22 RESEARCHES ON II. 



values of X, Y, may change likewise ; in Z, the sign of cos. a is always to be com- 

 bined with that of m, and as these signs change simultaneously, so there will be 

 no difference in the product m cos. a and in the terms depending on it. 



In order to ascertain more speedily the equations of equilibrium of a needle, let 

 us transfer the origin of the co-ordinate planes from to G, the middle point of 

 the needle which we shall suppose to be fixed. Let the new axes also be parallel 

 to the former ones, and let the axis OX be parallel with the axis of rotation, 

 around which the needle is movable : calling 



the co-ordinates of the point relative to the new origin G; L half the length of 

 the needle GA; d the angle of deviation of the needle from the plane of the circle, 

 we shall have 



Zo = GH, Yo = EG, and 



Xo = oa 



From these we obtain 



m= GK^HK—Ha= GL — EG=Lcos.d—Yo, 

 (2) n = KA = LA — LK= L sin. d — Z^, 

 p = — Xfl. 

 Likewise for the other pole 



m} = D G= — L COS. d — Yq, 

 7)}- = BD = — L sin. d — Z^, 



y = — Xo. 



These values must be substituted in XYZ to obtain the expression of the forces 

 referred to the new origin. 



For the purpose of avoiding a useless complication, we shall first consider a 

 simple declinating needle, and suppose the plane of the currents to be vertical. In 

 this case, if T represent the intensity of terrestrial magnetism parallel to the 

 magnetic meridian, and d^ the angle made by the circle with that meridian, the 

 components soliciting each pole parallel to OZ, Y, will be 



Tsin. d\ Tcos. d\ 



Now the whole system of forces acting on a declinating needle, movable on a 

 pivot without friction (marking them with the letters a, h, according as they belong 

 to either the one or the other pole), will be as follows : 



Parallel to F 



-f Ta COS. d\ — Th cos. d\ + Ya, — Yb, 



+ Ta sin. d}, — Th sin. d^, + Za, — Zh, 



parallel to 0^ 



parallel to OX 



Xa, — Xh. 



The last forces suppose the vertical magnetic component compensated by gravity, 

 which is effected by a suitable suspension of the needle. 



Multiplying each force by the distance of the pole to which it is applied from 

 the co-ordinate plane to which the direction of that force is parallel, and which 



