34 ON MAGNETISM. 



(a) The force on the right-hand pole of B tending 

 to increase x is 



dS c + b ,cos 6 x' 

 dr' r 



dr 



But since r 2 = (c + b cos x') z -f (e + & sin 6 y)*, 



ac 



.,, C+&COS0 X , , - , r ,. 



will = - , and therefore the force tending 



d8 dr 



to increase x will = -j- . -j- 

 dr dc 



__ 

 dc' 



To estimate the whole force on the needle in the di- 

 rection of x, the aggregate of the functions S for all the 

 various attractions and repulsions on the two poles must 

 be taken, and developed in the most convenient form 

 (that proceeding by inverse powers of c will always be 

 most convenient), and then it is only necessary to differ- 

 entiate with respect to c. 



(6) In like manner, the force tending to carry the 

 needle to the right is -y- . But if we suppose the 



ae 



center of the needle B to be really on the axis of x, it 

 is only necessary to retain e for the purpose of differen- 

 tiation, and then, after the differentiation, e may be 

 made = 0. It is evident that, in this case, it is only 

 necessary to develope S as far as the first power of e. 



(c) The force at right arg^s to the length of B, 

 tending to produce motion opposite to watch-hands, is 



force in x x sin 9 force in y x cos 0, 



