The Torsion Prohlein for Bodies of Revolution. 
151 
circumference of a circle got by rotating a point round the ^-axis in space 
of five-dimensions is Srr- . r^ (see Appendix). The condition for irratational 
motion will therefore lead to equation 3. 
Let us, therefore, consider the irratational motion, with symmetry about 
an axis, of an incompressible fluid in space of five dimensions. 
Let be the five co-ordinates, and let 0 be the velocity 
potential of the motion. That such a potential exists follows from the 
assumption that the motion is irratational. 
The components of velocity will be given by : 
fx. 
The condition for the inconipressibility of the fluid is 
SXi Sx.2 dX-^ ^ ' 
If we now choose the axis as the axis of symmetry, then 
7-2 = a;/ + x.^ ^ xl, 
where r is the distance of the point (x^^ x., x.^ x^ x-^) from the x^ axis. • 
Now, since there is symmetry about the .i-^-axis, we shall have 
0 = (j) (2;,r), 
where in the further development .^ is taken as the axis of symmetry, 
i. e. z = x^. 
Again, ^^^^ _ 
dx.y ~~ dr ' dx,2 
<^r2 ■ ~^ Sr \r rV 
with similar expressions for 
S'-(p S-ij) ^-(j, 
Sx.J^^ ^x^ ' ^x-^' 
Adding, we get : 
