Mr Hodgkinson on the Strength of' Materials. 355 



with his theory, that where the resistance to compression is considerably 

 greater than that to extension, the strength of the beam will be greater, 

 than if all the fibres acted by extension only : the only direGt contradiction 

 appears in the last case, where the resistance to compression is infinite, viz. 

 in a case which cannot occur, and which the theory in question is invented 

 on purpose to avoid. That it should fail here is obvious, because, in this 

 case, there are no forces acting under the neutral line, as the theory sup- 

 poses ; and it is singular Mr Hodgkinson did not perceive, that precisely 

 the same want of generality applies also to his theory, by taking the opposite 

 imaginary case, viz. of the material being infinitely inextensible ; for in 

 this case, the area of tension being zero, the breaking weight would be zero, 

 or taking any small area of tension, then the strength would be infinite, 

 both inconsistent results. 



The question is not, therefore, to be decided in this way, but by a refer- 

 ence to first principles, and it seems to be within narrow limits. Mr 

 Hodgkinson, following the theory of Coulomb, considers the resistance to 

 fracture as a single mechanical effort, measured by the area of tension, 

 multiplied by the tension on the unit of measure, and by the distance be- 

 tween the centre of tension and compression ; whence he deduces 

 Wsum of the forces in abd X Ff 

 ETc - 



_ sum of the forces in abd X (FP + fG ) 



W =: LC '~ 



T X D + T X A 

 W= j^ 



Whereas Mr Barlow considers the operation as compound, and estimates 

 the resistance to compression and tension separately. So that, denoting the 

 resistance to compression by R, his expression is 

 T X D + R X a 



W = LC 



In order to reduce this formula to a state proper for further investiga- 

 tion, it is necessary to establish the relation between the two parts, T 

 X D, and R X A, and he assumes them to be equal, observing " that it is 

 this equality only which determines the relation to take place about the 

 point P," whereas the other theory assumes T = R, which of course gives 

 Mr Hodgkinson's numerator, T X D + T X A. The question is thus 

 reduced to this very narrow limit, viz. Whether, from the nature of the ope- 

 rations, we ought to take T = R, or T X D = R X a, or whether the one 

 of these equalities may not belong to the first part, and the other to the 

 list part of the operations by which the fracture is produced. It is stated 

 by Mr Tredgold, that the neutral axis does not remain permanent during 

 the operation of fracture, but that the area of compression gradually in- 

 creases ; and if so, a different law must have place at the beginning and 

 end of the process. At all events, we have reduced the question to its nar- 

 rowest limit, and we leave it thus for the decision of those most conversant 

 with these matters, and proceed to an examination of the experimental 

 part of the paper. 



