514 WEBER ON THE MEASUREMENT 



length of a portion of each of the two conducting wires, to which 

 the elements of the current under consideration belong, calcu- 

 lated from a defined point of origin, proceeding in that direction 

 in which in the first the negative, in the second the positive elec- 

 tricity runs, we have 



cos = - ^', cos ©' = - ^ 

 ds^i a si 



""'^'- '""ds.dsl ^ ds.dsl ' 



If now these values be substituted, we have the fourth expression 

 of the force with which one element of the current acts upon the 

 other, viz. 



eie.ot'e' ( /I dr.,., dr.,., dd'i 



\2 ds„ds ""ds,, 



+ 



Siidsj/' 



' nil ' nil 



Now at that moment in which the electric masses alluded to 

 occur in the two elements « and «', the distances r^ r^, rn,, rmi, 

 have all the same value, which is expressed by r. Hence the 

 four expressions of the electro-dynamic force of the two elements 

 of the current a and a' become converted into the following : — 



ue . a.' e' , 



aauu' 



rr 



a e . «' e' 

 rr 



ae . a.' e' 

 rr 



(\ dr^dr, ddr, \ ,, . 



2 ds,dsl ~ ^ ds,ds;J' ■ • ' -^ '' 



, (I dr., dr., ddr,, \ ,„. 



aauu' .[- , '' ' — r -j — y^^ I, . . . . (2.) 



\2 ds,ds,l ds,ds„'/ ^ ' 



, (\ dr,,,dr,,, ddr,,, \ ,„ , 



.aauu'. I- , "' " — r-j — V^,),. • . • (3.) 



\2 ds„ds,l ds„ds,',J ^ ' 



ue.u'e' I /I dr„„dr,,,, ddr„„\ ,, , 



rr \2 ds„ds,' ds„d,' /' ^ ' 



from which we can construct the fifth expression, viz. (5.) : — 



_ «e«'e' '^,uu'.[- (^IJ^ _ ^!jA« , ^ni^in _ ^Lm^Lmi) 

 rr * 4 ' 'L.2\ds,dsl ds'ds,,' ds„ds„' ds„dsl ) 



_ ^ / ddr, _ ddr ,, ddr,,, _ ddr„„ y[^ 



\ds,dsl ds,ds„' ds„ds„' ds„ds,'/ J' 



The four variable distances r,, r„, r,,,, r„„, are now respectively 

 dependent upon the variable magnitudes of the paths s, and s,', 

 s, and s,,', s„ and s,,', s„ and s,', through which the moveable 

 masses to which they refer have passed in the two given con- 



