9 



on. We are, therefore, unable to obtain the value of this 

 moment, expressed as an explicit function of a and h, and 

 must have recourse to a different development. 



" Let the distance of any point of the iron bar from the 

 centre of the suspended magnet, CP', be denoted by R, 



R 2 = a 2 + (h - r') 2 , and p 2 = R 2 + r 2 - lar sin u. 

 Expanding — according to the inverse powers of R, and in- 

 tegrating, observing that M n = 0, when n is even, 

 frdmdm ^-[dm _ , . [dm' 3 . 5 . . „ (dm 



or, if we make 



"r dm dm . „ „ . „ 



— = A + i>a- sir w ; 



M 



f 



9 



in which, on account of the smallness of the distance of the 

 iron bar, the term containing sin 2 u may bear a very sensible 

 proportion to the whole. Accordingly, if we put, for abridg- 

 ment, 



Aa = MU, Ba^MUQ, 



the moment of the force exerted by the iron bar is 



M U cos u (1 + Q sin 2 u) ; 



and the equation of equilibrium therefore is 



U (1 4 Q sin 2 if) 1 = X tan u. 



"Let V$Y denote, as before, the change of U produced 

 by a small change of the earth's vertical force. Then, if we 



