150 
Transactions of the Royal Society of South Africa. 
Equations (1), (2), and (3) will still hold, even if rr, QO, zz, and rz do not 
vanish, provided only that the stresses, strains, etc., are independent of f. 
If, in addition, we put 
rr = 66 = zz = zr = o, 
it immediately follows from the equations of compatibility that 
21^, — U^— 0. 
Williers integrated 3 graphically in some special cases. 
The rest of this paper is concerned with the analytical and graphical 
determination of the stress-lines, stress-components, and displacement in 
certain selected cases. 
Section 1. 
Since we assume that the surface of revolution is free from external 
forces, the stress-lines cannot cut the surface anywhere. That is, in a 
meridian plane, the l)oundary lines of the body of revolution will themselves 
be stress-lines. The stress -lines are, however, given by = constant, and 
hence, if we can find any particular solution of equation (3), then any of 
the family of lines = constant may be taken as the boundary lines of the 
body of revolution in a meridian plane, provided, however, that has no 
singularities within the body. In such a ])ody the stresses and the dis- 
placement will be known, these being given by equations 1 and 2 
respectively. 
In all the cases considered, the body assumes at its two distant ends 
the shape of circular cylinders, and the distribution of the terminal couples 
over the end faces is determined in each case. 
As Saint- Venaut has shown, however, the problem in each case is inde- 
pendent of the distribution of the terminal couples, except at points near 
the distant ends, so that the results hold at all points sufliciently far from 
the ends to avoid the end effects, whatever be the terminal distribution. 
Section 2. 
Five- dimensional Hydro dynamical Analogy. 
If we consider equation 3, the similarity between it and the equation 
for the Stokes' stream function, 
i.e. ^/l H\ 5_ /I H\ ^ ^ 
hz\r' ("z I 8r\r' 8r / 
immediately strikes us, the only difference being that we have r" instead 
of r. 
If we further consider the physical meaning of the Stokes' stream 
function, then it is evident that if we deal with space of five-dimensions, 
the components of velocitv will be o-iven by — \ . and + i . since the 
