352 



MR WILLIAM SWAN ON OBSERVATIONS 



Hence 



sin (A + /) sin t] 



sin (A' + /) sin*/ 



Now if the inversion has been accurate, according to definition, we should have 



A' = 180° - A , and 9 ' = 180° - n . 

 Therefore 



sin (A + /) _ - 

 sin (A' - j) ~* 



a condition which, as * may have all values from 0° to 90°, can only be satisfied 

 by A = 90° or A = 270° ; that is, when the magnet is suspended with its magnetic 

 axis horizontal. Such is the position which the magnetic axis is made to assume 

 in practice, with more or less accuracy ; and it therefore follows, that the magnet 

 may remain in equilibrium after accurate inversion. 



Inaccurate Inversion of a Magnet. 



6. We have next to inquire what will be the errors occasioned by inaccurate 

 inversion. 



In fig. 1, let AB = a, BAZ = (3, BZ = 4^, AZB = fa ; 

 and after inversion, let 



6aZ = /3 3 , bZ = ■vh, , aZb = fa, 

 while a b = AB = a, 



Also, as before, let 



DN = 8 , DC = 8 1 , D c = 8, 



Then the observed angles for the magnetic meridian are 



8 1 = 8 + fa , 8 3 = 8 - fa 

 From which 



* = * A +*.)-* (fc - 0.) 

 = i (? x + ^ - € ; 

 where the error committed by taking the mean of 8 X and £ 3 for 8, is 



€ = i(fa- fa). 



Also in the triangles ABZ, abZ,we have 



. , sin a sin 8, . , sin a sin /5, 



m (p. = : — — !-!■ , sin 0, = -. — — t- 3 



sin-v}^ ^ 3 sm4/ 3 



sm 



from which equations, supposing the other angles to be known, fa , and 3 , may 

 be calculated, and e, the correction to be applied to 8, may be obtained. 



Method of ascertaining the relative positions of the Magnetic and Optical Axes in a 



Collimating Magnet. 



7. It thus appears that we can calculate the error in the determination of 

 magnetic declination due to imperfect inversion, provided we know the angles 

 a, (3, and 4,, both in the erect and inverted positions of the magnet. 



