152 
Transactions of the Royal Society of South Africa. 
Further, 
dz 
Jr] 
(4) 
(5) 
where and v^, are the components of velocity parallel to the z and r axis 
respectively. 
Equation (4) may also be directly obtained by considering the flow through 
an element h . h- rotated in the five- dimensional space round the ;s-axis. 
Equation 4 may also be written : 
^ ('-^ • + (r^ ■ V,) = 0. 
This is, however, the condition that 
r'v,. . clz — . . dr 
may be a complete dift'erenticil. A function \p therefore exists such that 
dz dr' 
I. e. 
1 H 
7-3 * dr 
V. = + i . 
dz J 
Since the motion is irratational, we have : 
dVa dVr 
(6) 
= 0, 
Of 
I. c. 
d / I ^v// \ 
dz \r' ' t"?; / dr \ ' dr 
dz 
5 /I 
(7j 
The stream -lines are given by 
dz _ dr 
v.. ~ V,. 
d^ -, . d^l , 
dz-^'-'-d;^-''' 
0, 
i. e. -ip — constant is the equation for the stream-lines. 
We note, then, that equations 3 and 7 are identical, and, further, that in 
the hydrodynamical problem \l> — const, represents the stream-lines, and in 
the elastic problem \l — const, represents the stress-lines. 
We shall, therefore, firstly draw up the velocity potentials and stream- 
functions for certain motions with symmetry about an axis in space of five 
dimensions. Since the surface of revolution of the body is free from 
external forces, therefore any of the stream-lines i// = const, may be taken 
