118 Proceedings of Royal 'Society of Edinburgh. [sess. 
and therefore consists of two shears ; the first of magnitude mr , 
with faces perpendicular to n and v, and the second of magnitude 
nrT V v, with faces perpendicular to v and V v. Assuming w to be 
the normal at a point of the curved surface, and therefore perpen- 
dicular to v, we have 
0 = rSw/x + So) V v 
= -iSvo>V(r 2 ) + ScoVv. 
Thus 
dv _d /r 2 \ 
dn dl\ 2 / 
( 69 ), 
where djdn denotes differentiation along the normal outwards, and 
djdl differentiation in the positive direction round the boundary. 
The surface traction on the plane end 
= 23V = + V V ). 
Hence the total moment round the axis 
= nrff (r 2 - rSfx V v)d,A 
' K r+// -v /A )> 
where dA is an element of area of the cross-section, and I is the 
moment of inertia of the cross-section round the axis. Thus the 
torsional rigidity is, as usual, n(l+ff^dA^. We leave the 
problem here to the theories of complex variables and Fourier’s 
theorem. 
It is in the general theories of electrostatics and electro- 
magnetism that I have found the methods now being defended 
the most powerful. I have been led to believe that there is in all 
the accepted theories which are based on general dynamical reason- 
ing an error of a somewhat serious character. I have also been led 
to a considerable modification and extension of Poynting’s theories. 
There are two reasons against giving these here. I have been told 
that this preliminary apology, as it may be termed, for my methods 
should be as short and simple as possible. Moreover, the greater 
part of my notes on this subject are at present inaccessible. I 
therefore limit myself in this branch to a single example. 
Maxwell has not investigated what are the general mechanical 
results of his electrostatic theory for crystallised dielectrics. 
