392 
MR. HODGKINSON’S EXPERIMENTAL RESEARCHES 
-f 
lire at its ends, so as to take the form A b c d e/B ; where all the curves 
Abe, ede, e/B, separated by the straight line AceB, would be equal, since 
the bar was supposed to be uniform. The curve having taken this form, 
suppose the points b and f to be rendered immoveable by some firm fixings 
at those points. This done, it is evident we may remove the parts near to 
A and B, without at all altering the curve bedef of the part of the pillar 
between b and f, and consider only that part. The part b f, which alone 
we shall have to consider, will be equally bent at all the points b, d,f. The 
parts c and e, too, are points of contrary flexure ; consequently the pillar is 
not bent in them. These points are unconstrained, except by the pressure 
which forces them together ; and the pillar might be reduced to any degree 
in them, provided they were not crushed or detruded by the compressing 
force. These points may then be conceived as acting like the rounded ends 
in the pillars of Table I. ; and the part c d e of the pillar, with its ends c and 
e supposed to be rounded, will be bearing the same weight as the whole pillar 
bedef, of double the length, with its ends b f firmly fixed. 
14. The theory of the strength of pillars, as given by Euler and La- 18 
grange*, and afterwards pursued by Poisson • f* and others, furnishes us with little 
information upon these subjects. According to that theory, the strength is inversely 
as the square of the length ; or a pillar, half the length of another of the same dia- 
meter, would have four times the strength. The results of my experiments give the 
strength nearer to three times, as a general value for the half length in cast iron. It 
is, however, very variable, as will be seen by the first and second tables, unless we 
restrict ourselves to the longer columns. 
The strength is much influenced, as has been previously observed, by the quantity 
of compression which the pillar sustains ; and, consequently, by the position of the 
neutral line when the pillar is bent. The strengths, too, are different in their defini- 
tion in the two cases. In the theory of Euler, the strength is estimated by the 
greatest weight which a pillar would bear without flexure ; whilst in the present ease, 
the estimate is formed upon the weight which would break the pillar by flexure. 
I have sought, on many occasions, but without success, to determine experimentally 
some fixed point, according to the definition of the continental theory. So far as I 
can see, flexure usually commences with very small weights, such as could be of 
little use to load pillars with in practice. It seems to be produced by weights much 
smaller than are sufficient to render it capable of being measured. I am, therefore, 
doubtful whether such a fixed point will ever be obtained, if indeed it exists. With 
respect to the conclusions of some writers, that flexure does not take place with less 
than about half the breaking weight, this, as is evident from my experiments, taken 
in general, could only mean large and palpable flexure ; and it is not improbable that 
the writers were in some degree deceived from their having generally used specimens 
* Acad, de Berlin, 1769 ; Collection Academique de Turin, Vol. 5. t Poisson, Mecanique, 2nd edit. 
