10 



write the equation of equilibrium under the abbreviated form 

 V- Xfu, and differentiate, and divide by Y= Xtan 0, 



Ftan 9 -= = f'u hi + fu -=-. 

 Y J J X 



„ ' j. tan u , ., 1 - Q sin 3 i< 



But /« = . — ^ . - ; and / w = — 5 — -r: — ,. . . — r^ , ne- 

 J 1 + Q sm 2 u J 'cos 2 w (1 + Q sm 2 w) 2 ' 



glecting the term 2 Q sin 4 r< in the numerator, as inconsiderable; 



wherefore putting, for abridgment, 



cotan o _ 1 1 - Q sm2 « 



P = F(l + Qsin 2 z<)' = cos 2 m ' l + Qsin 8 K- ' 



there is finally, 



SY ( ^ + SX\ 

 -=- = p I o e>w + tan u-^? j. 



" The coefficient j9, in this formula, is obtained by the me- 

 thod already explained. It has been shown that when a mag- 

 net is placed vertically, above or below the suspended magnet, 



M 



its inducing action on the iron bar = — (2-3 sin 2 <p), e and <p 



denoting as before ; and, as this force has the same effect as a 

 small change of the earth's vertical force, the effect upon the 



M 



suspended magnet is obtained by making 8 Y= — (2 - 3sin 2 0), 



SX= 0, in the preceding equation. Wherefore, kn denoting 



the corresponding change of angle, 



M 



— (2-3sm?<t>)=jjYSkn. 



Again, when the deflecting magnet is horizontal, and per- 

 pendicular to the magnetic meridian, its effect is the same as 

 that produced by a small change of the earth's horizontal 

 force, whose magnitude is given by the equation 



WIT--- M 



bX sm u + — - cosm = 0. 

 a 6 



Putting, therefore, ibr EX the value thus given, and making 



M 



— = XS kn : 



