ON THE STEENGTH OF PILLAES OF CAST lEON. 
855 
A 
B 
be uniform. The curve having taken this form, suppose it to be rendered immoveable 
at the points h andy, by some firm fixings at those points. This done, it is evi- 
dent we may remove the parts near to A and B, without at all altering the curve 
bcdefoi the part of the pillar between b and/*, and consider only that part. The 
part bf, which alone we shall have to consider, will be equally bent at all the 
points J, d, f. The points c and e too are points of contrary fiexure, 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 of 
the pillars in Table I. in the former research, and the part cde of the pillar, with 
its ends c and e rounded, will be bearing the same weight as the whole pillar 
bcdef of double the length with its ends 5, f firmly fixed. 
2. That the strain at all the points J, d,f\s, nearly equal, is shown by sub- 
sequent experiments, in Tables V. and VI., upon pillars varying in length from 
6 feet 3 inches to 5 feet and 2 feet 6 inches ; that fracture in some cases took 
place at all three points at the same time, and in others at two of them, showing 
the fracture to have commenced on opposite sides of the middle and of the ends, 
as might be inferred from the foregoing figure, or the last in Plate XXXII. 
3. That the strength of a pillar, with moveable or rounded ends, is equal to that of a 
pillar of the same diameter, and double the length, the ends being fiat or firmly fixed, 
may be inferred from the reasoning above ; but it is proved experimentally from various 
results in the abstract p. 391 of the former research, or as below. 
4. The strengths of pillars of the same diameter, but of different lengths, vary 
inversely as the I'Tth or l-6th power of the length, or in this case as 2‘’^ or 2*'® nearly, 
according to my experiments. Hence the strength of a pillar one-half the length of 
another is 3-249 or 3-031 times greater than it; or in other words, the strength of a 
pillar with fiat ends is somewhat more than three times as great as that of one of the 
same length and diameter with rounded ends. 
5. This is shoAvn in the abstract, p. 387 of the former research, from numerous expe- 
riments on solid pillars, with both ends rounded and both ends fiat, the mean ratio of 
their strengths being 1:3-167 
In the abstract, pages 856 and 857 of the present 
research, the mean ratio from the comparison of the 
strengths of four hollow pillars, with rounded ends and 
of the like number with fiat ends, was 1:3-077 
In the same abstract, solid pillars of the two forms, of 
wrought iron and of timber, gave from several experiments 
a mean ratio of 1:3-076. 
> Mean ratio 1 : 3-107. 
5 T 2 
