OF ELECTRO-DYNAMIC FORCES. 513 



the current under consideration belong, taken from a definite 

 point of origin, and proceeding in that direction in which, in the 

 first the positive, in the second the negative electricity runs, as 

 far as the element of the current mentioned, we obtain in the 

 same manner 



COS0=^', 003 0'=^. 



ddr,, dr., dr., 



cos e = r,, , ,", 4- -r^'-r^'r 



" ds,dsil dSj dsj 



On substituting these values, we have the following expression 

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

 the other : — 



ae.oe.'e' I /I dr,,dr„ ddr„ \ 



\2 dsi dsj " dsjds, / 



+ 



' II' II 



If, thirdly, we consider in the original expression ue .u' e' an 

 the product of the negative electrical masses — « e and — a! e' 

 into the two elements of the current, and uu' as the product of 

 their velocities — u and — u', and r^^^ denote the vai'iable distance 

 of these two moving masses, and lastly, s^^ and sj denote the 

 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 definite point of origin, and proceeding in that 

 direction in which the negative electricity runs, as far as the ele- 

 ment of the current under consideration, we have 



cos0= - 4^, cos0' = 4^' 

 dSfi d Sil 



ddr, I, dr,„dr 

 cos £ = — r 



iji 



'" d s,, d s,! d s„ d sj ' 



"11 "^^ "11 "' "11 " ";/ 



On substituting these values, we have a third expression for the 

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

 namely, 



a e . «' e' 



/I dvi .dr,, _ ddr,,, \ 

 \2 ds„dsj "'ds„ds,lJ' 



' III ' III 



In fine, if, fourthly, in the original expression we consider 

 — a e . «' e' as the product of the negative electric mass — « e of 

 the element of the current a into the positive electric mass «' e' 

 of the element of the current «', and — u u' as the product of 

 their velocities — u and u' ; if, moreover, rm, designate the vari- 

 able distance of these two moving masses, and s,, and sj the 



