148 
MINUTES OE PROCEEDINGS OE 
directed. Allow me, however, before stating what my proposition is, 
to shew yon that the longitudinal strain, in addition to the injury it 
causes in itself, also injuriously affects all the endeavours in the present 
systems of gun manufacture to meet the 
/?. Circumferential Strain . 
In Fig. 1, the solid lines represent the inner and outer surfaces of a 
hollow cylinder before extension—the dotted lines after extension. 
Supposing the most favourable case of the metal being perfectly dense, 
the area between the two solid lines will equal the area between the 
two dotted lines j and the area between the inner solid line and its 
dotted line will equal the area between the outer solid line and its 
dotted line. But as the outer area is further removed from the centre 
than the inner, its length is greater, and consequently its breadth less 
in proportion. The absolute amount of extension of the outer surface 
will thus be to that of the inner inversely as their radii. Owing, how¬ 
ever, to the greater length of outer area, it is capable of extending more 
instead of less in proportion to the radius. The total proportionate 
extension, therefore, of outer to inner surface is inversely as the squares 
of their radii; and as the tension increases directly with the extension, 
the tension or useful effect decreases as you recede from the centre, 
and is inversely as the square of the distance from the centre. 
If the metal be not dense, the decrease is even more rapid than this. 
Fig. 2 represents the tensions at different distances from the centre 
on the principle above stated. The horizontal distances are the dis¬ 
tances from the centre, the vertical distances are the tensions. Notice 
how rapidly the tension diminishes as the thickness increases, when the 
radius is small. 
The following equation shows the actual amount of useful effect 
which can be obtained by one thickness of metal, as compared with 
what it would be if all the metal could be made to do its work ;— 
Let r be the radius at any point, 
t be the tension at that point* 
a be the internal radius, 
b be the external radius, 
T (total useful 
s=s-between limits a and b 
r 
(f - a). 
If the tension were uniformly equal to that at a , it would give a total 
m 
m 
strength -g (b — a). Therefore 
actual tension 
ab a 0 
= r* bo 
total strength of metal m b 
that the value of any hollow cylinder may be represented by the ratio 
