Mr Hodgkinson an the Strength of Materials. 351 



Art. XXVIII.— ANALYSIS OF SCIENTIFIC BOOKS AND 



MEMOIRS.. 



I. On the Transverse Strain and Strength of Materials. By Mr Eatok 

 Hodgkinson. (Manchester Memoirs, vol. iv. p. 225. Lond- 1824.) • 



We slightly mentioned this article in our third number, and, at the same 

 time, stated our intention of examining it more at length in the present 

 Journal. It appears to have been read in March 1822, but the author, at 

 that time, not having seen Mr Barlow's " Essays on the Strength of Ma- 

 terials" <Sj-c. it was afterwards withdrawn, in order to make some notes 

 and additions considered necessary, in consequence of certain discrepancies 

 between the two theories, viz. the one adopted by Mr Barlow, and that by 

 Mr Hodgkinson ; of these we shall take some notice as we proceed, ob- 

 serving only in this place, that the points in dispute relate principally to 

 what may be called, the mechanism of the transverse strain, viz. to the 

 operations going on in the interior of a piece of timber at the moment of 

 rupture, as at the time of the greatest strain, and the relation between the 

 resistance of the timber, when strained transversely, and its power of di- 

 rect cohesion, but which, however, does not affect the computed strength 

 of timber as deduced from experiments made on the transverse strength ; 

 for all parties agree, that whatever may be the exterior operations, the 

 proportional strength of similar beams is directly as the breadth, and the 

 square of the depth, and inversely as the length, so that the point in ques- 

 tion may be considered as one rather of curious philosophical inquiry, 

 than of actual importance in these kind of experiments. 



Mr Hodgkinson, after a short introduction, commences his paper with 

 the following inquiry. Suppose a beam fixed firmly in a wall, and to 

 be then broken by a weight suspended at its other end, what are the circum- 

 stances attending the rupture ? admitting, in the first place, that the tim- 

 ber is wholly incompressible, and that it resists fracture by its tension only. 



Even this simple question has given rise to some controversies amongst 

 early writers on this subject. Galileo, who led the way in these researches, 

 assumed, that the fibres of the timber, from the lower edge of the beam, 

 about which it was supposed to revolve, to produce fracture, acted in such 

 a way, that their united effect was the same, as if the whole of their actions 

 took place at the centre of gravity of the surface; that is, he supposed, the 

 resistance of each fibre to be the same, at whatever distance it was situated 

 from the axis of motion, except so far as depended on its leverage. But 

 Mariotte, Leibnitz, and others, assert, and apparently with good reason, 

 that, according to the law, ut tensio sic vis, the resistance of each fibre is 

 proportionate to its tension, independent of the leverage, and, therefore, 

 introducing the latter, the effect of each will be as the square of its dis- 

 tance from the fulcrum. 



To meet these cases, Mr Hodgkinson assumes the resistance to vary as 

 an indeterminate power (v) of the distance of the fibres, and hence dedu- 

 ces a very simple and general formula, involving this indeterminate in- 

 dex v, to which, giving the different values 0, 1, &c. he arrives at the se- 



