THEORY OE ELECTROMAGNETISM. 
08 7 
their perfect naturalness , seems to me to receive illustration in the methods about to 
be described. 
The notation in the present paper will be mainly the same as that of my former 
paper on a “ Proposed Extension of the Powers of Quaternion Differentiation.”* 
As in that paper (which will for the future be referred to as “ the former paper”), 
a fixed position of all the matter in space will he taken as a standard of reference. 
Most of the following symbols have exactly the same meaning as before. 
p is the coordinate vector of any particle of matter in the standard position ; p the 
coordinate of the same particle in the present position, so that p may be regarded as 
a function of the independent variables, t (the time), and p. ch, dd denote elements 
of volume of the same particle in the standard and present positions ; ds, ds similar 
elements of surface ; and U^, UP the unit normals at ds, ds. In the present paper 
another notation will also be used, defined by 
dt = l Jvds, dt = UP ds .(1), 
whence 
JJv = U dt, ds = T d% .(2), 
and similarly for U dt', T df. This meaning of t is scarcely likely to clash with the 
usual summation meaning (which will also be freely used in the present paper), since 
in the present use the t will alw'ays be preceded by d, a combination that would be 
rare with the ordinary meaning. 
With this notation equations (2) and (3) of the former paper take the somewhat 
briefer form 
j \f)dp= jj<£Ve£SA.(3), 
jj<M2 = ds ..(4). 
In connection with these equations it is well to call attention to the following usual 
conventions which will be strictly adhered to. The right-handed system of rotation 
is adopted. Ur, or dt, when regarded, as in the last equation, as the normal of the 
boundary of any region, is always drawn from the region bounded. Thus, if UJv is 
regarded as the normal to the boundary of a dielectric at its junction with a con¬ 
ductor, it is drawn from the point of the bounding surface into the conductor. The 
positive direction, that of dp in equation (3), round the boundary of a surface, is that 
of positive rotation round a proximate positive normal, dt in equation (3). Thus the 
positive direction round the boundary of a magnetic shell whose positive normal is 
in the direction of magnetisation is that of the equivalent current. 
V will have the usual meaning with regard to p, and V' the same meaning with 
* ‘ Proceedings of the Royal Society of Edinburgh,’ 1890-91, p. 98. 
