24 REPORT— 1843. 
ae —be®, where a and D are constant quantities. He had found this law to obtain 
when the change produced in the body arose from extension or compression alone ; 
but when the change arose both from extension and compression, as in the flexure of 
a rectangular body, the force of a fibre was less than that due to perfect elasticity, as 
ax—ba* toa; or it was equal to az — ba", where « was the weight applied, and 
a, b constant quantities as before. 
In proof of these statements, Mr. Hodgkinson mentioned, that having remarked, 
in his experiments made for the British Association on the subject of hot and cold- 
blast iron, that the elasticity of bars, broken transversely, was injured much earlier 
than was generally assumed, he paid particular attention to this circumstance in his 
future experiments, and had bars so formed that he could separate the elasticity of 
extension from that of compression; by these bars, which were very long and of small 
depth, he perceived that J,th or jth of the breaking weight was sufficient to injure 
the elasticity. He mentioned the matter to his friend Mr. F airbairn (who was asso- 
ciated with him in the inquiry), soon after he had made the discovery, and Mr. Fair- 
bairn’s subsequent experiments, made to determine the strength of rectangular bars 
of iron, from all parts of the kingdom, were conducted in the same manner as Mr, 
Hodgkinson’s had been ; the deflexion and quantity of set, or defect of elasticity, from 
each weight being always observed. Mr. Fairbairn’s experiments were on bars cast 
one inch square and five feet long, and were made with the utmost care; Mr. Hodg- 
kinson has therefore adopted their results with respect to the “ set,” and taking means 
both from Mr. Fairbairn’s results and his own, on the same sort of bars, he has sought 
for the relation between the weights and the mean sets from those weights, these sets 
being the deflexions or deviations from the original form of the bar after the weights 
have been removed. To ascertain the relation above, Mr. Hodgkinson had curves de- 
scribed from the results of the experiments, making the sets the abscissee and the 
weights the ordinates, and the similarity in appearance of these curves to the common 
parabola led him carefully to examine whether they were not in reality represented 
by that curve. The examination was successful, the parabola was the curve, and the 
mean results of the observed set, together with the calculated ones, from equal addi- 
tions of weight, from 112 to 448 Ibs., derived from eighteen different kinds of iron, 
and about forty experiments, are below :— : 

Weights ...sceccsseserseeees eeceves «| 112 | 168 | 224 | 280} 3386 | 392} 448 
Mean sets, or defects of elasticity| "012 | ‘026 | -044 | ‘068 | -098 | +137 | "192 
Calculated sets from parabola ...| ‘012 | -026 | 046 | 075 | 104 } *142 | -186 

The following table contains the mean results from forty-four kinds of cast iron and 
from 90 to 100 experiments :— 

Weights ....sccsosssessoesecs 56 | 112 | 168 | 224] 280) 336 | 392 | 448 
Mean sets ©... ..sccenescscens 003 | -013 | -026 | -047 | -069 | :102 | +136 | ‘197 
Computed sets....ssssseeeee ‘003 | 012 | 027 | :047 | 072 | *102 | -138 | -181 
Mr. Hodgkinson made experiments on stone, timber and wrought iron, and ob- 
served the quantity of set in all. These different materials, when the results from 
them were constructed, all gave the form of the parabola, though less perfectly than 
in cast iron, as the experiments on them were but few. 
It appears from the above-stated experiments, and others that were made, that the 
sets produced in bodies are as the squares of the weights applied. Hence there is no 
weight, however small, that will not produce a set and permanent change in a body. 
All bodies when bent have the arrangement of their particles altered to the centre; 
and when bodies, as the axles of railway carriages, are alternately bent, first one way 
and then the opposite, at every revolution, we may expect that a total change in the - 
arrangement of their particles will ensue. It appears, too, from the results of these 
experiments, that all calculations hitherto made on the strength and elasticity of bodies 
have been only approximations. Mr. Hodgkinson stated that he laid the results of 
