The Torsion Problem for Bodies of Revolution. 159 
Further, from 10 the displacement ne will be given by 
Uq= - _ 4 + 
where the plane z — o is taken as the plane of zero displacement. 
If r^ is the radius of the body at 2; = + oc and — oc, then the moment 
of the terminal couples is given by 
Mo = 29rr^4. 
If and are the inner and outer radii at ^ = + oc and — oc, then 
M.'=: 27r (rV - '^V)- 
In general, then, we have . If in a meridian plane the boundaries of 
the body of revolution are given by 
4 (,2 + ,.2)4 
then the stress -components are given by 
rO = + 
- = const., 
- r2 
-^"^ + '^^■•(,2 + ,.)^' 
and the displacement ne by 
zr ( , h ) 
ne = -! Vo H ; 
the moment of the terminal couples being 
where r^' and i-^" are the inner and the outer radii of the body at the ends. 
2. Case. 
If we put v,j = 4 and k = — 1, i. e. change the axis of the doublet, then 
z 
(h. — — ^z -\r 
(.2 + r2)t 
,b.. = — 
{z^- + r^y. 
Proceeding as in the previous case we get the stream -lines C in Plate IX. 
In this case we have a circular cylinder which has been bulged in slightly 
as in Plate XII. 
The formulae for the stress-components, etc., follow from the general 
formulae of Case 1 by putting = 4, /i- = — 1, 
