G86 
MR. A. McAULAY ON THE MATHEMATICAL 
3. In choosing the coordinates of any mechanical system it is, of course, only 
necessary to take them so that when given they completely specify the instantaneous 
position of the system. But as in all ordinary dynamical problems, so in the general 
electromagnetic problem, there may be all the difference in the world between one set 
and another in respect to the simplicity of the investigations in which they are 
employed, and the amount of light they throw on the interdependence of the parts 
of the system. 
Now, I believe I am right in saying that all writers on the present subject take as 
the electric coordinates the three coordinates of the vector DcZs* for every element of 
volume d<s in space where D is the electric displacement at the point. According to 
the view just advanced that D is the measure of a property of the matter occupying 
the element ds, which is carried about with the matter, these are unsuitable coordi¬ 
nates. According to that view it is probable that the electric current is as intimately 
connected with the matter in which it resides, as is the electric displacement. It 
would seem to follow that the current components could not in general be considered 
as the rates of variation of the corresponding electric coordinates. 
4. Suppose all space split up into a series of elementary parallelepipeda which move 
with matter. Let A d% a ', A d%', A dX c r be the six vector faces of one such parallele¬ 
piped. We shall take for our electric coordinates the three quantities SD'c/2/, SD'dS/, 
SD 'dt c ', where D' is the electric displacement at the point, for every element in space. 
[The reason for the dashes will appear immediately.] 
Moreover, we assume that the same expressions, when 3)' is replaced by C', the 
current, are the rates of variation of the corresponding coordinates. In other words, 
the current C' at any point is defined by the equation 
SC' dt 
clt 
( 0 > 
where d%' is any vector element of surface which moves with matter, and d/dt denote 
differentiation with regard to the time which follows the motion of matter. Thus the 
whole current through any surface which moves with matter = the rate of variation 
of the whole displacement through that surface. 
B. Mathematical Machinery. 
5. As might be expected, the mathematical machinery that appears to be most 
convenient for investigating as fully as possible the consequences of these assump¬ 
tions, and others intimately connected with them, is novel. And I may remark in 
passing that what Professor Tait persistently and with complete justice emphasizes 
as one of the greatest boons that Quaternions grant to ungrateful physicists, viz., 
* Throughout this paper Mr. Heaviside’s practice of replacing Maxwell’s German letters by thick 
ordinary type is followed. 
