A NEW CUBBBST WEIGHER, ETC. 541 



APPENDIX B. 



On, the Forces bettveen Coils of Wire of Finite Section* 



The formula developed by J. VIRIAMU JONES gives the force when the windings of 

 the coils can be treated as infinitely tine helical filaments. In the ampere balance, 

 however, the wires are of finite thickness, and thus small corrections may become 

 necessary. 



If the force parallel to the axis experienced by a helical filament of radius A and of 

 fixed pitch and number of turns when carrying a current i be F, we have 



F =a xAd * <" 



where X is the magnetic force at right angles to the axis and B is measured round 

 the axis. If y be the co-ordinate, parallel to the axis, of one end of the helix, the 

 force on the helix in a magnetic field symmetrical about the axis is u function of A 

 and y, and we have 



dA* dy* Jo WA* dy* A (/A/ 



Now, if V be the magnetic potential of the magnetic field, V is symmetrical about 

 the axis, and hence satisfies LAPLACE'S equation 



But X = </V/</A, and hence, differentiating (3) with respect to A, 



_t/X.X 



. _ 



d A' dy> A </A A d A A' 

 Thus 



ePF . d*Y . f'A/X . X\ , fi 1 dF 



T-T-S + -r-r = l TT + T ) d " = T TT 

 rfA 1 dy* Jo\(/A A/ AJA 



-r-r 

 dy* 



This is similar to MAXWELL'S theoremt for mutual induction. 



Distribution of Current in the Wire. In default of any accurate knowledge of the 

 variations of specific resistance over the cross-section of the wire forming a helical 

 coil, it is impossible to accurately determine the distribution of current in the wire. 

 We shall, however, examine the case in which the specific resistance is uniform and 

 shall call the corresponding distribution of current the " natural " distribution. The 

 current density at any point may be taken as inversely proportional to the length of 



* For the major portion of the following treatment we arc indebted to Mr. Q. F. C. SEARLB, F.R8. 

 t MAXWELL, 'Electricity and Magnetism,' 3rd ed., vol. ii., 703. 



