232 Proceedings of the Royal Society of Edinburgh. [Sess^ 
is the flux per unit area due to the motion of the surface with velocity q. 
Hence ifj is the general operator giving the flux of momentum. The 
equation of rate of change of momentum per unit volume at a point whose 
velocity is q is 
i//\7 = ^VDB H- vQ . VDB 
= ^VDB + qv . DB + VDB . div q , (33) 
the first two terms giving the rate of change of density of momentum at 
the moving point, and the last term the rate of change due to expansion at 
the rate div q. 
This flux of momentum ifj is partly due to convection, and partly to be 
ascribed to a stress. It is interesting to note that if all the tubes were of 
one set, we could determine the stress by simply putting q equal to this 
velocity. We should then have H — VqD, and the stress would be 
cf>={E + VqB) . D - i(E + VqB)D 
= F.D-iFD 
==F.D + i(HB-ED) (34) 
In general the stress operator will be obtained by subtracting from 
the operator — E(Vd^B.qm) which gives the convective flux of momentum 
relative to a fixed point ; thus the stress is 
^ = ^, + 2(Vd^B.q^) (35)' 
= E . D + H . B - i ED - ^HB + ^Yq„,B . Vq^d,,, . B 
+ :§( Vq„,d^)B 
^ E . D + H . B - lED - JHB + 5Vq„,B . d^ - H . B + HB 
= ^{(E + Vq^B).d^}-iED + ~fHB (35) 
From (35)' we see that the stress coincides with Maxwell’s stress when 
there is no convection of momentum relative to the (so-called) fixed 
reference frame ; and from (35) that it consists in general of a quasi- 
tension equal to E-|-Vq^B per tube of the mtl\ set together with a hydro- 
static pressure J(ED — HB). The torque per unit volume is seen to be 
</>-</>’= S = — (E -h V q^B)dj^ 
-fSVd^Vq^B 
= -f ^Vq^Vd^B - 2V(Vq^d^)B 
= -f- i.Vq^\ d^B , ...... (36) 
the last expression being the rate of change of moment of momentum 
about a fixed point due to component of velocity perpendicular to the 
momentum, familiar in the hydrodynamics of the motion of bodies in 
a fluid. 
