II. ELECTRICAL RHEOMETRY. 7 



and the jjreceding expression for the force will become 



(4) ,Jil^, 



which is a very elegant expression for the elementary force, being in the direct 

 ratio of the differential element of the conic surface, having its vertex at the pole 

 and its base in the perimeter of the current. 



When the curve in which the current circulates is given, ds and A will be deter- 

 mined according to its nature, and if x, y, z are the co-ordinates of any point of the 

 current and p, m, n those of the pole parallel to three rectangular axes, we shall 

 have 



(5) r^ = [x — ])f -\- {y — m)- -\- {z — n)^ 



In order to find the three components X, Y, Z of the force <^, parallel to the 

 same axes, it will be sufficient to remember that ^ is directed along the normal to 

 the conic surface at ds. If therefore {Nx), [Ny), [Nz) indicate the angles made by 

 the said perpendicular with the three axes, we shall have 



(G) X= — ^cos. (Nx), F= — ^cos. {Ny),Z= — ^cos. (Nz). 

 The values of the cosines are very easily found by the analytical formulae of the 

 perpendiculars to the conic surfaces. Supposing that 



(7) f. = 

 is the equation to the conic surface, we shall have 



Cos. (Nx) = yr . ^ 



^ /c^ dx 



^ ^ A;' dz 

 where 



'^'=4(iy+(iy+(i)- 



If we take the plane of the curve as the co-ordinate plane xy, the condition of 

 all its points lying disposed in this plane, requires that after the differentiations we 

 should make 2 = in equation (8) . 



Therefore in this hypothesis we shall have 



-qv r (7s ^ >/ dy^ + dx^ 



^ ' I r" = {x — pY + [y — my + n'. 



The value of A will be found for every current, as will be shown hereafter. 



§ 2. DlFFEEENTIAL FOEMDL.E FOR CIRCULAR CURRENTS. 



We shall suppose the circular current to be lying in the plane XOY(Fig. 3), with 

 the origin of the co-ordinates at the centre of the circle, the axis OZ being perpen- 

 dicular to the plane of the circle. Let A be the pole of the magnet or the vertex 



