the magnetic 

 pole, which must be perpendicular to the plane passing through r and ds, will be 

 perpendicular to the plane touching the said conic surface in the point ds. 



Let therefore A (Fig. 2) be the pole or vertex of the cone, and ah the element ds 

 of the directrix MNQ, drawing to la a tangent hT, and from the vertex A a line 

 AP perpendicular to hT, which is tangent to the directrix in h, we have 



AP= Ah sin. AhP^ Ah sin. AhT, and making AP^ A, 

 we shall have 



A^T sin. V, whence sin. v = — , 



r 



and substituting the value of sin. v in (1), 



omitting the sign for the present, which will be restored positive or negative as 

 circumstances require. 



Now the quantity Ads is the double area of the infinitesimal sector of the conic 

 surface, having ds for its base and A for its altitude ; hence, if we call the whole 

 conic surface jS, we shall have 



Ads = i d/S, 



^ Pouillet, Traits de Physique, note Mathemat., liv. v. cbap. 1. 



