OF MAGNETIC DECLINATION. 351 



It will be observed, that unless the apparatus has been disturbed in the process 

 of inversion, the angle AOB will remain unaltered. It is otherwise with AOZ, 

 and BOZ, which do not necessarily remain unchanged, after the point of suspen- 

 sion has been transferred to z, or the magnet has been inverted. We may there- 

 fore define accurate inversion of a magnet to mean, that, a vertical straight line 

 being supposed to be rigidly connected with it, the magnet is turned round its 

 axis, until the line, moving along with it, becomes again vertical ; the inclina- 

 tions of the magnetic and optical axes to the line, thus remaining unchanged after 

 inversion. 



4. When the magnet has been inverted in the manner now defined, the mean of 

 the readings, in its erect and inverted positions, gives a correct value of the mag- 

 netic declination. It does not, however, follow, that no other position after inver- 

 sion would secure the same result, for it is obvious that a curve might be described 

 on the surface of the sphere, such that, for all points in it, the angle a Zb should 

 be equal to AZB ; and if the magnet were suspended from any of these points, we 

 should still have the desired condition fulfilled. We might also have the magnet 

 inverted, so that the arc AB revolved until it made an angle with ZA on the 

 opposite side of the line, and equal to ZAB. No importance, however, attaches 

 to the existence of these different modes of inversion ; for one method, answer- 

 ing the desired conditions, is sufficient, and the one which has been defined as 

 accurate inversion seems to be that most easily effected. 



5. It remains, however, to be shown that the magnet will remain in equili- 

 brium in the new position of accurate inversion. For this purpose let AA , 

 Fig. 2, be the magnetic axis, ZO a vertical line passing through the point of 

 suspension, and G the centre of gravity of the magnet, which, in the position of 

 equilibrium, will be in the same plane with AA' and ZO. The magnet is then 

 kept in equilibrium by the forces at A, A', the weight of the magnet acting at G, 

 and the tension of the suspension fibre acting in the line OZ. 



Let m = the force of free magnetism in either pole, 

 w - the weight of the magnet ; 



and put 



AO = a , A'0 = b, GO = c , 

 AZO = A z = / , AOZ = A , GOZ = „. 



Then we have 



m (a + b) sin (A + /) — w c sin r\ = 0. 



After inversion / remains unchanged, but A and n may be supposed to change to 

 \' and n ; while the new point of suspension may either be the former point 0, 

 or any other point in the line ZOz; and we shall in like manner obtain, 



m (a + b) sin (X' + i) — w c sin rj = ; 



