104 REPORT—1885. 
arbitrary coordinate axes and suppose that «, y, z are the coordinates of ds,, 
wl, y', z! those of ds,, then the z component of the force on ds, due to ds, 
is shown by Stefan to be equal to 
lat d, (a'—2x) d at dee! a it di | al —@ 
Vj AS, AS, { ™ ds, ds5 r +n ds, 7 ds maple ds, 7 ds, ak oP a cos « b 
with similar expressions for the force parallel to the axes of y and z. 
Here i, j are the currents through ds), ds, respectively, ¢ is m8 angle: 
between the elements of current, and 
n=4{a—b—c+2 2a} 
= —i{a—b+2c—d} 
g=4 fa+2b—c—d}. 
We see from this expression for the force parallel to # that the last 
term is the only one which does not vanish when integrated round two: 
closed circuits of which ds, and ds, are elements. So that the force will 
depend only upon q; the value of g will depend upon the units we adopt : 
in Stefan’s work qg is put equal to —1/2. 
This is the only condition to be got by considering the translatory 
force between two circuits; we can get another by considering the couple 
acting on the closed circuit, supposed rigid, of which ds, forms a part. 
For the z component N of this couple Stefan finds the expression 
= ly da! dy _dy' dz 
N= ijq || G74 cos edeydey — ‘gel {jo ae a} dads 
dss ds, dsy ds, 
But supposing the two circuits to have a potential 
ag cose | 
vy c r us) dsy, 
we can easily see that the couple 
1» 1 = 54 ' 
BP age Ye ey se {{ Lypael dy «dy dx 7 SiN 
=1) \| 73 COS € ds, ds, U7 || a, ‘Pa ds, — ds, da, f ds,dxo 3 
thus if two circuits have a potential 
P= o 
or substituting for p and q their values, 
2a+6+c—2d=0. 
If c=0 and d=0, as in Ampére’s theory, this relation becomes 
2a+b6=0, 
which is the same relation as Ampere deduced by finding the condition 
that the force due to a closed circuit on an element of current should be 
at right angles to the element, and Stefan has proved that on his theory 
the same condition leads to the equation 
P= 
i.e., the same cordition as the one which expresses that two closed circuits 
have a potential. 
