158 
Transactions of the Royal Society of South Africa. 
To draw the family of curves constant, we may firstly draw the curves 
= — = — p4 gii^4 Q 
sin^ 6 
p 
for equidistant values of ^//^ and and then obtain the curves ^//g = constant 
by vector addition. 
On Plate YIII the family A are the lines iho = constant, and the family B 
are the lines — constant. By vector addition we get the family C, whose 
equation is given by 
^3 = - p^) sin* B. 
Now, any of these stream -lines may be taken as the boundary lines of 
the body of revolution in a meridian plane, care being, however, taken that 
the singular points fall outside the body, the singular point being in this 
case at the origin. 
If we take ^^3 = 0 and — 1-2 (say), then we shall get a body such as 
shown in Plate X, the cavity being spherical. 
If r^ is the radius of the body of revolution at 2 = + x and — x , 
then r^ is given by 
If /-o is the radius of the body in the diametral plane z = 0, then ro is 
given by the equation 
TcJ' + i//3 . ro —1 = 0. 
From the two equations for r^ and we find 
r.i"' — r^^Vr, — 1 = 0, 
r-i r 
1 '1 
Hence we see that for great values of r^, r^ becomes equal to 
If, e. g., we take r-^ = 2, then r2 = 2-015 approximately. If, therefore, 
^3 = — 16 (say) be taken as the outer boundary line of the body of revo- 
lution in a meridian plane, then = - 16 approximates to a straight line 
parallel to the axis of the body, and hence we get the case of circular cylinder 
with a spherical " flaw " on its axis. 
Again, if we take ^//.^ = — O'l and ^.^ = — 1*2 as the boundary lines, 
then we get a body such as shown in Plate XI, which would correspond 
to a circular pipe bulged out uniformly along some diametral plane. 
For such bodies, then, the stress-components r9 and Oz are given by 
rO = r ''^ = I =z - 
