The Rev. H. Lloyd on the mutual Action of permanent Magnets. 161 

 If we develop the trinomial factor 



(a' - 2 or cos + A' ^ = o-^ ( 1 — 2 ^ cos (^ + -^,)7 



it is manifest that the quantity within the brackets will be expressed by a series 

 ascending by the powers of - ; and that accordingly the preceding integrals may 



Cv 



be developed in serial of the form 



m C ^^ . U, . U„ . U-t 



-|f^„+E + ^^ + ^^ + &e.), 

 a^\ a a^ a^ } 



in which the coefficient of the general term is 



U„=V 



\ qr'" dr. 



V being a function of the constant angle 0. Now, if the distribution of free 

 magnetism be symmetric on either side of the centre, the alternate coefficients, 

 U^, U^, U^,kc. vanish, the values of q being equal, with opposite signs, at the cor- 

 responding distances r = ± s. We have therefore, in this case, 



„ m fA. , Aj , A. , . \ 

 a' \ a * a^ ^ a^ ^ J 



(2) 



the two series descending according to the odd powers of a. 



When the length of the magnet is small, in comparison with the distance a, 

 these series converge rapidly, and, for most purposes, the first term affiards a 

 sufficient approximation to the actual value. We have then, approximately, 



X = ^, Y=^; (3) 



a^ a^ ^ ^ 



VOL. XIX. Y 



