102 REPORT—1885. 
element of current, ds’ its projection on the plane through its centre 
containing the straight current, 7 the distance of the element from the 
end of the straight current, and a the angle which the rectilinear 
current makes with the line joining its extremity to the elementary 
current. By taking the difference of two such rectilinear currents, 
Grassmann finds the action of an element (f) of current on another 
element (a) is a force at right angles to a’, the component of ain the 
plane containing / and the middle point of a and equal to 
.. dods' 
7] 
== SiON OS 
oa 
where 0 is the angle which 3 makes with 1, do the length of (3), andj the 
current flowing through it. 
The direction of the force is along AB, where A is the centre of the 
element (a) and B the point where the normal to a’ is cut by / produced 
in the direction of the current. 
If we treat this theory in the same way as we did Ampére’s on p. 
99 by considering the action of the component ay, B, of an element of 
current ds, on the components ay, [o, y2 of another element ds5, we see 
that Grassmann’s theory is equivalent to supposing that a, exerts no 
force on ao, (so, Or 72; that 3, exerts a force Aja, on ay at right 
angles to a, in the plane of ay, and a force Aj3,35 on fy, at right angles 
to it, that is, along the line joining the element, and that it exerts no 
force on Yo. 
Thus we see that Grassmann’s theory is equivalent to replacing 
Ampére’s assumption, that the force between two elements of current 
acts along the line joining them, by the assumption that two elements of 
current in the same straight line exert no force on each other. 
As a dynamical theory of electrodynamics, Grassmann’s theory is open 
to the same objection as Ampére’s, that it does not take into account the 
couples which may exist between the elements, and also to the additional 
objection that, according to it, the action of an element of current ds, on 
another element ds, is not equal and opposite to the action of ds, on 
ds,, so that the momentum of the two elements ds, and ds, will not 
remain constant, and, as the theory does not take into account the sur- 
rounding ether, there is no way of explaining what has become of the 
momentum lost or gained by the elements. As a piece of geometrical 
analysis, however, the theory is very elegant and worthy of the author of 
the ‘ Ausdehnungslehre.’ 
From the way in which Grassmann’s theory was developed we see 
that between closed circuits it must give the same forces as Ampére’s; for 
unclosed circuits this is not the case, and Grassmann, at the end of the 
paper quoted above, mentions a case where the two theories would give 
opposite results, assuming that unclosed streams exist. Suppose we have 
a magnet ws and an unclosed current AB in the same plane as the 
magnet and passing through its middle point, then if Ampére’s theory 
be true, the magnet will twist in one direction; if Grassmann’s, it will twist 
in the opposite. This depends upon the change, according to Ampere’s 
theory, of the force between two parallel elements from attraction to repul- 
sion, when they make the angle with the line joining them at less than 
sin-!1/,/3, while according to Grassmann’s theory, there is no such 
change. 
