404 
MR. HODGKINSON’S EXPERIMENTAL RESEARCHES 
All those values of x are in excess of the part of the general mean they are com- 
pared with. We might, therefore, obtain a rough approximation to the strength of 
short pillars by calculating, by the formula before used, the strength of the pillar; 
and then taking such a portion of this calculated strength as the length of the pillar 
is of what it would have been if it were 30 times the diameter. This easy rule would 
nearly in all cases give the strength somewhat under the real strength, as appears 
from the instances we have cited, and others we might take from the abstract. 
More correct mode of estimating the strength. 
41. We have seen, that when pillars are reduced in length beyond a certain extent, 
there is a reduction in their strength from that which is given by the rules for longer 
pillars ; and this falling off is nearly in proportion to the reduction in the length of 
the pillar. Another mode of considering the matter will, it is conceived, throw some 
light upon the reason of the conclusion in the last article, and furnish better means 
of estimating the strengths. 
The two last pillars, in the abstract last given (Art. 39.), broke by being crushed, 
all the others having been broken by flexure. One broke with 48T tons per square 
inch; and the other with 51*7- The mean result of the experiments, upon the re- 
sistance to crushing in this metal, gave 49 tons per square inch nearly (Art. 55.). 
Taking a mean between the results of such pillars in the abstract as have their length 
the same number of times the diameter nearly, we have as below, in experiments 3, 
4 and 5, a pressure of 1 9* 1 6 tons answering to a value of x = 75370lbs. ; a mean from 
the experiments 6, 7, 8 and 9, gives a pressure of 23‘97 tons per inch, answering to a 
value of x = 53770 lbs. ; and a mean between the results in the 10th and 11th expe- 
riments, gives 34-9 for the pressure per inch, and the value of x = 32241. 
We therefore see that of the crushing weight reduced the value of x from 
23*97 . 34*9 
98922 to 75370 ; 49 - reduced it to 53770; and ^ reduced it to 32241. In other 
words, when the pressure is a little more than two-thirds of the crushing weight, the 
strength to bear flexure is reduced to less than one-third ; when the pressure is a little 
under half of the crushing weight, the value of x is somewhat more than half its first 
value ; and when the pressure is less, the reduction is diminished. Comparing the 
weights, per square inch, with all the values of x, in the two columns, we find, amidst 
many anomalies, that the weights increase in some such proportion as the values of x 
decrease ; and the latter become very small where the pressure approaches to the 
crushing force. Considering then the pillar as having two functions, one to support 
the weight and the other to resist flexure, it follows, that when the material is incom- 
pressible (supposing such to exist), or when the pressure necessary to break the pil- 
lar is very small, on account of the greatness of its length compared with its lateral 
dimensions, then the strength of the whole transverse section of the pillar will be 
employed in resisting flexure ; when the breaking pressure is one-half of what would 
