412 
MR. HODGKINSON’S EXPERIMENTAL RESEARCHES 
and an enlarged middle, was stronger than a uniform pillar of the same weight. There 
seemed little doubt that the same would be the case in hollow pillars, or that a hollow 
pillarwith an enlarged middle, having the metal everywhere of uniform thickness, would 
be stronger than a hollow uniform pillar of the same weight and thickness of metal. 
To form these pillars with much regularity in small experimental castings, would have 
been difficult ; but some idea of the influence of increasing the strength of the middle, 
might be obtained by reducing the thickness of the uniform pillars, near to the ends, 
leaving the middle the same. The pillars, in experiments 7 and 8. of Table VIII., in 
which the ends were rounded, together with one or two others, were reduced in this 
manner. The pillar was reduced to half its thickness of metal, in a band two or three 
inches broad, near to each end, and to three-fourths of its thickness, in a similar band, 
about half the distance between the middle and each end. These pillars did not break 
in the reduced parts; and from the results it would seem doubtful, whether the strength 
had been in any degree lessened by the partial reduction of thickness. If this reduc- 
tion had been carried uniformly on, decreasing from the ends to the middle of the 
pillar, leaving it the same there, the pillar would but have had three-fourths of the 
metal in it, and its strength would, perhaps, have been but little decreased. 
Of the first ten pillars in Table IX., being those with their ends flat, one half were 
reduced by having portions cut away from each, as before described. These pillars 
never broke in the reduced parts first, except in one instance, and that was where the 
thickness, half-way between the middle and the ends, had been reduced somewhat 
more than one-fourth, which was the quantity ordered to be taken away from that part. 
The calculated values in Table IX., taking a mean from the five reduced pillars, 
give 3060 ; and from the five which were not reduced, 3308 ; the latter being -f- 3 - 
stronger than the former, leaving it doubtful whether reducing the strength toward 
the ends, in pillars with flat ends, effects any saving of metal. The experiments in 
Table VII., upon solid pillars, with discs upon the ends, show a saving, however, 
when the strength in the middle is much greater than toward the ends. The short 
pillars in Table X. will be referred to further on. 
In Table XL, the first two pillars were from the same model. They were about 
inches diameter in the middle, and If inch near to the ends, which had strong 
discs upon them turned flat, to give them a firm bearing ; the pillars were hollow, as 
before. The weight of the former was 36 lbs., and of the latter 36f lbs., and their 
breaking weights 28667 and 27491, mean 28079 lbs. From a mean between experi- 
ments 5, 6, 7? Table IX., it appears that a hollow uniform pillar 2 - 02 inches external 
diameter (the mean of the above nearly), and weighing 36^ lbs., required 305 18 lbs. 
to break it. This seems to show that the uniform cylinder is the stronger*. 
* This conclusion is in agreement with what may be deduced from Euler’s theoretical values of the strengths 
of uniform cylindrical solid pillars, and of those in the shape of a truncated cone (Berlin Memoirs, 1757); his 
formulae for these being P = 
a 2 D 4 
. p, and P' 
a 2 D' 2 E 2 
A 2 d* A 2 d 4 ^ 
These values are to one another, cateris paribus, as D 4 to D ,a E 2 ; where D is the diameter of the uniform 
