694 
MR A. McAULAY OR THE MATHEMATICAL 
the variation of geometrical coordinates an element ds of surface where there is finite 
surface density of electricity be moved from P to P', then in general the element of 
matter will by the variation of the geometrical coordinates only be entirely deprived of 
its charge, for this charge will be left behind at P. This result is, to say the least, 
an unfortunate one, and to be avoided, if by legitimate means it is possible. Still 
more disastrous results are arrived at if we assume that the components of D for every 
element of matter are the electric coordinates, for then the charge in the whole of 
space is varied by a mere variation of the geometrical coordinates. 
The legitimate way out of the difficulty seemed to be to assert that these electric 
coordinates, though theoretically permissible, were very unsuitable. To find suitable 
ones it was natural to use the principle that the electric coordinates must be such that 
the variation in the geometrical coordinates does not alter the charge of any portion 
of matter . This is, of course, ensured by assuming that SD 'dX is unaltered by 
variation of the geometrical coordinates, and from this it is but a step to the asser¬ 
tion that SD 'dX is itself a suitable electric coordinate. 
Intimately connected with this question of the independent variation of geometrical 
and electrical coordinates is that of the correct expression for an electric current in 
(say) an arbitrarily moving fluid. It is not necessary to present all the reasons that 
occurred to me for the form already described (§ 4) as these are sufficiently indicated 
in the above considerations of variation of coordinates. 
D. An Analogy. 
15 . The resemblances and differences between the present fundamental assumptions 
and what I take to be Maxwell’s, are, perhaps, more clearly brought out by 
analogy. 
I will first describe what I understand to be the analogy which Maxwell allows 
himself throughout his theory, in order more closely to realise the interdependence of 
the various physical quantities considered, and as an aid to memory. The analogy 
contemplates the whole of space as being filled with an incompressible liquid. In 
dielectrics the liquid is, as it were, held in elastic meshes, in the form of closed cells, 
so that if it be displaced it tends to return to its original position. In the ideal 
conductor there are no such meshes, or rather there are meshes which do not 
form closed cells, so that the liquid can move through them, but is resisted while in 
motion. An actual body which admits some conduction, but behaves also like a 
dielectric will be typified by meshes which allow a slow leakage of the liquid. Now 
suppose into any space we introduce from some external source more liquid. This 
foreign liquid will be what is called the electric charge of that space, and it may be 
measured (since the liquid is incompressible) by the surface integral over the 
boundary of the space considered of the displacement of the original liquid outwards. 
Thus, <£ electric displacement ” is represented in the analogy by a flux of the liquid. 
