90 Galvanism , from Galvani to Ohm. 



Now, by (3), the resolved part of F along ds' must vanish when 

 integrated round the circuit s, i.e. it must be a complete 

 differential when dr is taken to be equal to - ds. That is to- 

 say, 



^(ds.ds')(r.ds') (ds . r) (ds'. r) 2 



/o-3 .f\> 



must be a complete differential ; or 



must be a complete differential ; and therefore 



7 A B iA 



d '^ = -- 5 ( dS ' r )> 



3 ^ B J 



or ~2^" dr = r* dT ' 



or B = - I A. 



Thus finally we have 



F = Constant x ii'i || (ds . ds') - - 5 (ds . r)(ds'. r) 



This is Ampere's formula : the multiplicative constant depends 

 of course on the units chosen, and may be taken to be - 1. 



The weakness of Ampere's work evidently lies in the 

 assumption that the force is directed along the line joining the- 

 two elements : for in the analogous case of the action between 

 two magnetic molecules, we know that the force is not directed 

 along the line joining the molecules. It is therefore of interest 

 to find the form of F when this restriction is removed. 



For this purpose we observe that we can add to the expression 

 already found for F any term of the form 



0(r) . (ds . r) . ds', 

 where 0(r) denotes any arbitrary function of r ; for since 



this term vanishes when integrated round the circuit s ; and it 



