II. ELECTRICAL RHEOMETRY. 21 



§ 6. Discussion of the Formul.?:, and Equilibrium of a Magnetic Needle. 



Formulae (15, § 3) and (13, 14, § 5) contain the theory of forces acting on a 

 magnetic pole of a needle, and it is very easy to deduce from them also the law of 

 equilibrium of a needle, supported by a silk fibre or a pivot. De la Rives's floating 

 ring, and the instrument called comjMss of tangents, are only particular cases thereof, 

 and the theory of the common galvanometer is also dependent on the same. In 

 the case of a floating ring, the magnet is fixed and the currents are movable ; the 

 contrary takes place in galvanometers, where the current is fixed and the needle is 

 movable. This being the most interesting case, we shall consider it principally, as 

 it will be very easy to apply the conclusions to the other. The case of a circular 

 current having been more generally resolved, we shall discuss this first. 



Let us consider a needle A B (Fig. 6) supported at its middle point C either by a 

 silk fibre or a pivot : let us pass a plane through its natural position of equilibrium 

 and that which it takes by the action of the currents, and let DE he the inter- 

 section of this plane with the plane of the currents X Y. It is evident that each 

 pole is solicited by three forces X, Y, Z, as well as by terrestrial magnetism and 

 gravity, and the needle will be in equilibrio whenever the sum of their moments 

 taken relatively to three orthogonal planes passing through C satisfy the condi- 

 tions required in mechanics for the equilibrium of a free body, if the needle be only 

 suspended by a silk fibre ; or for a body movable on a point or an axis, if the needle 

 be sup2»rted by a pivot or axis. 



In order to find the direction of the forces X, Y, and ascertain the direction of 



their resultant, and their sign, let us join as before (§ 2) the centre of the circle 



with the pole A and project the line OA = Z on the plane YX. Calling s the 



length of this projection and u the angle made by it with the axis Y, according 



to the denominations of that paragraph, we shall have 



m ^ s COS. u, p ^ s . sin. u, 



from which 



±- = tang, u; 

 m 



therefore u ^ a, and a is the angle made by the projection of I on the plane XF 

 with the axis OY; this arbitrary quantity is then fully determined. 

 Therefore making for the sake of brevity 



(1) Q = iM5 (2 E' — 2 V F' — c- E^), 



we can regard the forces X, Y, as components of a force Q acting parallel to the 

 plane XOY and to the projection s, passing by the pole through which the resultant 

 of all the forces passes. The force Q is evidently the component of the whole 

 resultant ^ resolved in the direction of a plane passing through the pole of the 

 needle and the axis Z of the circle, and perpendicularly to that axis. According 

 to the position of the poles sin. a and cos. a may change their signs, and so the 

 6 



