408 
MR. HODGKINSON’S EXPERIMENTAL RESEARCHES 
sustains a considerable diminution in its power of resistance to fracture by flexure, in 
consequence of a partial crushing, or crippling of the material. Suppose c d 2 — the 
crushing force of the pillar (d being its diameter), or that pressure which would cause 
rupture in it, if it were too short to break by flexure ; and ncd 2 that part of this pressure, 
which is the utmost it would, as flexible, sustain without apparent crippling or crushing. 
Then, since the strengths to resist fracture by flexure in pillars, with both ends rounded 
^376 (^3 '55 
and both flat, were 33379 -jty, an d 98922 respectively (Art. 36. 38.), we have 
i L 
these quantities each equal to n c d 2 , in the cases where short pillars are bearing the 
greatest weights they can sustain without any apparent crushing. Whence, 
In pillars with rounded ends. 
d 3 ' 76 
33379 -jyj- = n c d 2 ; /. I = 
176 
/ 33379 \ 17 ■, 17 
V 
nc 
) X d 
In pillars with flat ends, 
d 3 ' 55 
98922 — yy = ncd 2 ; 
i 
In the former of these cases, l varies somewhat faster than as the first power of 
the diameter, and in the second, somewhat slower ; the two showing that, in the case 
of pillars equally loaded to resist crushing by the weight, the length to the diameter 
will be nearly in a constant ratio, or the pillars must be similar. 
46. Strength of long uniform Pillars, as dependent on their weight, the length being 
constant. 
d m 
T n general, the strength varies as ; where d and l are the diameter and 
length ; and m, n, the constants on which the strength depends. If the length be 
the same, the strength varies as d m ; and since the weight q of the pillar is as the square 
1 m 
of the diameter or directly as the area, d oc q* ; whence strength oc q 2 . 
376 
In pillars with rounded ends, m — 3‘76 .\ strength oc q 2 oc q VB8 . 
3-55 
In pillars with flat ends, m — 3*55 .*. strength oc q 2 OC q V77b - 
These values were used in comparing the strengths of pillars of equal weight in 
Tables VI. and VII., and in other places. 
It will be borne in mind that this proposition applies only to pillars above a cer- 
tain length, as in Articles 36 and 38. 
Hollow Pillars of Cast Iron, the Low Moor, No. 3. 
47. The preceding observations have been made upon solid pillars only, as these, 
on account of their greater simplicity, are not only more easy to be cast than hollow 
ones, but present fewer difficulties in experimenting upon, and in deducing conclu- 
