144 On the Induction of Electric Currents. 



The second equation represents a repulsive force acting 

 from the disk ; and the third equation represents a force acting 

 towards the axis of the disk. 



These results are equivalent to those obtained by Maxwell 

 and observed by Arago. (see art. 669). 



If the magnet were not sufficiently long to enable us to 

 leave the action of the negative pole out of account, the mag- 

 netic potential of the currents excited by the latter will be 

 found by changing m into —m,h into h + a, where a is the 

 length of the magnet ; and the combined effect of the two 

 poles will be obtained by adding the results. 



It is, however, simpler to place a short magnet, of length 2c, 

 at right angles to the axis of revolution, and consider the 

 combined effect due to both poles. The magnetic potential 

 of the currents due to the action of the negative pole will be 

 obtained in this case by changing m into — m, and <j> into 

 7T + ; whence the expressions for the magnetic forces acting 

 on the positive pole will be found from equations (10) by 

 changing m into 2m, and n into 2n + 1, and effecting the 

 summation with respect to n from oo to 0. 



The calculation can then be effected as follows : — Expand 

 the Bessel's functions in powers of Xc ; the expressions for 

 the forces will then assume the form of series every term of 

 which is of the form 



V>e- 2kh d\ 



f 



W\?+(2n + l)W 

 where p and n are positive integers. Now 



i 





where q = 2(2n + l)h(o/R. Hence, by differentiating with 

 respect to h, the first-mentioned integral can be expressed in 

 terms of the sine and cosine integrals, the values of which 

 have been calculated by Mr. Grlaisher, whence numerical - 

 values of the magnetic forces might be found. 



If we neglect powers of c higher than the fifth, the mag- 

 netic forces in the <f>, z, and p directions will be found to be 

 respectively equal to 



/.« x d e- 2Kh ( XV , 5\V \ mB4* C* \ 7 e-*>*dk 



mK J R 2 X 2 + a> 2 V~ 4 + T927^ + ~64~J WxF+%? ; 



X 3 e- 2A * / a X 2 c 4s 



\ 6 e~ 2kh d\ 

 R 2 X 2 + 9o> 2 * 



f« X 2 6 -2x/, / x v 5XV 5 \,^ 3ma> 2 c 5 r 



