INTENSITY OF THE EARTH'S MAGNETIC FORCE. 251 



deflecting magnet, and in the line perpendicular to it, respectively, 

 their moment to turn the magnet round its point of suspension 

 will be 



r denoting the distance of the particle m of the suspended magnet 

 from its centre, and the angle which the line connecting this 

 particle and the centre of the deflecting magnet makes with the 

 axis of the latter. Now, I have elsewhere shown* that, if we 

 include the terms involving the fifth power of the distance, the 

 values of X and T are 



T = ~sin 9 Jf+f=^(5cos 2 9-l) ; 



a being the distance of the particle m of the suspended magnet 

 from the centre of the deflecting magnet, and M and 1T 3 denoting, 

 respectively, the integrals corresponding to \mrdr,\mr*dr, for the 

 deflecting magnet, taken between the limits r = \ length. Now 



sin 9 = - sin 



so that, extending the approximation to the term involving the 

 fifth power of the distance only, we must make sin 9 = 0, cos 9 = 1, 

 in the coefficient of that term. The preceding expressions are 

 thus reduced to 



and, substituting, the moment of these forces to turn the magnet is 



^ilff* sin * cos 9 cos i/, + (2 - 3 8in 2 9) sin^ + ^~ sin^J ; 

 or, eliminating by the relation between it and i//, 



* Transactions of the Royal Irish Academy, vol. xix. p. 163. 



