Mr Hodgkinson on the Strength of' Materials. 353 



and Ff one arm of a bended lever, while the length of Gf is the other, 

 (the points F and G being supposed to be connected by the chain FG, mere- 

 ly to give the lever the appearance of greater strength.) And to obtain 

 the strength of the body we shall have 



W X ,/&== sum of the forces in aid X F/'. 



,„, ,.,. sum of the forces in abd X Ff . . A ~ . . 



Whence W= s — ? r- ==; -, where the deflection of the 



the length JG 



beam is neglected ; or introducing that, as in Art. 9, we have W (the 



wei^hO = smn of tlie foices '» ahd x (FP + PQ _ T X D +' F X A 

 ° Length x cosine oi Deflection L x C 



where T = sum of the forces rising from tension, D and a the distances 

 of the centres of tension and compression from the neutral line, L the 

 length of the piece, and C the cosine of its deflection- 



" A necessary consequence of this reasoning is, that the sums of the 

 forces of extension and compression are just equal to one another:* For 

 the weight \V, acting in the direction of GW, parallel to the surface of 

 fracture adbc, can have no influence in pushing the piece toward or draw- 



• " The mode of reasoning adopted above has been objected to by Mr Barlow, who 

 conceives that the forces in F and/, or those of extension and compression, instead 

 of being equal, should be inversely as their distances from the neutral line, or that 

 the forces in F X PF = forces in/ X Pf, and that these taken collectively are 

 = the rectangle under the weight and the length of the beam, which is sup- 

 posed to turn as on a pivot round the neutral line. Whence L X W = the forces 

 in F X PF X forces in/ X P/= twice the forces in F X PF. The mode of 

 estimating the strengths of bodies, as deduced from this, is very simple and easy ; 

 it is in effect this : — Find the neutral line — suppose that the fulcrum — estimate the 

 strength of the area of tension, as was dons in incompressible bodies, and double 

 that for the answer. 



But this rule, it appears to me, contains within itself a fundamental error which 

 will become very apparent by the following consideration It is supposed to be Ge- 

 neral whatever the situation of the neutral line may be. Let then the body be in- 

 compressible ; the neutral line will in that case be extremely near the edge, and 

 the strength as estimated by this rule will be double what from incontestable prin- 

 ciples it ought to be : a consequence which the ingenious author could have had no 

 idea of, when he proposed this theory. The error too will be found to exist, though 

 in a less degree, in almost every other position of the neutral line, and may be very 

 plainly seen, it' we estimate by this rule, strengths of bodies that suffer a slight com- 

 pression, and compare the results with the known strengths of the same bodies, if 

 they had been wholly incompressible. 



For example : — The strength of an incompressible joist, broke by a weight hung 



2sbn~ 

 at one end, being - , (articles 4th and 9th,) where s is the longitudinal 



strength of the fibres in a unity of surface, b the breadth of the piece, a its depth, I 

 its length, and v the index of extension : the strength of a compressible one, ac- 



cording to Mr Barlow, will he———, where d is the depth of the area of tension, 



and the rest as before. 



VOL. III. NO. II. OCTOBER 1825. Z 



