ON ELECTRICAL THEORIES. 105 
Stefan shows that, from the consideration of the action of closed 
circuits on elements of other circuits or of themselves, it is impossible to. 
get any other relation between the quantities a, b, c, d, so that we have 
only two relations between the quantities a, b, c, d, and thus two of them 
must be indeterminate. 
We may give any values we please to these quantities, provided they 
satisfy these two relations ; if we put c = 0, d= 0 we get Ampére’s theory ; 
if a = 0,c = 0, Grassmann’s ; and we can get a number of other theories by 
giving different values to these quantities. 
Stefan’s theory is open to the same objection as Ampére’s, since it 
does not take into account the couples which one element may produce 
on another. He also limits the generality of his theory by supposing that 
the force between two elements of currents in one plane is in that plane. 
Korteweg’s Theory. 
According to this theory, the forces between two elements of current 
are the same as in Stefan’s theory ; Korteweg, however, considers in 
addition the couples which one element may produve on another. 
If we use the notation we adopted in discussing Stefan’s theory, we 
have, considering the force on dsy, a force 
- {aajay + 03, Bo} 
along the line joining the elements, and a force 
= {cao + das/3,} 
parallel to the axis of 7. 
In addition.to these forces, Korteweg supposes that from the action of 
a, on 3, there is a couple whose axis is parallel to the axis of z equal to 
fa 1Po, 
and from the action of a, on yy a couple on y, whose axis is parallel to. 
the axis of y and equal to 
—fajy2; 
from the action of /3, on a, there is a couple on a, whose axis is parallel 
to the axis of z and equal to 
92149, 
and from the action of (, on y, there is a couple on y, whose axis is. 
parallel to the line joining the elements and equal to 
hay. 
If we now take arbitrary co-ordinate axes, the forces on the element ds.. 
are the same as those given by Stefan’stheory. The couples, however, are 
different. The component parallel to the axis of «of the couple on 
ds, is given by the equation 
L=|* dr (: elas, a) — a—d—c dr dr 
1 1 
sas : — (y'z —z'y) 
7 ds, ds ds, we ds, ds 4 : 
' Crelle, xe. p. 49, 1881. 
