414 
MR. HODGRINSON’S EXPERIMENTAL RESEARCHES 
Co7nputed Strength of Hollow Cast-Iron Cylindrical Pillars. 
49. In solid cylinders of the same material, the strength to resist breaking varies 
according to some constant power n of the diameter ; the length of the pillar and 
form of the ends being the same. Thus, strength oc D K , where D is the diameter. 
In cylinders with rounded ends, my experiments give n = 3*76 nearly ; and in those 
with flat ends, n — 3"55 nearly. 
According to the theory of Euler, the strength of a hollow cylinder, to resist in- 
cipient flexure, varies as the difference of the fourth powers of the radii, or of the dia- 
meters, external and internal, or as D 4 — d 4# , where D and d are the external and in- 
ternal diameters'!-. 
It appeared probable that this ratio, modified by changing the indices to the re- 
sults of the experiments upon solid pillars, might agree with those upon hollow ones. 
This was tried, and, with great satisfaction, found to answer. 
We have, therefore, 
Strength as D 3 ' 6 — d 3 ' 6 , in pillars with rounded ends ; 
Strength as D 3JJ — d 3 55 , in pillars with flat ends. 
To determine the strength of a solid pillar of a given length and one inch diameter, 
calling that strength x, and putting iv for the breaking weight of a hollow pillar of 
the same length whose external and internal diameters are D and d, we have 
D 376 - d 376 : l 376 - 0 376 : : w : x. 
Whence 
w 
x = for rounded ends. 
X = D 3-55 rf 3 - 55 ’ for flat ends. 
By these formulae, the strength of a solid cylinder, 1 inch diameter and 7 feet 6f 
inches long, has been calculated from all our longest hollow cylinders, and inserted 
with the results of the experiments from Tables VIII. and IX. 
The value of x thus obtained from each of the experiments, must, it is evident, be 
nearly constant, if the assumption respecting the indices be correct. 
I have extracted these values from the different experiments and given them in the 
following Table, and also a fraction indicating the magnitude of the difference between 
the result from each experiment and the general mean, and consequently the error 
which would arise from calculating by the general mean. 
* Poisson’s Theorem gives P the strength, = — a ~ ^9 — rf~^ ^ ^ where l = the length, n — 3T416, 
a = a quantity, which is equivalent to the modulus of elasticity in weight per unit of section ; and g' , g the 
external and internal semi- diameters. But (y' 2 + y 2 ) (g Ul — y 2 ) = y' 4 — y 4 ; and all the other quantities are 
constant, according to the assumption in the text. 
t Poisson, Mecanique, vol. i. p. 620. 
