II. ELECTRICAL RHEOMETRY. 



§ 1. General FoRMULJi; for the Attraction of a Magnetic Pole on an 

 Electric Current. 



Let us consider an electric current circulating in a close curve, MNQ (Fig. 1), 

 which we shall suppose to be plane for the sake of simplicity: let P be a magnetic 

 pole, and ha any element ds of the current itself : it is required to find the expres- 

 sion of the force P on that element. 



It is well known that Coulomb, inquiring into the law of attraction of two mag- 

 netic needles, found that their action could be represented very accurately by four 

 forces acting in the inverse ratio of the squares of the distances ; two being attrac- 

 tive, and two repulsive. This law was also received as certain by Poisson, who 

 made it the basis of his calculations on magnetism, and found it to agree with 

 experiment. Savary, in his memoir on the apj)lication of the calculus to electro- 

 dynamic phenomena,' following Ampere's steps, and considering magnets as electro- 

 dynamic coils, was led to the same conclusion as Coulomb. Savary's calculus, 

 however, being only approximate, might have left a doubt whether this was exactly 

 true beyond the limits of his approximation. And indeed the experiments showed 

 a little deviation from the result of the calculus, because the centres of elementary 

 forces, viz., the poles in magnets, are never exactly at the ends of the bar, but a 

 little inside. Now, besides the reasons for this difiei'ence assigned by Ampere and 

 Nobili, relative to the disposition of the currents in the body of the bar, Mr. Plana 

 assured me that, pushing the approximations beyond the limits of the calculations 

 of Savary, the centre of action is really found to be a little inward, a result which 

 agrees with observation. We may therefore be assured that " the action of a 

 magnetic pole is a force acting in inverse ratio of the squares of the distances." 

 Besides, Ampere and Savary" proved that the action of an electro-dynamic cylinder 

 or coil on one element of an electric current, can be reduced to two forces acting in 

 the inverse ratio of the squares of the distances of each pole from the element, and 

 in the direct ratio of the sine of the angle comprised between the direction of the 

 element itself and the line measuring its distance from the pole. As to the direc- 

 tion of this force, if P be one of the poles of the magnet, and ah = ds the element 

 of the curve described by the current, drawing a tangent ht to ds and a plane {Pbt) 

 passing through the pole P, the resultant of the forces exerted by P on ds will be 

 perpendicular to this plane at h; representing therefore hy v the angle Pht, and by 

 r the distances Ph, the expression for the resultant will be 



,-,. h sin. vds 

 (1) ,^ = , 



' Mdmoire sur 1' Application du Calcul aux Ph(5nom6nes Electro-Dynamiques. Bachelier, Paris, 1S23. — 

 See also Ampere's Recueil d'Observations Electro-Dynamiques, page 325; and Annales de Ch. et Phy. xxii. 

 page 91. 



" TWorie des Ph^nomenes EleotroDynamiques, pp. 9G, 98. 

 2 



